1. Introduction
Every elliptic curve over 
${{\mathbb {Q}}}$ can be uniquely represented as 
$E_{AB}:y^{2}=x^{3}+Ax+B$, where 
$A$ and 
$B$ are integers such that there is no prime 
$p$ with 
$p^{4}\mid A$ and 
$p^{6}\mid B$, and such that 
$\Delta (A,B):=-4A^{3}-27B^{2}\neq 0$. Given an elliptic curve 
$E$ over 
${{\mathbb {Q}}}$, we denote its algebraic rank by 
$r(E)$ and its analytic rank by 
$r_{{\rm an}}(E)$. The Birch and Swinnerton-Dyer (BSD) conjecture asserts that these two quantities are equal, that is, we have 
$r(E)=r_{{\rm an}}(E)$.
Foundational conjectures of Goldfeld [Reference GoldfeldGol79] (in the case of families of quadratic twists of elliptic curves) and Katz and Sarnak [Reference Katz and SarnakKS99] (for the full family of elliptic curves) assert that a density of 
$50\,\%$ of elliptic curves have rank 
$0$, and that 
$50\,\%$ have rank 
$1$, and that the average rank of elliptic curves is 
$1/2$. Both these conjectures are formulated through a study of the associated family of the 
$L$-functions 
$L_E(s)$ attached to the elliptic curves 
$E$. The behavior of 
$L_E(s)$ at and near the critical point is used to control the distribution of analytic ranks, which, assuming the BSD conjecture, can be used to give heuristics for the distribution of the algebraic ranks.
The most natural way to order a family of 
$L$-functions is by their conductors, which, in this case of 
$L$-functions of elliptic curves, is equal to the levels of the associated modular forms. Thus in the Goldfeld and Katz–Sarnak conjectures, it is implicitly assumed that elliptic curves are ordered by their conductors. However, when studying two-parameter families of elliptic curves, the curves 
$E_{AB}$ are usually ordered by their (naive) height 
$H(E_{AB})={{\rm max}}\{4|A|^{3},27B^{2}\}$.Footnote 1 Assuming the generalized Riemann hypothesis, Brumer [Reference BrumerBru92], Heath-Brown [Reference Heath-BrownHea04], and Young [Reference YoungYou06] proved the successively better bounds of 2.3, 2, and 
$25/14$ on the average analytic ranks of elliptic curves when ordered by height. On the algebraic side, Bhargava and the second named author [Reference Bhargava and ShankarBS15] proved that the average rank of elliptic curves, when ordered by height, is bounded by 
$0.885$.
If elliptic curves are instead ordered by conductor, even asymptotics for the number of curves are not known. The discriminant 
$\Delta (E_{AB})$ of 
$E_{AB}$ is (up to absolutely bounded factors of 
$2$ and 
$3$) 
$-4A^{3}-27B^{2}$. The conductor 
$C(E_{AB})$ of 
$E_{AB}$ is (again, up to bounded factors of 
$2$ and 
$3$) the product over all primes 
$p$ dividing 
$\Delta (E_{AB})$ of either 
$p$ or 
$p^{2}$ depending on whether 
$E_{AB}$ has multiplicative or additive reduction at 
$p$. Building on the work of Brumer and McGuinness [Reference Brumer and McGuinnessBM90] on the family of elliptic curves ordered by discriminant, Watkins [Reference WatkinsWat08] gives heuristics suggesting that the number of elliptic curves with conductor bounded by 
$X$ grows as 
$\sim cX^{5/6}$ for an explicit constant 
$c$. Lower bounds of this magnitude are easy to obtain, but the best known upper bound is 
$O(X^{1+\epsilon })$ due to work of Duke and Kowalski [Reference Duke and KowalskiDK00].
The difficulties in determining precise upper bounds are twofold. First, it is difficult to rule out the possibility of many elliptic curves with large height but small discriminant. Second, it is difficult to rule out the possibility of many elliptic curves with large discriminant but small conductor. It is interesting to note here that the second difficulty is exactly a nonarchimedean version of the first. Indeed, curves 
$E_{AB}$ with large height and small discriminant correspond to pairs 
$(A,B)$ of integers, where 
$4A^{3}$ and 
$-27B^{2}$ are unusually close as real numbers. On the other hand, curves 
$E_{AB}$ with large discriminant and small conductor correspond to pairs of integers 
$(A,B)$ such that 
$4A^{3}$ and 
$-27B^{2}$ are unusually close as 
$p$-adic numbers.
In this paper we focus on studying the second difficulty while entirely sidestepping the first. To this end, we let 
${{\mathcal {E}}}$ denote the set of elliptic curves 
$E$ over 
${{\mathbb {Q}}}$ that satisfy the following properties.
1. The
$j$-invariant $j(E)$ of $E$ satisfies $j(E)\ll \log \Delta (E)$.2.
$E$ has good reduction at $2$ and $3$.
Recall that the 
$j$-invariant 
$j(E_{AB})$ for the elliptic curve 
$E_{AB}$ equals 
$A^{3}/\Delta (E_{AB})$ up to a fixed constant. The condition 
$j(E_{AB}) \ll \log \Delta (E_{AB})$ then implies that 
$A^{3} \ll \Delta (E_{AB})^{1+\epsilon }$ and also 
$B^{2}\ll \Delta (E_{AB})^{1+\epsilon }$ from the definition of 
$\Delta (E_{AB})$. In other words, the first of the above two conditions excludes all elliptic curves 
$E$ with 
$\Delta (E)\ll H(E)^{1-\epsilon }$ and is critical for our results. According to the Brumer–McGuinness heuristics [Reference Brumer and McGuinnessBM90], only a negligible number of elliptic curves are being excluded by the assumption of this property, but this is unproven. The second property is a technical assumption at 
$2$ and 
$3$. Our method also works with other local reduction conditions at 
$2$ and 
$3$, as long as there is a uniform bound on the contribution to 
$\Delta (E)$ at 
$2$ and 
$3$, with the only change being the leading constant in Theorem 1.1 below. We will in fact have to further restrict our families of elliptic curves. We define the families
\begin{align*}
{{\mathcal{E}}}_{{\rm sf}}&:= \biggl\{E\in
{{\mathcal{E}}}:\frac{\Delta(E)}{C(E)} \mbox{ is
squarefree} \biggr\},\\ {{\mathcal{E}}}_\kappa&:= \{
E\in{{\mathcal{E}}}:\beta_E\leq\kappa\},
\end{align*}
for every 
$\kappa >1$, where the Szpiro constant 
$\beta _E$ is defined to be 
$\log |\Delta (E)|/\log (C(E))$. When ordered by conductor, the family 
${{\mathcal {E}}}_\kappa$ conjecturally contains 
$100\,\%$ of elliptic curves with good reduction at 
$2$ and 
$3$, and 
${{\mathcal {E}}}_{{\rm sf}}$ conjecturally contains a positive proportion of elliptic curves. We prove the following result determining asymptotics for these families of elliptic curves, ordered by their conductors.
Theorem 1.1 Let 
$1<\kappa <7/4$ be a positive constant. Then we have
\begin{equation} \begin{aligned}
\#\{E\in {{\mathcal{E}}}_{{\rm sf}}: C(E) < X\}& \sim
\frac{1+\sqrt{3}}{60\sqrt{3}}
\frac{\Gamma(1/2)\Gamma(1/6)}{\Gamma(2/3)}
\cdot\prod_{p\geq 5} \biggl
(1+\frac{1}{p^{7/6}}-\frac{1}{p^{2}}-\frac{1}{p^{13/6}}
\biggr) \cdot X^{5/6},\\ \#\{E\in
{{\mathcal{E}}}_\kappa: C(E) < X\}& \sim
\frac{1+\sqrt{3}}{60\sqrt{3}}
\frac{\Gamma(1/2)\Gamma(1/6)}{\Gamma(2/3)}
\cdot\prod_{p\geq 5}
\biggl[\biggl(1-\frac{1}{p}\biggr)\biggl(1+\frac{1}{p^{5/3}}
+\frac{1}{p^{11/6}}+\frac{1}{p^{17/6}}\biggr)\\ & \quad
+\frac{1}{p}\biggl(1-\frac{1}{p}\biggr)
\biggl(1-\frac{1}{p^{1/6}}\biggr)^{{-}1}\biggl(1+\frac{2}{p}-\frac{2}{p^{3/2}}\biggr)\biggr]
\cdot X^{5/6},\\
\#\{E\in{{\mathcal{E}}}:|\Delta(E)| < X\}& \sim
\frac{1+\sqrt{3}}{60\sqrt{3}}
\frac{\Gamma(1/2)\Gamma(1/6)}{\Gamma(2/3)}
\cdot\prod_{p\geq 5} \biggl (1-\frac{1}{p^{10}}
\biggr)X^{5/6}. \end{aligned}
\end{equation}
We expect Theorem 1.1 to hold for all 
$\kappa$. Furthermore, since the abc conjecture implies that for 
$\kappa >6$ [Reference OesterléOes88], all but finitely many curves in 
${{\mathcal {E}}}$ belong to 
${{\mathcal {E}}}_\kappa$, we expect these asymptotics to also hold for the family 
${{\mathcal {E}}}$. We note that the Euler factors appearing in Theorem 1.1 arise naturally from the densities of elliptic curves over 
${{\mathbb {Q}}}_p$ with fixed Kodaira symbol. These densities are computed in Theorem 1.6.
Our next main result is on the distribution of ranks of elliptic curves in 
${{\mathcal {E}}}_{{\rm sf}}$. As in [Reference Bhargava and ShankarBS15], we study the ranks of these elliptic curves via their 
$2$-Selmer groups. Recall that the 
$2$-Selmer group 
${{\rm Sel}}_2(E)$ of an elliptic curve 
$E$ over 
${{\mathbb {Q}}}$ is a finite 
$2$-torsion group which fits into the exact sequence
\begin{equation} 0\to E({{\mathbb{Q}}})/2E({{\mathbb{Q}}})\to{{\rm Sel}}_2(E)\to \unicode{x0428}_E[2]\to 0, \end{equation}
where 
$\unicode{x0428}_E$ denotes the Tate–Shafarevich group of 
$E$. Our result regarding the 
$2$-Selmer groups of elliptic curves in 
${{\mathcal {E}}}_{{\rm sf}}$ is as follows.
Theorem 1.2 When elliptic curves in 
${{\mathcal {E}}}_{{\rm sf}}$ are ordered by their conductors, the average size of their 
$2$-Selmer groups is 
$3$.
Theorem 1.2 has the following immediate corollary.
Corollary 1.3 When elliptic curves in 
$E\in {{\mathcal {E}}}_{{\rm sf}}$ are ordered by their conductors, their average 
$2$-Selmer rank is at most 
$1.5$; thus, their average rank is at most 
$1.5$ and the average rank of 
$\unicode{x0428}_E[2]$ is also at most 
$1.5$.
Corollary 1.3 provides evidence for the widely held belief that the distribution of the ranks of elliptic curves is the same regardless of whether the curves are ordered by height or conductor. Moreover, as expected, the average size of the 
$2$-Selmer groups of curves in 
${{\mathcal {E}}}_{{\rm sf}}$ is the same as the average over all elliptic curves ordered by height obtained in [Reference Bhargava and ShankarBS15, Theorem 1.1]. We remark that our methods are flexible enough to recover versions of Theorems 1.1 and 1.2 where the families 
${{\mathcal {E}}}_{{\rm sf}}$ and 
${{\mathcal {E}}}_\kappa$ are restricted so that the curves in them satisfy any large set of local conditions. This result is stated in Theorem 7.1.
Uniformity estimates
The key new ingredients required for proving the main results are ‘uniformity estimates’ or ‘tail estimates’. These are upper bounds on the number of elliptic curves in our families whose discriminants are large compared to their conductors. For the proof of Theorem 1.2, we additionally need bounds on the sum of the sizes of the 
$2$-Selmer groups of elliptic curves in 
${{\mathcal {E}}}_{{\rm sf}}$ with large discriminant and small conductor. To this end, we prove the following result for the family 
${{\mathcal {E}}}_{{\rm sf}}$.
Theorem 1.4 For positive real numbers 
$X$ and 
$M$, we have
\[ \# \biggl \{(E,\sigma):E\in{{\mathcal{E}}}_{{\rm sf}}, C(E) < X, \frac{\Delta(E)}{C(E)} > M, \sigma\in{{\rm Sel}}_2(E) \biggr\} \ll_\epsilon \frac{X^{5/6+\epsilon}}{M^{1/6}}. \] We note that up to the power of 
$X^{\epsilon }$, this is expected to be the optimal bound.
For the family 
${{\mathcal {E}}}_\kappa$, we prove the following result.
Theorem 1.5 Let 
$\kappa <7/4$ and 
$\delta >1$ be positive constants. Then there exists a positive constant 
$\theta$, depending only on 
$\delta$ and 
$\kappa$, such that for every 
$X>0$, we have
\[ \#\{E\in{{\mathcal{E}}}_\kappa:C(E) < X, \beta_E\geq\delta\} \ll_{\epsilon} X^{5/6-\theta+\epsilon}. \] Weaker versions of these uniformity estimates have been obtained previously. In [Reference Bhargava and ShankarBS15, Theorem 2.13], the authors bound the average number of 
$2$-Selmer elements in elliptic curves 
$E$, where the height 
$H(E)$ is bounded and 
$\Delta (E)$ is divisible by the square of a large prime. These estimates were used to obtain asymptotics on the number of elliptic curves with bounded height and squarefree discriminant, as well as to compute the average size of the 
$2$-Selmer groups of these elliptic curves. In a different direction, Fouvry, Nair, and Tenenbaum [Reference Fouvry, Nair and TenenbaumFNT92] prove bounds on the number of elliptic curves 
$E$ with large Szpiro constant 
$\beta _E$, when these curves are ordered by height or discriminant. Strikingly, they do not require any assumption bounding 
$\beta _E$ from above! However, neither of these results in [Reference Bhargava and ShankarBS15] or [Reference Fouvry, Nair and TenenbaumFNT92] allows for the elliptic curves in question to be ordered by conductor.
The main difficulty we face in proving Theorems 1.4 and 1.5 is that the heights and discriminants of curves with bounded conductor can grow very rapidly. Indeed, the height (and discriminant) of 
$E\in {{\mathcal {E}}}_{{\rm sf}}$ with 
$C(E)=X$ can be as large as 
$X^{2}$. As a consequence, the error term arising from the use of the Ekedahl sieve, a key input in the uniformity estimates of [Reference Bhargava and ShankarBS15], is 
$O(X^{4/3})$, which is much too large. Subsequent improvements to the Ekedahl sieve by Taniguchi and Thorne [Reference Taniguchi and ThorneTT20], in which the sieve is combined with equidistribution methods, are also insufficient for our purposes.
Our new methods, in addition to obtaining Theorems 1.4 and 1.5, also yield a significantly stronger version of [Reference Bhargava and ShankarBS15, Theorem 2.13], bounding the number of 
${{\rm GL}}_2({{\mathbb {Z}}})$-orbits on integral binary quartic forms 
$f$ with bounded height, such that the discriminant of 
$f$ is divisible by the square of a large prime. This result is stated as Theorem 6.5, and is proved in § 6, using as input the results of § 4. This improvement has several applications. First, it is a necessary ingredient to obtain a power-saving error term for the count of the average size of the 
$2$-Selmer group of elliptic curves when ordered by height. Second, as we prove in a forthcoming note, this improved estimate leads to a proof of Lenstra's conjecture [Reference Ash, Brakenhoff and ZarrabiABZ07, Conjecture 1.1] in the degree-4 case.
Outline of the proofs
We now describe the proofs of our main theorems. We study the ratios 
$\Delta (E)/C(E)$ of elliptic curves 
$E:y^{2}=f(x)$ in our families by considering the associated family of cubic rings 
$R_f:={{\mathbb {Z}}}[x]/f(x)$ and cubic algebras 
$K_f:={{\mathbb {Q}}}[x]/f(x)$ over 
${{\mathbb {Q}}}$. Let 
${{\mathcal {O}}}_f$ denote the ring of integers of 
$K_f$. Then 
$R_f$ is a suborder of 
$K_f$. Define the invariants
\begin{align*} Q(E)&:= [{{\mathcal{O}}}_f:R_f],\\ D(E)&:= {{\rm Disc}}(K_f) \end{align*}which satisfy the relation
\[ \Delta(E)={{\rm Disc}}(R_f)=Q(E)^{2}D(E). \]
For primes 
$p$, we let 
$C_p(E)$, 
$\Delta _p(E)$, 
$Q_p(E)$, and 
$D_p(E)$ denote the 
$p$-parts of 
$C(E)$, 
$\Delta (E)$, 
$Q(E)$, and 
$D(E)$, respectively. The local invariants 
$C_p(E)$, 
$\Delta _p(E)$, 
$Q_p(E)$, and 
$D_p(E)$ depend only on the Kodaira symbol of 
$E$. The starting point of our proof is a determination of these local invariants along with a computation of the density of elliptic curves over 
${{\mathbb {Q}}}_p$ with fixed Kodaira symbol.
Theorem 1.6 Fix a prime 
$p\geq 5$ and a Kodaira symbol 
$T$. Let 
$E:y^{2}=f(x)$ be an elliptic curve over 
${{\mathbb {Z}}}_p$ such that the Kodaira symbol of 
$E$ is 
$T$. Then the local invariants of 
$E$ are as given in Table 1. Furthermore, there exists an element 
$t\in {{\mathbb {Z}}}_p$ such that coefficients of 
$f(x+t)=x^{3}+ax^{2}+bx+c$ are as given in the second column of Table 1. Finally, the density of all elliptic curves with Kodaira symbol 
$T$ is as given in the last column.
Local invariants of elliptic curves.
These density computations are straightforward, and indeed many of them are implicit in the work of Watkins [Reference WatkinsWat08, § 3.2]. However, we include a proof since our use of a 
${{\mathbb {G}}}_a$-action on the space of monic cubic polynomials simplifies the computations.
We use three different techniques to prove the estimates of Theorems 1.4 and 1.5. First, we fix a prime 
$p\geq 5$ and a Kodaira symbol 
$T$. The set of elliptic curves that have Kodaira symbol 
$T$ at 
$p$ is cut out by certain congruence conditions 
$S$ modulo 
$q$, some power of 
$p$. Working modulo 
$q$, we compute the Fourier transform of the characteristic function of 
$S$. An application of Poisson summation then yields baseline estimates for the number of elliptic curves with bounded height having Kodaira symbol 
$T$ at 
$p$.
Our next two techniques average over primes 
$p$ in a crucial way. Suppose that 
$E:y^{2}=f(x)$ is an elliptic curve in 
${{\mathcal {E}}}_{{\rm sf}}$ such that the ratio 
$\Delta (E)/C(E)$ is large. Then for any prime 
$p\geq 5$, the Kodaira symbol of 
$E$ at 
$p$ is 
${{\mathrm I}}_0$, 
${{\mathrm I}}_1$, 
${{\mathrm I}}_2$, 
${{\mathrm I}}{{\mathrm I}}$, or 
${{\mathrm I}}{{\mathrm I}}{{\mathrm I}}$. We prove that either the discriminant of the algebra 
$K_f$ is small, or the shape of the ring of integers 
${{\mathcal {O}}}_f$ of 
$K_f$ is very skewed. The work of Bhargava and Harron [Reference Bhargava and HarronBH16] proves that the shapes of rings of integers are equidistributed in the family of cubic fields. Furthermore, the forthcoming thesis of Chiche-Lapierre [Reference Chiche-LapierreChi19] determines asymptotics for the number of cubic fields such that the shapes of their ring of integers are constrained to lie within 
$0$-density sets. Using ideas from these works, we prove bounds on the number of possible cubic algebras 
$K_f$ corresponding to elliptic curves in 
${{\mathcal {E}}}_{{\rm sf}}$ with bounded conductor, along with bounds on the average sizes of the 
$2$-torsion subgroups 
${{\rm Cl}}_2(K_f)$ of the class groups of 
$K_f$. In combination with the work of Brumer and Kramer [Reference Brumer and KramerBK77], relating the size of 
${{\rm Sel}}_2(E)$ to 
$\#{{\rm Cl}}_2(K_f)$, we deduce Theorem 1.4.
One may view the smallness of the discriminant of 
$K_f$ and the skewness of the ring of integer 
${{\mathcal {O}}}_f$ above as archimedean constraints on the cubic algebra 
$K_f$ which allowed us to obtain enough savings to count elliptic curves in 
${{\mathcal {E}}}_{{\rm sf}}$ and their 
$2$-Selmer elements. However, primes 
$p$ with Kodaira symbol 
${{\mathrm I}}{{\mathrm V}}$ or 
${{\mathrm I}}_n$ with 
$n\geq 3$ do not impose any of these constraints as 
$D_p(E)=C_p(E)$ and 
$\Delta _p(E)$ is large. These reduction types do not appear for elliptic curves in 
${{\mathcal {E}}}_{{\rm sf}}$ but they do for the family 
$E_\kappa$. This is the main obstruction to obtaining 
$2$-Selmer averages for the family 
$E_\kappa$. These primes do impose 
$p$-adic conditions in the sense that 
$p$ divides both the index 
$Q(E)$ of 
$R_f$ in 
${{\mathcal {O}}}_f$ and the discriminant 
$D(E)$ of 
$K_f$. To exploit this, we proceed as follows. The set of integer monic traceless cubic polynomials 
$f$ with 
$p\mid Q(E_f)$ embeds into the space of binary quartic forms with a rational linear factor. This embedding 
$\sigma$ is defined in (20). The group 
${{\rm PGL}}_2$ acts on the space of binary quartic forms, and the ring of invariants for this action is freely generated by two polynomials 
$I$ and 
$J$. Restricting to the space of reducible binary quartic forms gives an additional invariant 
$Q$. Explicitly, if 
$g(x,y)$ is a binary quartic form with coefficients in 
${{\mathbb {Q}}}$, and 
$g(\alpha ,\beta )=0$, then define
\[ Q(g(x,y),[\alpha:\beta])=\frac{g(x,y)}{\beta x-\alpha y}(\alpha,\beta). \]
This new invariant 
$Q$ is an exact analogue of the 
$Q$-invariants used in [Reference Bhargava, Shankar and WangBSW16] to compute the density of polynomials with squarefree polynomials. As there, for every fixed root 
$[\alpha :\beta ]\in {{\mathbb {P}}}^{1}({{\mathbb {Z}}})$, the discriminant polynomial on the space of integer binary quartic forms 
$g$ with 
$g(\alpha ,\beta )$ is reducible, and in fact divisible by 
$Q^{2}$. We also define
\[ D(g(x,y),[\alpha:\beta]):=\Delta(g)/(Q(g(x,y),[\alpha:\beta]))^{2}. \]
Our embedding 
$\sigma$ satisfies 
$Q(E)=Q(\sigma (E))$ and 
$D(E)=D(\sigma (E))$. Then the required estimates on elliptic curves 
$E\in {{\mathcal {E}}}_\kappa$ with large 
$\Delta (E)/C(E)$ translate to estimates on the number of 
${{\rm PGL}}_2({{\mathbb {Z}}})$-orbits on integral reducible binary quartic forms with bounded height and large 
$Q$- and 
$D$-invariants. We prove the required estimates by fibering over roots, and then combining geometry of numbers methods with the Ekedahl sieve.
This paper is organized as follows. In §§ 2 and 3 we work locally, one prime at a time. Theorem 1.6 is proved in § 2, while the Fourier coefficients corresponding to a fixed Kodaira symbol are computed in § 3. The computation of the Fourier coefficients is then used to obtain estimates (see Theorem 3.1) on curves with fixed Kodaira symbols at finitely many primes. We prove bounds on the number of cubic fields 
$K$, weighted by 
$|{{\rm Cl}}_2(K)|$, in § 4, and obtain estimates on the number of reducible integer binary quartic forms with large 
$Q$- and 
$D$-invariants in § 5. The results of §§ 3–5 are combined in § 6 to prove the uniformity estimates Theorems 1.4 and 1.5. Finally, in § 7, we prove the main results, Theorems 1.1 and 1.2.
2. Reduction types of elliptic curves
Throughout this section we fix a prime 
$p\geq 5$. Let 
$U$ denote the space of monic cubic polynomials. Then for any ring 
$R$, we have
\[ U(R)=\{x^{3}+ax^{2}+bx+c:a,b,c\in R\}. \]
We denote the space of traceless elements of 
$U$ (i.e. 
$a=0$ in the above equation) by 
$U_0$. The group 
${{\mathbb {G}}}_a$ acts on 
$U$ via 
$(t\cdot f)(x)=f(x+t)$. Given any element 
$f\in U({{\mathbb {Z}}}_p)$, there exists a unique element 
$\gamma \in {{\mathbb {Z}}}_p$ such that 
$f_0(x)=(\gamma \cdot f)(x)$ belongs to 
$U_0({{\mathbb {Z}}}_p)$. Thus we may identify the quotient space 
${{\mathbb {Z}}}_p\backslash U({{\mathbb {Z}}}_p)$ with 
$U_0({{\mathbb {Z}}}_p)$. We denote the Euclidean measures on 
$U({{\mathbb {Z}}}_p)$ and 
$U_0({{\mathbb {Z}}}_p)$ by 
$dg=da\,db\,dc$ and 
$df=db\,dc$, respectively, where 
$da$, 
$db$ and 
$dc$ are Haar measures on 
${{\mathbb {Z}}}_p$ normalized so that 
${{\mathbb {Z}}}_p$ has volume 
$1$. Then the change-of-measure formula for the bijection
\begin{equation} \begin{aligned} {{\mathbb{Z}}}_p\times U_0({{\mathbb{Z}}}_p) & \to U({{\mathbb{Z}}}_p) \\ (t,f(x)) & \mapsto g(x)=(t\cdot f)(x)=f(x+t) \end{aligned} \end{equation}
is 
$dt\,df=dg$, where 
$dt$ is again the Haar measure on 
${{\mathbb {Z}}}_p$ normalized so that 
${{\mathbb {Z}}}_p$ has volume 
$1$.
Given an element 
$f(x)\in U({{\mathbb {Z}}}_p)$ such that the discriminant 
$\Delta (f)$ is nonzero, we consider the elliptic curve 
$E_f$ over 
${{\mathbb {Q}}}_p$ with affine equation 
$y^{2}=f(x)$. An element 
$f(x)\in U({{\mathbb {Z}}}_p)$ with nonzero discriminant is said to be minimal if 
$\Delta (f)=\Delta (E_f)$. Equivalently, 
$f(x)$ is minimal if 
$f_0(x)=x^{3}+Ax+B$, the unique element in 
$U_0({{\mathbb {Z}}}_p)$ in the 
${{\mathbb {Z}}}_p$-orbit of 
$f$, does not satisfy 
$p^{4}\mid A$ and 
$p^{6}\mid B$. Another equivalent condition is that the roots of 
$f_0(x)$ are not all multiples of 
$p^{2}$. We denote the set of minimal elements in 
$U({{\mathbb {Z}}}_p)$ by 
$U({{\mathbb {Z}}}_p)^{{\rm min}}$, and denote 
$U({{\mathbb {Z}}}_p)^{{\rm min}}\cap U_0({{\mathbb {Z}}}_p)$ by 
$U_0({{\mathbb {Z}}}_p)^{{\rm min}}$. The map 
$f\mapsto E_f$ is then a natural surjective map from 
${{\mathbb {Z}}}_p\backslash U({{\mathbb {Z}}}_p)^{{\rm min}}$ (equivalently 
$U_0({{\mathbb {Z}}}_p)^{{\rm min}}$) to the set of isomorphism classes of elliptic curves over 
${{\mathbb {Q}}}_p$.
The twisting-by-
$p$ map is a natural involution on the set of isomorphism classes of elliptic curves over 
${{\mathbb {Q}}}_p$. This yields a natural involution 
$\sigma$ on 
${{\mathbb {Z}}}_p\backslash U({{\mathbb {Z}}}_p)^{{\rm min}}$. If 
$f\in U({{\mathbb {Z}}}_p)^{{\rm min}}$ such that 
$f_0(x) = x^{3}+Ax+B$ with 
$p^{2}\nmid A$ or 
$p^{3}\nmid B$, then we say 
$f$ is small, and in this case 
$\sigma (f)_0(x) = \sigma (f_0)(x)=x^{3} + p^{2}Ax + p^{3}B$. Otherwise, if 
$f_0(x) = x^{3}+Ax+B$ with 
$p^{2}\mid A$ and 
$p^{3}\mid B$, then we say 
$f$ is large and in this case, 
$\sigma (f)_0(x)=\sigma (f_0)(x) = x^{3} + p^{-2}Ax + p^{-3}B$. We have 
$\Delta (E_{\sigma (f)}) = p^{6}\Delta (E_f)$ if 
$f$ is small and 
$\Delta (E_{\sigma (f)}) = p^{-6}\Delta (E_f)$ otherwise. Let 
$U({{\mathbb {Z}}}_p)^{{\rm sm}}$ denote the set of small elements 
$f\in U({{\mathbb {Z}}}_p)$.
Let 
$E$ be an elliptic curve over 
${{\mathbb {Q}}}_p$, and let 
$\mathscr {X}$ be a minimal proper regular model of 
$E$ over 
${{\mathbb {Z}}}_p$. For brevity, we will say that 
$T$, the Kodaira symbol associated to the special fiber of 
$\mathscr {X}$, is the Kodaira symbol of 
$E$. Define the index of 
$E$ by 
${{\rm ind}}(E):=\Delta (E)/C(E)$. Then the index of 
$E$ is 
$1$ if and only if the Kodaira symbol of 
$E$ is 
${{\mathrm I}}_0$ (when 
$E$ has good reduction), 
${{\mathrm I}}_1$, or 
${{\mathrm I}}{{\mathrm I}}$. Given 
$f\in U({{\mathbb {Z}}}_p)^{{\rm min}}$, we define the index of 
$f$ to be 
${{\rm ind}}(f):={{\rm ind}}(E_f)$. We also define two other invariants associated to elements 
$f\in U({{\mathbb {Z}}}_p)^{{\rm min}}$. Let 
$K_f$ denote the cubic étale algebra 
$K_f:={{\mathbb {Q}}}_p[x]/f(x)$, let 
${{\mathcal {O}}}_f$ denote the ring of integers of 
$K_f$, and let 
$R_f$ denote the cubic ring 
${{\mathbb {Z}}}[x]/f(x)$. We define
\begin{align*} Q_p(f)&:= [{{\mathcal{O}}}_f:R_f],\\ D_p(f)&:= {{\rm Disc}}(K_f). \end{align*}
These quantities are clearly invariant under the action of 
${{\mathbb {Z}}}_p$ on 
$U({{\mathbb {Z}}}_p)$ and satisfy the equation
\[ \Delta(f)=\Delta(R_f)=D_p(f)Q_p(f)^{2}. \]The following lemma will be used repeatedly in the proof of Theorem 1.6 to deduce maximality of cubic rings.
Lemma 2.1 Let 
$p$ be a prime. Let 
$f(x) = ux^{3} + pax^{2} + pbx + pc$ be a monic cubic polynomial with 
$a,b,c\in {{\mathbb {Z}}}_p$ and 
$u\in {{\mathbb {Z}}}_p^{\times }$. Then the ring 
$R_f = {{\mathbb {Z}}}_p[x]/(f(x))$ is maximal in 
$K_f = {{\mathbb {Q}}}_p[x]/(f(x))$ if and only if 
$p\nmid c$.
Proof. We may assume without loss of generality that 
$u=1$. Under the Delone–Faddeev correspondence, the ring 
$R_f$ is isomorphic to the cubic ring associated to the binary cubic form 
$x^{3} + pax^{2}y+pbxy^{2}+pcy^{3}$ using the normal basis 
$\{1,x,x^{2}+bx+c\}$. The paragraph following Lemma 13 in [Reference Bhargava, Shankar and TsimermanBST13] then implies that 
$R_f$ is nonmaximal if and only if there exists 
$r\in {{\mathbb {Z}}}_p$ such that 
$p$ divides the linear coefficient of 
$f(x+r)$ and 
$p^{2}$ divides the constant coefficient of 
$f(x+r)$. Any such 
$r$ with 
$p$ dividing the constant coefficient of 
$f(x+r)$ must be in 
$p{{\mathbb {Z}}}_p$, in which case the linear coefficient of 
$f(x+r)$ is always divisible by 
$p$ and the constant coefficient of 
$f(x+r)$ is congruent to 
$pc$ mod 
$p^{2}.$ Therefore, 
$R_f$ is nonmaximal if and only if 
$p\mid c.$
We now prove Theorem 1.6.
Proof of Theorem 1.6 Let 
$f\in U({{\mathbb {Z}}}_p)^{{{\rm min}}}$. A direct computation shows that there always exists 
$t\in {{\mathbb {Z}}}_p$ such that 
$t\cdot f$ has coefficients satisfying the congruence conditions detailed in one of the rows of column two of Table 1. We now assume that 
$f$ satisfies one set of congruence conditions detailed in column two of Table 1. Carrying out Tate's algorithm (as detailed in [Reference SilvermanSil94, pp. 366–368]) for each congruence type in column two verifies for us the first four columns of Table 1. Finally, looking at 
$f_0 \in U_0({{\mathbb {Z}}}_p)^{{{\rm min}}}$ in the 
${{\mathbb {Z}}}_p$-orbit of 
$f$ yields that the Kodaira symbol of 
$E_f$ is 
${{\mathrm I}}_n$, 
${{\mathrm I}}{{\mathrm I}}$, 
${{\mathrm I}}{{\mathrm I}}{{\mathrm I}}$, or 
${{\mathrm I}}{{\mathrm V}}$ if and only if 
$f \in U({{\mathbb {Z}}}_p)^{{{\rm sm}}}$.
It remains for us to verify the last three columns of Table 1. We will begin by verifying the fifth and sixth column, leaving the density computation to Proposition 2.2. We first deal with small elliptic curves, that is, when the associated Kodaira symbol is 
${{\mathrm I}}_n$, 
${{\mathrm I}}{{\mathrm I}}$, 
${{\mathrm I}}{{\mathrm I}}{{\mathrm I}}$, or 
${{\mathrm I}}{{\mathrm V}}$. Without loss of generality, we assume that 
$f(x)$ satisfies the congruences in one of the rows of column two (as mentioned above, we may do this by replacing 
$f$ by a 
${{\mathbb {Z}}}_p$-translate). The result is clear if 
$E_f$ has good reduction, which happens precisely when 
$\Delta _p(E_f)=1$.
First assume that 
$E_f$ has additive reduction, in which case 
$C_p(E)=p^{2}$. Then the Kodaira symbol of 
$E_f$ is 
${{\mathrm I}}{{\mathrm I}}$, 
${{\mathrm I}}{{\mathrm I}}{{\mathrm I}}$, or 
${{\mathrm I}}{{\mathrm V}}$. In the first case, Lemma 2.1 yields that 
$R_f$ is indeed the maximal order, thereby verifying the fifth and sixth columns for elliptic curves having Kodaira symbol 
${{\mathrm I}}{{\mathrm I}}$. In the second case, Lemma 2.1 also yields that 
$R_f$ is nonmaximal. As the discriminants of 
$\mathcal {O}_f$ and 
$R_f$ differ by a perfect square, it follows that 
$Q_p(f) = D_p(f) = p$, thereby verifying the fifth and sixth columns for elliptic curves having Kodaira symbol 
${{\mathrm I}}{{\mathrm I}}{{\mathrm I}}$. In the third case, we write 
$a=pa_1$, 
$b=p^{2}b_2$, and 
$c = p^{2}c_2$, with 
$p \nmid c_2$. We see that 
$R_f$ is a suborder of index 
$p$ of the maximal order 
${{\mathcal {O}}}$ corresponding to the binary cubic form 
$px^{3}+pax^{2}y+pb_2xy^{2}+c_2y^{3}$ with 
$\Delta _p({{\mathcal {O}}})=p^{2}$. Note the maximality of 
${{\mathcal {O}}}$ follows from Lemma 2.1 applied to the polynomial 
$g(x)=c_2x^{3} + pb_2x^{2}y + paxy^{2} +p$ as 
$R_g\simeq {{\mathcal {O}}}.$ This confirms the fifth and sixth columns of Table 1 in the case where 
$E_f$ has additive reduction and 
$f\in U({{\mathbb {Z}}}_p)^{{\rm sm}}$.
Next, assume that 
$E_f$ has multiplicative reduction. In this case, 
$f$ mod 
$p$ has one simple root (necessarily defined over 
${{\mathbb {F}}}_p$) and one root with multiplicity 
$2$. Hensel's lemma implies that 
$f(x)$ factors as a product of a linear polynomial 
$\ell (x)$ and a quadratic polynomial 
$q(x)$ (we make no assumptions about whether 
$q(x)$ is reducible or not) over 
${{\mathbb {Z}}}_p$. Therefore, 
$D_p(f)$ must equal either 
$p$ or 
$1$,Footnote 2 depending on whether 
$\Delta _p(f) = n$ is odd or even. It follows that 
$D_f = p^{n \pmod 2}$ and 
$Q_p(f) = p^{\lfloor {n}/{2} \rfloor }$. We have now verified columns five and six for all small elliptic curves.
We now turn to large elliptic curves. Let 
$E$ be a large elliptic curve over 
${{\mathbb {Z}}}_p$. Let 
$E'$ denote the twist of 
$E$ by 
$p$. Then the Kodaira symbol of 
$E'$ is 
${{\mathrm I}}_n$, 
${{\mathrm I}}{{\mathrm I}}$, 
${{\mathrm I}}{{\mathrm I}}{{\mathrm I}}$, or 
${{\mathrm I}}{{\mathrm V}}$, depending on whether the Kodaira symbol of 
$E$ is 
${{\mathrm I}}_n^{*}$, 
${{\mathrm I}}{{\mathrm V}}^{*}$, 
${{\mathrm I}}{{\mathrm I}}{{\mathrm I}}^{*}$, or 
${{\mathrm I}}{{\mathrm I}}^{*}$, respectively. Let 
$y^{2}=f(x)$ be a model for 
$E'$, where the coefficients of 
$f(x)=x^{3}+ax^{2}+bx+c$ satisfy the congruence conditions of Table 1. Then 
$y^{2}=g(x)=x^{3}+pax^{2}+p^{2}bx+p^{3}c$ is a model for 
$E$. Furthermore, 
$K_g=K_f$ and 
$R_g$ has index 
$p^{3}$ in 
$R_f$. It follows that the local invariants of 
$E$ are as in Table 1. Theorem 1.6 follows the density computations in the following proposition.
Proposition 2.2 The density of elliptic curves over 
${{\mathbb {Z}}}_p$ having a fixed Kodaira symbol is as in Table 1.
Proof. Let 
$T$ be a fixed Kodaira symbol. Let 
$U({{\mathbb {Z}}}_p)^{(T)}$ (respectively, 
$U_0({{\mathbb {Z}}}_p)^{(T)}$) denote the set of elements 
$f \in U({{\mathbb {Z}}}_p)^{{\rm min}}$ (respectively, 
$f \in U_0({{\mathbb {Z}}}_p)^{{\rm min}}$) such that 
$E_f$ has Kodaira symbol 
$T$. Then the density of elliptic curves with Kodaira symbol 
$T$ is 
${{\rm Vol}}(U_0({{\mathbb {Z}}}_p)^{(T)})={{\rm Vol}}(U({{\mathbb {Z}}}_p)^{(T)})$, where the equality holds since 
${{\mathbb {Z}}}_p\cdot U_0({{\mathbb {Z}}}_p)^{(T)}=U({{\mathbb {Z}}}_p)^{(T)}$ and the Jacobian change of variables of the map (3) is 
$1$.
We start with Kodaira symbol 
${{\mathrm I}}_0$. The set 
$U({{\mathbb {Z}}}_p)^{({{\mathrm I}}_0)}$ consists of those 
$f\in U({{\mathbb {Z}}}_p)$ such that 
$f(x)\pmod {p}$ has three distinct roots in 
$\bar {{{\mathbb {F}}}}_p$. Denote these roots by 
$\alpha _1$, 
$\alpha _2$, and 
$\alpha _3$. Either the 
$\alpha _i$ all belong to 
${{\mathbb {F}}}_p$, or 
$\alpha _1\in {{\mathbb {F}}}_p$ and 
$\alpha _2,\alpha _3$ are a pair of conjugate elements in 
${{\mathbb {F}}}_{p^{2}}\backslash {{\mathbb {F}}}_p$, or the 
$\alpha _i$ are conjugate elements in 
${{\mathbb {F}}}_{p^{3}}\backslash {{\mathbb {F}}}_p$. Thus, we have
\begin{align*} {{\rm Vol}}(U({{\mathbb{Z}}}_p)^{({{\mathrm I}}_0)})&= \frac{p(p-1)(p-2)}{6p^{3}} +\frac{p(p^{2}-p)}{2p^{3}}+\frac{p^{3}-p}{3p^{3}}\\ &= 1-\frac{1}{p}, \end{align*}as required.
Second, we consider the Kodaira symbol 
${{\mathrm I}}_n$ for 
$n\geq 1$. Suppose 
$f(x)\in U({{\mathbb {Z}}}_p)^{{{\mathrm I}}_n}$. Then 
$f(x)$ has exactly one double root modulo 
$p$. We therefore have 
$f(x) = g(x)(x-\alpha )$, where 
$g(x)$ has a double root modulo 
$p$, and 
$p \nmid g(\alpha )$. Clearly, we have 
$\Delta _p(g)=\Delta _p(f)=p^{n}$, since 
$\Delta _p(E_f)=p^{n}$. We write the quadratic factor 
$g(x)$ in unique form as 
$g(x)=(x+\beta )^{2}+\gamma$. The discriminant condition translates to 
$p^{n}\parallel \gamma$, and the condition that 
$p\nmid g(\alpha )$ translates to 
$p\nmid (\alpha +\beta )$. Therefore, every element of 
$U({{\mathbb {Z}}}_p)^{{{\mathrm I}}_n}$ can be expressed uniquely in the form
\[ ((x+\beta)^{2}+\gamma)(x-\alpha)=x^{3}+(2\beta-\alpha)x^{2} +(\beta^{2}-2\alpha\beta+\gamma)x-\alpha\beta^{2}-\alpha\gamma, \]
such that 
$p^{n}\parallel \gamma$ and 
$p\nmid (\alpha +\beta )$. The Jacobian change of variables for the map 
$(\alpha ,\beta ,\gamma )\mapsto (a,b,c)$ is 
$-2(\alpha +\beta )^{2}-2\gamma$, which is always a unit. Thus, we have
\begin{align*} {{\rm Vol}}(U({{\mathbb{Z}}}_p)^{{{\mathrm I}}_n})&= {{\rm Vol}}(p^{n}{{\mathbb{Z}}}_p\backslash p^{n+1}{{\mathbb{Z}}}_p){{\rm Vol}}(\{(\alpha,\beta)\in{{\mathbb{Z}}}_p^{2}:p\nmid(\alpha+\beta)\})\\ &= (p-1)^{2}/p^{n+2}, \end{align*}as required.
Third, we consider the Kodaira symbols 
${{\mathrm I}}{{\mathrm I}}$, 
${{\mathrm I}}{{\mathrm I}}{{\mathrm I}}$, and 
${{\mathrm I}}{{\mathrm V}}$. If 
$f\in U_0({{\mathbb {Z}}}_p)$ is such that the Kodaira symbol of 
$E_f$ is one of the three above, then 
$f(x)=x^{3}+Ax+B$ has a triple root modulo 
$p$, which implies that 
$p$ divides 
$A$ and 
$B$. By examining the second column of Table 1, we see that the Kodaira symbol of 
$E_f$ is 
${{\mathrm I}}{{\mathrm I}}$ if and only if 
$p\mid A$ and 
$p\parallel B$; 
${{\mathrm I}}{{\mathrm I}}{{\mathrm I}}$ if and only if 
$p\parallel A$ and 
$p^{2}\mid B$; and 
${{\mathrm I}}{{\mathrm V}}$ if and only if 
$p^{2}\mid A$ and 
$p^{2}\parallel B$. Hence the volumes of 
$U_0({{\mathbb {Z}}}_p)^{(T)}$, for 
$T={{\mathrm I}}{{\mathrm I}}$, 
${{\mathrm I}}{{\mathrm I}}{{\mathrm I}}$, and 
${{\mathrm I}}{{\mathrm V}}$, are respectively 
$(p-1)/p^{3}$, 
$(p-1)/p^{4}$, and 
$(p-1)/p^{5}$, as required.
Finally, we turn to the large Kodaira symbols, that is, those corresponding to large elliptic curves. Consider the map
\begin{align*} \sigma: U({{\mathbb{Z}}}_p)^{{\rm sm}}&\to U({{\mathbb{Z}}}_p)\\ x^{3}+ax^{2}+bx+c&\mapsto x^{3}+pax^{2}+p^{2}bx+p^{3}c. \end{align*}
Clearly, if 
$S\subset U({{\mathbb {Z}}}_p)$ is any measurable set, then 
${{\rm Vol}}(\sigma (S))=p^{-6}{{\rm Vol}}(S)$. Furthermore, we set 
$\sigma ({{\mathrm I}}_n) = {{\mathrm I}}_n^{*}$, 
$\sigma ({{\mathrm I}}{{\mathrm I}}) = {{\mathrm I}}{{\mathrm V}}^{*}$, 
$\sigma ({{\mathrm I}}{{\mathrm I}}{{\mathrm I}}) = {{\mathrm I}}{{\mathrm I}}{{\mathrm I}}^{*}$, and 
$\sigma ({{\mathrm I}}{{\mathrm V}})={{\mathrm I}}{{\mathrm I}}^{*}$. Then 
$\sigma$ sends 
$f$ of Kodaira symbol 
$T$ to 
$\sigma (f)$ of Kodaira symbol 
$\sigma (T)$. Moreover, we have 
$\sigma (t\cdot f)=(pt)\cdot \sigma (f)$. Hence we have
\[ \sigma(U({{\mathbb{Z}}}_p)^{(T)})=\sigma({{\mathbb{Z}}}_p\cdot U_0({{\mathbb{Z}}}_p)^{(T)}) =(p{{\mathbb{Z}}}_p)\cdot\sigma(U_0({{\mathbb{Z}}}_p)^{(T)}). \]
Fix any 
$g\in U_0({{\mathbb {Z}}}_p)^{(\sigma (T))}$. There exists 
$t\in {{\mathbb {Z}}}_p$ such that the coefficients of 
$t\cdot g$ are as in the second column of Table 1. Hence there exists 
$f\in U({{\mathbb {Z}}}_p)^{(T)}$ with 
$\sigma (f)=t\cdot g$. Then 
$\sigma (f_0)$ is 
${{\mathbb {Z}}}_p$-equivalent to 
$g$. Since 
$\sigma (f_0)$ and 
$g$ both belong to 
$U_0({{\mathbb {Z}}}_p)$, we must have 
$\sigma (f_0)=g$. Hence we have 
$\sigma (U_0({{\mathbb {Z}}}_p)^{(T)})=U_0({{\mathbb {Z}}}_p)^{(\sigma (T))}$. Therefore, we have
\begin{align*} {{\rm Vol}}(U({{\mathbb{Z}}}_p)^{(\sigma(T))})&= {{\rm Vol}}({{\mathbb{Z}}}_p\cdot U_0({{\mathbb{Z}}}_p)^{(\sigma(T))})\\ &= p\cdot {{\rm Vol}}(p{{\mathbb{Z}}}_p\cdot U_0({{\mathbb{Z}}}_p)^{(\sigma(T))})\\ &= p\cdot {{\rm Vol}}(\sigma(U({{\mathbb{Z}}}_p)^{(T)}))\\ &= p^{{-}5}{{\rm Vol}}(U({{\mathbb{Z}}}_p)^{(T)}). \end{align*}This concludes the proof of Proposition 2.2, and thus of Theorem 1.6.
Theorem 1.6 has the following immediate corollary, which will be useful in what follows.
Corollary 2.3 Let 
$p\geq 5$ be a prime. The density of elliptic curves 
$E$ over 
${{\mathbb {Q}}}_p$ with good, multiplicative, or additive reduction, such that 
${{\rm ind}}(E)=\Delta _p(E)/C_p(E)=p^{k}$, is as given in Table 2.
$p$-adic densities of elliptic curves with given index.
3. Fourier coefficients of polynomials with fixed Kodaira symbol
Let 
$p\geq 5$ be a prime, and let 
$U({{\mathbb {Z}}}_p)^{{\rm min}}$ and 
$U({{\mathbb {Z}}}_p)^{{\rm sm}}$ be as in § 2. Recall that to each 
$f(x)\in U({{\mathbb {Z}}}_p)^{{\rm min}}$ we associate the Kodaira symbol of the elliptic curve 
$E_f$. By (the proof of) Theorem 1.6, an element 
$f(x)\in U({{\mathbb {Z}}}_p)^{{\rm min}}$ belongs to 
$U({{\mathbb {Z}}}_p)^{{\rm sm}}$ and satisfies 
$\Delta (f) \neq C(f)$ if and only if the Kodaira symbol of 
$f$ is 
${{\mathrm I}}{{\mathrm I}}{{\mathrm I}}$, 
${{\mathrm I}} {{\mathrm V}}$, or 
${{\mathrm I}}_n$ for 
$n \geq 2$. Denote the set of polynomials 
$f(x)\in U({{\mathbb {Z}}})$ such that 
$f\in U({{\mathbb {Z}}}_p)^{{\rm min}}$ for all primes 
$p$ by 
$U({{\mathbb {Z}}})^{{\rm min}}$. Given 
$f(x)\in U({{\mathbb {Z}}})^{{\rm min}}$ and a prime 
$p$, we say that the Kodaira symbol of 
$f$ at 
$p$ is 
$T$, the Kodaira symbol of 
$f(x)$ considered as an element in 
$U({{\mathbb {Z}}}_p)^{{\rm min}}$.
Let 
$\Sigma$ be a set consisting of the following data: a finite set 
$\{p_1,\ldots ,p_k\}$ of primes 
$p_i\geq 5$ along with a Kodaira symbol 
$T(p_i)$ which is 
${{\mathrm I}}{{\mathrm I}}{{\mathrm I}}$, 
${{\mathrm I}}{{\mathrm V}}$ or 
${{\mathrm I}}_{n\geq 2}$ associated to each prime 
$p_i$ in the set. We say 
$f\in U({{\mathbb {Z}}})$ has splitting type 
$\Sigma$ if 
$f$ has Kodaira symbol 
$T(p_i)$ at each prime 
$p_i$ in 
$\Sigma$. Let 
$U({{\mathbb {Z}}})_\Sigma$ denote the set of elements 
$f\in U({{\mathbb {Z}}})$ with splitting type 
$\Sigma$. Given such a collection 
$\Sigma$, we define the constant 
$Q(\Sigma )$ to be 
$\prod _{p_i}p_i^{a_i}$, where 
$a_i=1$ if 
$T(p_i)$ is 
${{\mathrm I}}{{\mathrm I}}{{\mathrm I}}$ or 
${{\mathrm I}}{{\mathrm V}}$, and 
$a_i=\lfloor n/2 \rfloor$ if 
$T(p_i)$ is 
${{\mathrm I}}_n$. Note that if 
$f\in U({{\mathbb {Z}}})_\Sigma$, then 
$Q(\Sigma )\mid Q(f)$. We define 
$m_{T}(\Sigma )$ to be the product of all primes 
$p$ such that 
$T(p) = T$. We also define 
$m_{{\rm odd}}(\Sigma )$ to be the product of all primes 
$p$ in 
$\Sigma$ such that 
$\sigma (p)={{\mathrm I}}_n$ for some odd integer 
$n$. Finally, we define 
$\nu (\Sigma )$ to be the product over the primes 
$p$ in 
$\Sigma$ of the density 
$\nu (T_p)$, that is, the 
$p$-adic volume of the set of elements in 
$U({{\mathbb {Z}}}_p)^{{{\rm min}}}$ having Kodaira symbol 
$T(p)$.
Define the height function 
$H$ on 
$U({{\mathbb {R}}})$ to be
\[ H(x^{3}+ax^{2}+bx+c):= {{\rm max}}\{|a|^{6},|b|^{3},|c|^{2}\}. \]
The goal of this section is to obtain a bound on the number of elements in 
$U({{\mathbb {Z}}})$ that have bounded height and specified Kodaira symbols 
${{\mathrm I}}{{\mathrm I}}{{\mathrm I}}$, 
${{\mathrm I}}{{\mathrm V}}$ or 
${{\mathrm I}}_{n\geq 2}$ at finitely many primes. We prove the following theorem.
Theorem 3.1 Let 
$\Sigma$ be as above and, for every Kodaira symbol 
$T$, denote 
$Q(\Sigma )$, 
$m_{{\rm odd}}(\Sigma )$, and 
$m_T(\Sigma )$ by 
$Q$, 
$m_{{\rm odd}}$, and 
$m_T$, respectively. Then we have
\[ \#\{f\in U({{\mathbb{Z}}})_\Sigma:H(f) < Y\}\ll_\epsilon \frac{Y}{Q^{2}m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\mathrm I}}{{\mathrm V}}}^{2}m_{{\rm odd}}}+\frac{Qm_{{\rm odd}}}{m_{{{\mathrm I}}{{\mathrm V}}}} Y^{\epsilon}, \]
where the implied constant is independent of 
$Y$ and 
$\Sigma$.
Theorem 3.1 will be used in § 6.2, where we sum over all possible reduction types such that 
$m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}$, 
$m_{{{\mathrm I}}{{\mathrm V}}}$, 
$m_{{{\rm even}}}$, 
$m_{{{\rm odd}}}$, and 
$Q$ belong to specified dyadic ranges, to prove Theorem 1.5.
This section is organized as follows. First, in § 3.1, we recall some preliminary results from Fourier analysis. In particular, the ‘twisted Poisson summation’ formula of Proposition 3.2 will be our main tool in proving Theorem 3.1. Also, in (6), we determine how the action of 
${{\mathbb {G}}}_a$ on 
$U$ changes the Fourier coefficients of functions. Next, in § 3.2, we compute the Fourier coefficients of a slightly modified version of the characteristic functions of the set of monic polynomials having Kodaira symbol 
$T$, for 
$T={{\mathrm I}}{{\mathrm I}}{{\mathrm I}}$, 
${{\mathrm I}}{{\mathrm V}}$, and 
${{\mathrm I}}_{n\geq 2}$. Finally, in § 3.3, we use these computations and the twisted Poisson summation formula to prove Theorem 3.1.
3.1 Preliminary results from Fourier analysis
We fix a positive integer 
$N$ with 
$(N,6)=1$, and consider the space 
$\widehat {U({{\mathbb {Z}}}/N{{\mathbb {Z}}})}$ dual to 
$U({{\mathbb {Z}}}/N{{\mathbb {Z}}})$. We write elements 
$\chi \in \widehat {U({{\mathbb {Z}}}/N{{\mathbb {Z}}})}$ as triples 
$\chi =(\check {a},\check {b},\check {c})\in ({{\mathbb {Z}}}/N{{\mathbb {Z}}})^{3}$, and view 
$\chi$ as the character given by
\begin{equation} \chi(x^{3}+ax^{2}+bx+c)= e \biggl (\frac{\check{a}\cdot a+\check{b}\cdot b+\check{c}\cdot c}{N} \biggr), \end{equation}
where 
$e(x) := \exp (2\pi ix)$. Given a function 
$\phi :U({{\mathbb {Z}}}/N{{\mathbb {Z}}})\to {{\mathbb {C}}}$, we have the Fourier dual 
$\hat {\phi }:\widehat {U({{\mathbb {Z}}}/N{{\mathbb {Z}}})}\to {{\mathbb {C}}}$ defined to be
\[ \hat\phi(\chi):=\sum_{f\in U({{\mathbb{Z}}}/N{{\mathbb{Z}}})}\phi(f)\chi(f), \]and Fourier inversion yields the equality
\[ \frac{1}{N^{3}}\sum_\chi\hat{\phi}(\chi)\overline{\chi(f)}=\phi(f). \] The additive group 
${{\mathbb {Z}}}/N{{\mathbb {Z}}}$ acts on the space 
$U({{\mathbb {Z}}}/N{{\mathbb {Z}}})$ via the action 
$(r\cdot f)(x)=f(x+r)$. Identifying 
$U({{\mathbb {Z}}}/N{{\mathbb {Z}}})$ with the coefficient space 
$({{\mathbb {Z}}}/N{{\mathbb {Z}}})^{3}$, we write the action explicitly:
\[ r\cdot (a,b,c)=((a+3r),(b+2ra+3r^{2}),(c+rb+r^{2}a+r^{3})). \]
Given a function 
$\phi :U({{\mathbb {Z}}}/N{{\mathbb {Z}}})\to {{\mathbb {C}}}$ and an element 
$r\in {{\mathbb {Z}}}/N{{\mathbb {Z}}}$, we define 
$r\cdot \phi :U({{\mathbb {Z}}}/N{{\mathbb {Z}}})\to {{\mathbb {C}}}$ to be 
$(r\cdot \phi )(f):=\phi ((-r)\cdot f)$. We also define an action of 
${{\mathbb {Z}}}/N{{\mathbb {Z}}}$ on 
$\widehat {U({{\mathbb {Z}}}/N{{\mathbb {Z}}})}$ by
\begin{equation} r\cdot\chi:= ((\check{a}+2r\check{b}+r^{2}\check{c}),(\check{b}+r\check{c}),\check{c}), \end{equation}
for 
$\chi =({{\check {a}}},{{\check {b}}},{{\check {c}}})$. Then we have
\begin{align} \widehat{r\cdot\phi}(\chi)&=
\sum_f (r\cdot\phi)(f)\chi(f) = \sum_f \phi(({-}r)\cdot
f)\chi(f) = \sum_f \phi(f)\chi(r\cdot f)\nonumber\\ &=
\sum_{f=(a,b,c)} \phi(f)e \biggl(\frac{\check{a}a+\check{b}(b+2ra)+\check{c}(c+rb+r^{2}a)}{N}\biggr)
e\biggl(\frac{3\check{a}r+3\check{b}r^{2}+\check{c}r^{3}}{N}\biggr)\nonumber\\
&= e
\biggl(\frac{3\check{a}r+3\check{b}r^{2}+\check{c}r^{3}}{N}\biggr)
\sum_{f=(a,b,c)} \phi(f)e
\biggl(\frac{(\check{a}+2r\check{b}+r^{2}\check{c})a+
(\check{b}+r\check{c})b+\check{c}c}{N}\biggr)\nonumber\\ &=
\Psi_r(\chi)\hat{\phi}(r\cdot\chi),
\end{align}
where we set
\[ \Psi_r(\chi):=e \biggl(\frac{3\check{a}r+3\check{b}r^{2}+\check{c}r^{3}}{N}\biggr). \] Note that if we identify elements 
$\chi =(\check {a},\check {b},\check {c})\in \widehat {U({{\mathbb {Z}}}/N{{\mathbb {Z}}})}$ with binary quadratic forms
\[ P_\chi(x,y):=\check{a}x^{2}+2\check{b}xy+\check{c}y^{2}, \]
then the action of 
${{\mathbb {Z}}}/N{{\mathbb {Z}}}$ on 
$\widehat {U({{\mathbb {Z}}}/N{{\mathbb {Z}}})}$ in (5) corresponds exactly to the natural action
\[ P_{r\cdot \chi}(x,y)=P_\chi(x,y+rx). \]
We define 
$\Delta _2(\chi )=\check {b}^{2}-\check {a}\check {c}$. Then 
$\Delta _2$ is invariant under the action of 
${{\mathbb {Z}}}/N{{\mathbb {Z}}}$. Throughout the rest of this section, we will thus identify the space 
$\widehat {U({{\mathbb {Z}}}/N{{\mathbb {Z}}})}$ with the space 
$V_2({{\mathbb {Z}}}/N{{\mathbb {Z}}})$, where 
$V_2={{\rm Sym}}_2(2)$ is the space of binary quadratic forms with middle coefficient a multiple of 
$2$.
Finally, we recall the following ‘twisted Poisson summation’.
Proposition 3.2 Let 
$\psi :U({{\mathbb {R}}})\to {{\mathbb {R}}}$ denote a smooth function with bounded support. Let 
$\phi :U({{\mathbb {Z}}}/N{{\mathbb {Z}}})\to {{\mathbb {R}}}$ be any function. Then, for every positive real number 
$Y$, we have
\[ \sum_{(a,b,c)\in U({{\mathbb{Z}}})}\psi \biggl ( \frac{a}{Y^{1/6}},\frac{b}{Y^{1/3}},\frac{c}{Y^{1/2}} \biggr)\phi(a,b,c)= \frac{Y}{N^{3}}\sum_{\chi=(\check{a},\check{b},\check{c})\in \widehat{U({{\mathbb{Z}}})}}\hat{\psi} \biggl (\frac{Y^{1/6}\check{a}}{N},\frac{Y^{1/3}\check{b}}{N}, \frac{Y^{1/2}\check{c}}{N} \biggr) \hat\phi(\check{a},\check{b},\check{c}). \]
The 
$\hat \psi$ on the right-hand side is the usual Fourier transform over 
${{\mathbb {R}}}$ and so decays faster than any polynomial.
Proof. Fix a complete set of representatives in 
$U({{\mathbb {Z}}})$ for the quotient map 
$U({{\mathbb {Z}}})\rightarrow U({{\mathbb {Z}}}/N{{\mathbb {Z}}})$. We abuse notation and identify these representatives with the corresponding elements in 
$U({{\mathbb {Z}}}/N{{\mathbb {Z}}}).$ Then
\begin{align*}
& \sum_{(a,b,c)\in U({{\mathbb{Z}}})}\psi \biggl (
\frac{a}{Y^{1/6}},\frac{b}{Y^{1/3}},\frac{c}{Y^{1/2}}
\biggr)\phi(a,b,c)\\
&\quad = \sum_{(a_1,b_1,c_1)\in
U({{\mathbb{Z}}}/N{{\mathbb{Z}}})}\phi(a_1,b_1,c_1)\sum_{(a_2,b_2,c_2)\in
U({{\mathbb{Z}}})}\psi \biggl (
\frac{a_1+Na_2}{Y^{1/6}},\frac{b_1+Nb_2}{Y^{1/3}},\frac{c_1+Nc_2}{Y^{1/2}}
\biggr)\\
&\quad = \sum_{(a_1,b_1,c_1)\in U({{\mathbb{Z}}}/N{{\mathbb{Z}}})}\phi(a_1,b_1,c_1)\frac{Y^{1/6}}{N}e
\biggl(\frac{a_1\check{a}}{N}\biggr)\frac{Y^{1/3}}{N}e
\biggl(\frac{b_1\check{b}}{N}\biggr)\frac{Y^{1/2}}{N}e
\biggl(\frac{c_1\check{c}}{N}\biggr)\nonumber \\ &\qquad
\times \sum_{\chi=(\check{a},\check{b},\check{c})\in
\widehat{U({{\mathbb{Z}}})}}\hat{\psi} \biggl
(\frac{Y^{1/6}\check{a}}{N},\frac{Y^{1/3}\check{b}}{N},
\frac{Y^{1/2}\check{c}}{N} \biggr)\\ &\quad =
\frac{Y}{N^{3}}\sum_{\chi=(\check{a},\check{b},\check{c})\in
\widehat{U({{\mathbb{Z}}})}}\hat{\psi} \biggl
(\frac{Y^{1/6}\check{a}}{N},\frac{Y^{1/3}\check{b}}{N},
\frac{Y^{1/2}\check{c}}{N} \biggr)
\hat\phi(\check{a},\check{b},\check{c}),
\end{align*}where the second equality follows from Poisson summation.
3.2 Bounds on Fourier coefficients
Let 
$p\geq 5$ be a fixed prime. The conditions imposed by the choice of Kodaira symbol 
$T$ being equal to 
${{\mathrm I}}{{\mathrm I}}{{\mathrm I}}$, 
${{\mathrm I}}{{\mathrm V}}$ or 
${{\mathrm I}}_{n\geq 2}$ are defined via congruence conditions modulo 
$N=N_p(T)$, where 
$N$ is 
$p^{2}$, 
$p^{2}$ or 
$p^{n}$, respectively. Hence, when we refer to an element 
$f$ having one of the above Kodaira symbols, we will be implicitly assuming that 
$f$ belongs to 
$U({{\mathbb {Z}}}/N{{\mathbb {Z}}})$, where 
$N$ is the appropriate power of 
$p$. Naturally, in this context, we will also assume that elements 
$\chi$ belong to 
$\widehat {U({{\mathbb {Z}}}/N{{\mathbb {Z}}})}$, and represent them as triplets 
$(\check {a},\check {b},\check {c})\in ({{\mathbb {Z}}}/N{{\mathbb {Z}}})^{3}$.
For a Kodaira symbol 
$T\in \{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}},{{\mathrm I}}{{\mathrm V}},{{\mathrm I}}_{\geq 2}\}$, we define the set 
${{\mathcal {S}}}_0(T)$ to be
\[ \begin{array}{rl} \{x^{3}+ax^{2}+bx+c:p\mid a;p\mid b,p^{2}\mid c\}\subset U({{\mathbb{Z}}}/p^{2}{{\mathbb{Z}}}) & \mbox{if } T={{\mathrm I}}{{\mathrm I}}{{\mathrm I}},\\ \{x^{3}+ax^{2}+bx+c:p\mid a;p^{2}\mid b,p^{2} \mid c\}\subset U({{\mathbb{Z}}}/p^{2}{{\mathbb{Z}}}) & \mbox{if } T={{\mathrm I}}{{\mathrm V}},\\ \{x^{3}+ax^{2}+bx+c:p^{n}\mid b,p^{2n}\mid c\}\subset U({{\mathbb{Z}}}/p^{2n}{{\mathbb{Z}}}) & \mbox{if } T={{\mathrm I}}_{2n},\\ \{x^{3}+ax^{2}+bx+c:p^{n+1}\mid b,p^{2n+1}\mid c\}\subset U({{\mathbb{Z}}}/p^{2n+1}{{\mathbb{Z}}}) & \mbox{if } T = {{\mathrm I}}_{2n+1}.\\ \end{array} \]
From the second column of Table 1, it follows that every element having Kodaira symbol 
$T$ is contained within some 
${{\mathbb {G}}}_a$ translate of 
${{\mathcal {S}}}_0(T)$. Let 
$\Phi _{0,T}$ denote the characteristic function of 
${{\mathcal {S}}}_0(T)$, and define the function 
$\Phi _{T}$ by
\[ \Phi_{T} = \sum_{r\in{{\mathbb{Z}}}/M{{\mathbb{Z}}}} r\cdot\Phi_{0,T}, \]
where 
$M=M_p(T)$ is 
$p$ if 
$T = {{\mathrm I}}{{\mathrm I}}{{\mathrm I}}, {{\mathrm I}}{{\mathrm V}}$, 
$p^{n}$ if 
$T = {{\mathrm I}}_{2n}$, and 
$p^{n+1}$ if 
$T = {{\mathrm I}}_{2n +1}$. The next lemma, determining the Fourier transforms of the sets 
$\Phi _{0,T}$, follows quickly from the definitions.
Lemma 3.3 Let 
$p\geq 5$ be a prime number. Let 
$T$ be one of the three Kodaira symbols, and let 
$N=N_p(T)$ denote the appropriate power of 
$p$. For 
$\chi =(\check {a},\check {b},\check {c})\in \widehat {U({{\mathbb {Z}}}/N{{\mathbb {Z}}})}$, we have
\[\begin{array}{@{}ll}
|\widehat{\Phi_{0,{{\mathrm I}}{{\mathrm I}}{{\mathrm
I}}}}(\chi)|= \left\{ \begin{array}{@{}ll} p^{2} & \mbox{if }
p\mid\check{a}, p\mid\check{b},\\ 0 &
\mbox{otherwise;} \end{array} \right.
& |\widehat{\Phi_{0,{{\mathrm I}}{{\mathrm V}}}}(\chi)|=
\left\{ \begin{array}{@{}ll} p & \mbox{if }
p\mid\check{a},\\ 0 &
\mbox{otherwise;}
\end{array}\right.\\ |\widehat{\Phi_{0,{{\mathrm
I}}_{2n}}}(\chi)|= \left\{ \begin{array}{@{}ll} p^{3n} &
\mbox{if } p^{2n}\mid\check{a},
p^{n}\mid\check{b},\\ 0 &
\mbox{otherwise;} \end{array} \right.
& |\widehat{\Phi_{0,{{\mathrm I}}_{2n+1}}}(\chi)|= \left\{
\begin{array}{@{}ll} p^{3n+1} & \mbox{if }
p^{2n+1}\mid\check{a}, p^{n}\mid\check{b},\\ 0 &
\mbox{otherwise.} \end{array} \right.
\end{array}\]As an immediate consequence, (6) yields the inequality
\begin{equation}
|\widehat{\Phi_T}(\chi)|\leq p^{k_T}r_T(\chi),
\end{equation}
where 
$k_T$ is 
$2$, 
$1$, 
$3n$, or 
$3n+1$ depending on whether 
$T$ is 
${{\mathrm I}}{{\mathrm I}}{{\mathrm I}}$, 
${{\mathrm I}}{{\mathrm V}}$, 
${{\mathrm I}}_{2n}$, or 
$I_{2n+1}$, respectively, and 
$r_T(\chi )$ is the number of 
$r\in \{0,\ldots ,M-1\}$ such that 
$(r\cdot \chi )$ belongs to the support of 
$\widehat {\Phi _{0,T}}$. To bound 
$\widehat {\Phi _T}(\chi )$, it then remains to bound 
$r_T(\chi )$.
Lemma 3.4 We have the following assertions.
1. Let
$T={{\mathrm I}}{{\mathrm I}}{{\mathrm I}}$. Then $r_T(\chi )=0$ unless $p\mid \Delta _2(\chi )$. In that case, $r_T(\chi )=1$ if $p\nmid \chi$ and $r_T(\chi )=p$ otherwise.2. Let
$T={{\mathrm I}}{{\mathrm V}}$. Then $r_T(\chi )\leq 2$ if $p\nmid \chi$ and $r_T(\chi )=p$ otherwise.3. Let
$T={{\mathrm I}}_{2n}$. Then $r_T(\chi )=0$ unless $\chi$ is ${{\mathbb {G}}}_a$-equivalent to some element $(0,p^{n+i}\check {b},p^{j}\check {c})$, for $i$ and $j$ nonnegative integers and $p\nmid {{\check {b}}}{{\check {c}}}$. Then $r_T(\chi )\ll p^{{{\rm min}}(i,\lfloor j/2\rfloor )}$.4. Let
$T={{\mathrm I}}_{2n+1}$. Then $r_T(\chi )=0$ unless $\chi$ is ${{\mathbb {G}}}_a$-equivalent to some element $(0,p^{n+i}\check {b},p^{j}\check {c})$, for $i$ and $j$ nonnegative integers and $p\nmid {{\check {b}}}{{\check {c}}}$. Then $r_T(\chi )\ll p^{{{\rm min}}(i,\lceil j/2\rceil )}$.
Proof. We prove the above lemma in the case where 
$T={{\mathrm I}}_{2n}$. Assume that 
$\chi$ is 
${{\mathbb {G}}}_a$-equivalent to 
$(0,p^{n+i}{{\check {b}}},p^{j}{{\check {c}}})$, for 
$i$ and 
$j$ nonnegative integers and 
$p\nmid {{\check {b}}}{{\check {c}}}$. Note that the entry 
$p^{j}{{\check {c}}}$ does not change under the 
${{\mathbb {G}}}_a$-action. Then, by definition, we have
\[ r_T(\chi)=\#\{r\in{{\mathbb{Z}}}/p^{n}{{\mathbb{Z}}}:p^{n}\mid rp^{j}, p^{2n}\mid 2p^{n+i}r{{\check{b}}}+r^{2}p^{j}{{\check{c}}}\}. \]
Write 
$r\in {{\mathbb {Z}}}/p^{n}{{\mathbb {Z}}}$ as 
$r=sp^{k}+p^{n}{{\mathbb {Z}}}$ with 
$p\nmid s$. Then the condition on 
$r$ translates to
\[ p^{n}\mid p^{j+k},\quad p^{2n}\mid (2{{\check{b}}}p^{n+i+k}+s{{\check{c}}}p^{j+2k}). \]
We consider two possible cases. First assume that 
$p^{2n}$ divides both 
$2{{\check {b}}}p^{n+i+k}$ and 
$s{{\check {c}}}p^{j+2k}$. Then we have 
$k\geq {{\rm max}}(n-i, n-\lfloor j/2\rfloor )$, which implies that there are 
$p^{{{\rm min}}(i,\lfloor j/2\rfloor )}$ choices for 
$r$. Otherwise, we have 
$n+i+k=j+2k=:\ell <2n$, and 
$p^{2n-\ell }\mid 2{{\check {b}}}+s{{\check {c}}}$. In this case, 
$s$ is determined modulo 
$p^{2n-\ell }$, which implies that there are 
$p^{n-k-(2n-\ell )}=p^{\ell -n-k}$ choices for 
$r$. Note that 
$\ell -n-k=i$. Furthermore, we have 
$j\!+2k=\!\ell <2n$, from which it follows that 
$2(\ell \!-n-\!k)=2j+2k-2n<j$. This proves the lemma in the case where 
$T={{\mathrm I}}_{2n}$. The other three cases are similar, and we omit the proof.
3.3 Proof of Theorem 3.1
Let 
$\Sigma$ be as before, that is, a finite set consisting of primes 
$p\geq 5$ and a Kodaira symbol 
$T_p={{\mathrm I}}{{\mathrm I}}{{\mathrm I}}$, 
${{\mathrm I}}{{\mathrm V}}$, or 
${{\mathrm I}}_{n\geq 2}$ for each prime 
$p$ in this set. For each prime 
$p$ of 
$\Sigma$, set 
$N_p:=N_p(T_p)$ and set 
$m_{{{\rm odd}},p}$ to be 
$p$ if 
$T_p={{\mathrm I}}_{2n+1}$ and 
$1$ otherwise. Set 
$Q_p$ to be the 
$Q$-invariant associated to 
$T_p$ in Table 1. We define the quantities 
$N=N(\Sigma )$, 
$Q=Q(\Sigma )$, and 
$m_{{\rm odd}}=m_{{\rm odd}}(\Sigma )$ to be the product over all primes 
$p$ of 
$\Sigma$ of 
$N_p$, 
$Q_p$, and 
$m_p$, respectively. Note that 
$N = Q^{2}m_{{{\rm odd}}}$. Since 
$Q^{2}$ divides 
$\Delta (f)$ for any 
$f$ with splitting type 
$\Sigma$, we may assume that 
$Q$ and also 
$N$ are bounded above by some fixed power of 
$Y$.
Then elements with splitting type 
$\Sigma$ are defined via congruence conditions modulo 
$N$. Let 
$\phi :U({{\mathbb {Z}}}/N{{\mathbb {Z}}})\to {{\mathbb {R}}}$ denote the characteristic function of elements with splitting type 
$\Sigma$. Let 
$\psi :U({{\mathbb {R}}})\to {{\mathbb {R}}}_{\geq 0}$ be a smooth compactly supported function such that 
$\psi (f)=1$ for 
$H(f)\leq 1$. We have
\begin{align} \#\{f\in U({{\mathbb{Z}}})_\Sigma:H(f) < Y\}&\leq \sum_{(a,b,c)\in U({{\mathbb{Z}}})_\Sigma}\psi \biggl (\frac{a}{Y^{1/6}},\frac{b}{Y^{1/3}},\frac{c}{Y^{1/2}} \biggr) \phi(a,b,c)\nonumber\\ &= \frac{Y}{N^{3}} \sum_{\chi=(\check{a},\check{b},\check{c})\in \widehat{U({{\mathbb{Z}}})}}\hat{\psi} \biggl (\frac{Y^{1/6}{{\check{a}}}}{N},\frac{Y^{1/3}{{\check{b}}}}{N},\frac{Y^{1/2}{{\check{c}}}}{N} \biggr) \hat\phi({{\check{a}}},{{\check{b}}},{{\check{c}}})\nonumber\\ &= S_0+S_{{{\check{a}}}{{\check{c}}}=0}+S_{\Delta_2=0}+S_{{\neq} 0}, \end{align}
where 
$S_0$ is the contribution of the term 
$\chi =0$, 
$S_{{{\check {a}}}{{\check {c}}}=0}$ is the contribution from the nonzero terms 
$\chi$ with 
${{\check {a}}}{{\check {c}}}=0$, 
$S_{\Delta _2=0}$ is the contribution from nonzero terms 
$\chi$ with 
$\Delta _2(\chi )=0$, and 
$S_{\neq 0}$ is the contribution from the terms 
$\chi$ with 
${{\check {a}}}{{\check {c}}}\Delta _2(\chi )\neq 0$. We bound each of these quantities in turn.
To begin with, since 
$\hat {\phi }(0)/N^{3}=\nu (\Sigma )$ and 
$\psi$ is compactly supported, we have
\begin{equation} S_0=\frac{Y}{N^{3}}\hat{\psi}(0)\hat{\phi}(0)\ll \nu(\Sigma)Y \ll \frac{Y}{Q^{2}m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\mathrm I}}{{\mathrm V}}}^{2}m_{{\rm odd}}}, \end{equation}
by Table 1. To bound 
$S_{{{\check {a}}}{{\check {c}}}=0},$ 
$S_{\Delta _2=0}$ and 
$S_{\neq 0}$, we have the following immediate consequence of (7) and Lemma 3.4.
Corollary 3.5 With notation as above, let 
$\chi = ({{\check {a}}},{{\check {b}}},{{\check {c}}})\in \widehat {U({{\mathbb {Z}}})}$ with 
$\hat {\phi }(\chi )\neq 0$. Let 
$A$ be the largest divisor of 
$m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\mathrm I}}{{\mathrm V}}}$ dividing 
${{\check {a}}}$, 
${{\check {b}}}$ and 
${{\check {c}}}$. For each prime 
$p$ with 
$T_p = I_{2n}$ or 
$T_p = I_{2n+1}$ for some 
$n\geq 1$, let 
$k_p$ be the nonnegative integer with 
$p^{2n+k_p}\mid \mid \Delta _2(\chi )$. Then
\begin{equation} \hat{\phi}(\chi) \ll A m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}^{2} m_{{{\mathrm I}}{{\mathrm V}}} \prod_{T_p = I_{2n}} p^{3n+k_p/2} \prod_{T_p = I_{2n+1}} p^{3n+1+k_p/2}. \end{equation} Since 
$\hat \psi$ decays faster than any polynomial, it suffices to consider characters 
$\chi = ({{\check {a}}},{{\check {b}}},{{\check {c}}})$ such that
\[ {{\check{a}}}\ll N^{1+\epsilon}/Y^{1/6}, \quad {{\check{b}}}\ll N^{1+\epsilon}/Y^{1/3}, \quad {{\check{c}}}\ll N^{1+\epsilon}/Y^{1/2}. \]
We consider 
$S_{{{\check {a}}}{{\check {c}}}=0}$ first. Fix a divisor 
$A$ of 
$m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\mathrm I}}{{\mathrm V}}}$ and a nonnegative integer 
$k_p$ for every prime 
$p$ with 
$T_p = I_{\geq 2}$. The number of characters 
$\chi = (0,{{\check {b}}},{{\check {c}}})$ such that 
$A$ is the largest divisor of 
$m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\mathrm I}}{{\mathrm V}}}$ dividing 
$\chi$ and 
$m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\mathrm I}}{{\mathrm V}}}\mid \Delta _2(\chi )$ and 
$p^{2n+k_p}\mid \mid \Delta _2(\chi )$ for every prime 
$p$ with 
$T_p = I_{2n}$ or 
$T_p = T_{2n+1}$ is
\[ \ll_\epsilon \frac{N^{1+\epsilon}}{m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\mathrm I}}{{\mathrm V}}}Y^{1/3}} \frac{N^{1+\epsilon}}{AY^{1/2}} \prod_{T_p = I_{2n}\,or\,I_{2n+1}} p^{{-}n-k_p/2}. \]
The number of choices for 
$A$ and the 
$k_p$ is 
$\ll Y^{\epsilon }$. Combining with the bound (10), we have
\[
\frac{Y}{N^{3}}\sum_{\substack{{{\check{b}}}\ll N^{1+\epsilon}/Y^{1/3}\\
{{\check{c}}}\ll N^{1+\epsilon}/Y^{1/2}} }\hat{\phi}(0,{{\check{b}}},{{\check{c}}}) \ll_\epsilon
\frac{Y^{1/6+\epsilon}}{N}m_{{{\mathrm I}}{{\mathrm
I}}{{\mathrm I}}}\prod_{T_p = I_{2n}} p^{2n}\prod_{T_p =
I_{2n+1}} p^{2n+1} = \frac{Y^{1/6+\epsilon}}{m_{{{\mathrm
I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\mathrm I}}{{\mathrm
V}}}^{2}}. \]
To bound the sum of 
$\hat {\phi }({{\check {a}}},{{\check {b}}},0)$, we need a slight refinement. Fix again a divisor 
$A$ of 
$m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\mathrm I}}{{\mathrm V}}}$ and a nonnegative integer 
$\ell _p$ for every prime 
$p$ with 
$T_p = I_{\geq 2}$. Suppose 
$\chi = ({{\check {a}}},{{\check {b}}},0)$ with 
$p^{n+\ell _p}\mid \mid {{\check {b}}}$ for every prime 
$p$ with 
$T_p = I_{2n}$ or 
$T_p = T_{2n+1}$. In order for 
$\widehat {\Phi _{T_p}}(\chi )\neq 0$ at these primes 
$p$, we need also 
$p^{n+\ell _p}\mid {{\check {a}}}$ by Lemma 3.4. If we further require that 
$A$ is the largest divisor of 
$m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\mathrm I}}{{\mathrm V}}}$ dividing 
$\chi$ and 
$m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\mathrm I}}{{\mathrm V}}}\mid \Delta _2(\chi )$, then the number of such 
$\chi$ is
\[ \ll_\epsilon \frac{N^{1+\epsilon}}{AY^{1/6}}\frac{N^{1+\epsilon}}{m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\mathrm I}}{{\mathrm V}}}Y^{1/3}} \prod_{T_p = I_{2n}\,or\,I_{2n+1}} p^{{-}2n-2\ell_p}, \]
and for any such 
$\chi$, we have
\[ \hat{\phi}(\chi) \ll A m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}^{2} m_{{{\mathrm I}}{{\mathrm V}}} \prod_{T_p = I_{2n}} p^{3n+\ell_p} \prod_{T_p = I_{2n+1}} p^{3n+1+\ell_p}. \]Combining these two bounds gives
\[ \frac{Y}{N^{3}} \sum_{\substack{{{\check{a}}}\ll N^{1+\epsilon}/Y^{1/6}\\ {{\check{b}}}\ll N^{1+\epsilon}/Y^{1/3}}} \hat{\phi}({{\check{a}}},{{\check{b}}},0) \ll_\epsilon \frac{Y^{1/2+\epsilon}}{N}m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}\prod_{T_p = I_{2n}} p^{n}\prod_{T_p = I_{2n+1}} p^{n+1} = \frac{Y^{1/2+\epsilon}}{Qm_{{{\mathrm I}}{{\mathrm V}}}}. \]Hence, we have
\begin{equation} S_{{{\check{a}}}{{\check{c}}}=0} \ll_\epsilon \frac{Y^{1/6+\epsilon}}{m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\mathrm I}}{{\mathrm V}}}^{2}} + \frac{Y^{1/2+\epsilon}}{Qm_{{{\mathrm I}}{{\mathrm V}}}}. \end{equation} Next, we consider 
$S_{\Delta _2=0}$. Let 
$\chi =(\check {a},\check {b},\check {c})\in \widehat {U({{\mathbb {Z}}})}$ with 
$\Delta _2(\chi )=0$. Fix a divisor 
$A$ of 
$m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\mathrm I}}{{\mathrm V}}}$ and nonnegative integers 
$\ell _p$ for each prime 
$p$ with 
$T_p=I_{\geq 2}$. Suppose 
$A$ is the largest divisor of 
$m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\mathrm I}}{{\mathrm V}}}$ dividing 
$\chi$ and 
$p^{\ell _p}\mid \mid {{\check {c}}}$ for all 
$p$ with 
$T_p = I_{\geq 2}$. Then, similar to the case of 
$\hat {\phi }({{\check {a}}},{{\check {b}}},0)$, we also need 
$p^{\ell _p}\mid {{\check {a}}}$ in order that 
$\widehat {\Phi _{T_p}}(\chi )\neq 0$. As 
${{\check {b}}}^{2} = {{\check {a}}} {{\check {c}}}$, it follows that 
$p^{\ell _p}\mid {{\check {b}}}$, in which case
\[ \hat{\phi}(\chi) \ll A m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}^{2} m_{{{\mathrm I}}{{\mathrm V}}} \prod_{T_p=I_{2n}} p^{3n+\lceil {\ell_p}/{2}\rceil} \prod_{T_p = T_{2n+1}}p^{3n+1+\lceil{\ell_p}/{2}\rceil}. \]
Once we fix 
${{\check {b}}}$, the number of choices for pairs 
$({{\check {a}}},{{\check {c}}})$ such that 
${{\check {b}}}^{2} = {{\check {a}}}{{\check {c}}}$ is 
$\ll N^{\epsilon }$. Therefore the number of such characters 
$\chi$ is
\[ \ll_\epsilon \frac{N^{1+\epsilon}}{AY^{1/3}}\prod_{T_p = I_{2n}\,or\,I_{2n+1}} p^{-\ell_p}. \]Combining these two bounds gives
\begin{equation} S_{\Delta_2=0}\ll_\epsilon \frac{Y^{2/3+\epsilon}}{N^{2}}m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}^{2} m_{{{\mathrm I}}{{\mathrm V}}}\prod_{T_p=I_{2n}} p^{3n} \prod_{T_p = T_{2n+1}}p^{3n+1}=\frac{Y^{2/3+\epsilon}}{Qm_{{\rm odd}}m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\mathrm I}}{{\mathrm V}}}^{2}}. \end{equation} Finally, we turn to 
$S_{\neq 0}$. Once again, we fix a divisor 
$A$ of 
$m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\mathrm I}}{{\mathrm V}}}$ and a nonnegative integer 
$k_p$ for each prime 
$p$ with 
$T_p = I_{\geq 2}$. The number of characters 
$\chi =({{\check {a}}},{{\check {b}}},{{\check {c}}})$ such that 
$A$ is the largest divisor of 
$m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\mathrm I}}{{\mathrm V}}}$ dividing 
$\chi$, 
$m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\mathrm I}}{{\mathrm V}}}\mid \Delta _2(\chi )$, and 
$p^{2n+k_p}\mid \mid \Delta _2(\chi )$ for any prime 
$p$ with 
$T_p = I_{2n}$ or 
$T_p = T_{2n+1}$ is
\[ \ll_\epsilon Y^{\epsilon} \frac{N^{1+\epsilon}}{AY^{1/3}} \frac{N^{2+\epsilon}}{m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\mathrm I}}{{\mathrm V}}}Y^{2/3}}\prod_{T_p = I_{2n} \text{ or }I_{2n+1}}p^{{-}2n-k_p}. \]
Indeed, the above bounds the number of pairs 
$({{\check {b}}},\Delta _2(\chi ))$ satisfying the desired divisibility conditions, and given 
${{\check {b}}}$ and 
$\Delta _2(\chi )$, there are 
$Y^{\epsilon }$ choices for 
${{\check {a}}}$ and 
${{\check {c}}}$. Combining with (10) then gives
\begin{equation} S_{{\neq} 0} \ll_\epsilon Y^{\epsilon} m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}\prod_{T_p=I_{2n}}p^{n}\prod_{T_p=I_{2n+1}}p^{n+1} = \frac{Qm_{{\rm odd}}}{m_{{{\mathrm I}}{{\mathrm V}}}}Y^{\epsilon}. \end{equation}Theorem 3.1 now follows from (8), (9), (11)–(13), and the inequality of arithmetic and geometric means.
4. The family of cubic fields with prescribed shapes
A cubic ring is a commutative ring with unit that is free of rank 3 as a 
${{\mathbb {Z}}}$-module. Given a cubic ring 
$R$, the trace 
${{\rm Tr}}(\alpha )$ of an element 
$\alpha \in R$ is the trace of the linear map 
$\times \alpha :R\to R$. The discriminant 
${{\rm Disc}}(R)$ of 
$R$ is then the determinant of the bilinear pairing
\[ R\times R\to{{\mathbb{Z}}},\quad (\alpha,\beta):={{\rm Tr}}(\alpha\beta). \]
Given a nondegenerate cubic ring 
$R$, that is, a cubic ring 
$R$ with nonzero discriminant, we then consider the cubic étale algebras 
$R\otimes {{\mathbb {Q}}}$ over 
${{\mathbb {Q}}}$ and 
$R\otimes {{\mathbb {R}}}$ over 
${{\mathbb {R}}}$. There are two possibilities for 
$R\otimes {{\mathbb {R}}}$, namely, 
${{\mathbb {R}}}^{3}$ and 
${{\mathbb {R}}}\oplus {{\mathbb {C}}}$. We have 
$R\otimes {{\mathbb {R}}}\cong {{\mathbb {R}}}^{3}$ when 
${{\rm Disc}}(R)>0$ (equivalently, when the signature of 
$R\otimes {{\mathbb {Q}}}$ is 
$(3,0)$) and 
$R\otimes {{\mathbb {R}}}\cong {{\mathbb {R}}}\oplus {{\mathbb {C}}}$ when 
${{\rm Disc}}(R)<0$ (equivalently, when the signature of 
$R\otimes {{\mathbb {Q}}}$ is 
$(1,2)$).
The ring 
$R$ embeds as a lattice into 
$R\otimes {{\mathbb {R}}}$ with covolume 
$\sqrt {{{\rm Disc}}(R)}$. As regarded as this lattice, the element 
$1\in R$ is part of any Minkowski basis, and so the first of the successive minima of 
$R$ is simply 
$1$. Let 
$\ell _1(R)\leq \ell _2(R)$ denote the other two successive minima of 
$R$. We define the skewness of 
$R$ by
\[ {{\rm sk}}(R):=\ell_2(R)/\ell_1(R). \] Given a field 
$K$, we denote the ring of integers of 
$K$ by 
${{\mathcal {O}}}_K$, and the class group of 
$K$ by 
${{\rm Cl}}(K)$. For positive real numbers 
$X$ and 
$Z$, let 
${{\mathcal {R}}}_3^{\pm }(X,Z)$ denote the set of cubic fields 
$K$ that satisfy the following two bounds: 
$X\leq \pm {{\rm Disc}}({{\mathcal {O}}}_K)<2X$ and 
${{\rm sk}}({{\mathcal {O}}}_K)>Z$. Set 
${{\mathcal {R}}}_3(X,Z)$ to be the union 
${{\mathcal {R}}}_3^{+}(X,Z)\cup {{\mathcal {R}}}_3^{-}(X,Z)$. In this section, we prove the following result.
Theorem 4.1 Let 
$X$ and 
$Z$ be positive real numbers. Then
\[ \sum_{K\in{{\mathcal{R}}}_3(X,Z)}|{{\rm Cl}}(K)[2]| \ll X/Z,\]
where the implied constants are independent of 
$X$ and 
$Z$.
This section is organized as follows. In § 4.1 we recall the parametrization of cubic rings and of 
$2$-torsion elements in the class groups of cubic rings, in terms of integral orbits for the action of 
${{\rm GL}}_2({{\mathbb {Z}}})$ on 
${{\rm Sym}}^{3}({{\mathbb {Z}}}^{2})$ and of 
${{\rm GL}}_2({{\mathbb {Z}}})\times {{\rm SL}}_3({{\mathbb {Z}}})$ on 
${{\mathbb {Z}}}^{2}\otimes {{\rm Sym}}^{2}({{\mathbb {Z}}}^{3})$, respectively. In §§ 4.2 and 4.3, we then prove Theorem 4.1 using these parametrizations in conjunction with geometry-of-numbers methods.
4.1 The parametrization of cubic rings and the $2$-torsion in their class groups
In this section, we recall two parametrizations: first, the parametrization of cubic rings, due to Levi [Reference LeviLev14], Delone and Faddeev [Reference Delone and FaddeevDF40] and Gan, Gross, and Savin [Reference Gan, Gross and SavinGGS02]; and second, Bhargava's parametrization [Reference BhargavaBha04] of elements in the 
$2$-torsion subgroups of cubic rings. Let 
$V_3={{\rm Sym}}^{3}(2)$ denote the space of binary cubic forms. We consider the twisted action of 
${{\rm GL}}_2$ on 
$V_3$ given by
\[ (\gamma\cdot f)(x,y):=\frac{1}{\det\gamma}f((x,y)\cdot\gamma), \]
for 
$\gamma \in {{\rm GL}}_2$ and 
$f(x,y)\in V_3$. Then we have the following result.
Theorem 4.2 [Reference LeviLev14, Reference Delone and FaddeevDF40, Reference Gan, Gross and SavinGGS02]
There is a natural bijection between the set of 
${{\rm GL}}_2({{\mathbb {Z}}})$-orbits on 
$V_3({{\mathbb {Z}}})$ and the set of cubic rings.
We collect some well-known facts about the above bijection (for proofs and a more detailed discussion, see [Reference Bhargava, Shankar and TsimermanBST13, § 2]). For an integral binary cubic form 
$f$, we denote the corresponding cubic ring by 
$R_f$. The bijection is discriminant preserving, that is, we have 
$\Delta (f)={{\rm Disc}}(R_f)$. The ring 
$R_f$ is an integral domain if and only if 
$f$ is irreducible over 
${{\mathbb {Q}}}$. The group of automorphisms of 
$R_f$ is isomorphic to the stabilizer of 
$f$ in 
${{\rm GL}}_2({{\mathbb {Z}}})$.
The bijection of Theorem 4.2 can be explicitly described as follows. Given a cubic ring 
$R$, consider the map 
$R/{{\mathbb {Z}}}\to \wedge ^{2}(R/{{\mathbb {Z}}})\cong {{\mathbb {Z}}}$ given by 
$r\mapsto r\wedge r^{2}$. This map is easily seen to be a cubic map and gives the binary cubic form corresponding to 
$R$. In fact, this map yields the finer bijection
\begin{equation} V_3({{\mathbb{Z}}}) \longleftrightarrow \{(R,\omega,\theta)\}, \end{equation}
where 
$R$ is a cubic ring and 
$\langle \omega ,\theta \rangle$ is a basis for the two-dimensional 
${{\mathbb {Z}}}$-module 
$R/{{\mathbb {Z}}}$. More explicitly, if 
$f(x,y) = ax^{3} + bx^{2}y + cxy^{2} + {{\rm d}\kern 0.05em y}^{3}$ with 
$a\neq 0$, then 
$R$ is the cubic subring of 
${{\mathbb {Q}}}[x]/(f(x,1))$ with 
${{\mathbb {Z}}}$-basis 
$\{1,\omega ,\theta \}$ where 
$\omega = ax$ and 
$\theta = ax^{2} + bx + c$. Conversely, the integral binary cubic form corresponding to 
$(R,\omega ,\theta )$ is 
$f(x,y)$, where
\begin{equation} (x\omega+y\theta)\wedge(x\omega+y\theta)^{2}=f(x,y)(\omega\wedge\theta). \end{equation}
It is easily seen that the actions of 
${{\rm GL}}_2({{\mathbb {Z}}})$ on 
$V_3({{\mathbb {Z}}})$ and on the set of triples 
$(R,\omega ,\theta )$ agree. Here the latter action is given simply by the natural action of 
${{\rm GL}}_2({{\mathbb {Z}}})$ on the basis 
$\{\omega ,\theta \}$ of 
$R/{{\mathbb {Z}}}$.
Let 
$f$ be an integral binary cubic form, and let 
$(R,\omega ,\theta )$ be the corresponding triple. Fix an element 
$\alpha =n+a\omega +b\theta$ of 
$R$, where 
$n$, 
$a$, and 
$b$ are integers and 
$(a,b)\neq (0,0)$. The ring 
${{\mathbb {Z}}}[\alpha ]$ is a subring of 
$R$ having finite index denoted 
${{\rm ind}}(\alpha )$. It follows from (15) that we have
\begin{equation} {{\rm ind}}(\alpha)=f(a,b). \end{equation}
Clearly 
${{\rm ind}}(\alpha )={{\rm ind}}(\alpha +n)$ for 
$n\in {{\mathbb {Z}}}$. Finally, we note that the bijections of Theorem 4.2 and (15) continue to hold if 
${{\mathbb {Z}}}$ is replaced by any principal ideal domain [Reference Bhargava, Shankar and WangBSW15, Theorem 5].
Next, we describe the parametrization of 
$2$-torsion ideals in the class groups of cubic rings. Let 
$W$ denote the space 
$2\otimes {{\rm Sym}}^{2}(3)$ of pairs of ternary quadratic forms. For a ring 
$S$, we write elements 
$(A,B)\in W(S)$ as a pair of 
$3\times 3$ symmetric matrices with coefficients in 
$S$. The group 
$G_{2,3}={{\rm GL}}_2\times {{\rm SL}}_3$ acts on 
$W$ via the action 
$(\gamma _2,\gamma _3)\cdot (A,B):=(\gamma _3A\gamma _3^{t},\gamma _3B\gamma _3^{t})\gamma _2^{t}$. We have the resolvent map from 
$W$ to 
$V_3$ given by
\begin{align*} W&\to V_3\\ (A,B)&\mapsto \det(Ax+By). \end{align*}
The resolvent map respects the group actions on 
$W$ and 
$V_3$: we have
\begin{equation} {{\rm Res}}((\gamma_2,\gamma_3)\cdot (A,B))=(\det\gamma_2)(\gamma_2\cdot{{\rm Res}}(A,B)). \end{equation} The following result parametrizing 
$2$-torsion ideals in cubic rings is due to Bhargava [Reference BhargavaBha04, Theorem 4].
Theorem 4.3 [Reference BhargavaBha04]
There is a bijection between 
${{\rm GL}}_2({{\mathbb {Z}}})\times {{\rm SL}}_3({{\mathbb {Z}}})$-orbits on 
$W({{\mathbb {Z}}})$ and equivalence classes of triples 
$(R,I,\delta )$, where 
$R$ is a cubic ring, 
$I\subset R$ is an ideal of 
$R$ having rank 
$3$ as a 
${{\mathbb {Z}}}$-module, and 
$\delta$ is an invertible element of 
$R\otimes {{\mathbb {Q}}}$ such that 
$I^{2}\subset (\delta )$ and 
$N(I)^{2}=N(\delta )$. Here two triples 
$(R,I,\delta )$ and 
$(R',I',\delta ')$ are equivalent if there exist an isomorphism 
$\phi :R\to R'$ and an element 
$\kappa \in R\otimes {{\mathbb {Q}}}$ such that 
$I'=\phi (\kappa I)$ and 
$\delta '=\phi (\kappa ^{2}\delta )$. Moreover, the ring 
$R$ of the triple corresponding to a pair 
$(A,B)$ is the cubic ring corresponding to 
${{\rm Res}}(A,B)$ under the Delone–Faddeev parametrization.
When 
$R=R_f$ is the maximal order in a cubic field 
$K$, the above result gives a bijection between the set of 
${{\rm GL}}_2({{\mathbb {Z}}})\times {{\rm SL}}_3({{\mathbb {Z}}})$-orbits on the set of pairs 
$(A,B)\in W({{\mathbb {Z}}})$ with resolvent 
$f$, and the set of equivalence classes of pairs 
$(I,\delta )$, where 
$I$ is an ideal of 
$R$, 
$\delta \in K$, and 
$I^{2}=(\delta )$. The latter set is termed the 
$2$-Selmer group
\[ 1\to R^{{\times}}/(R^{{\times}})^{2}\to{{\rm Sel}}_2(K)\to{{\rm Cl}}(K)[2]\to 1, \]
where 
$R^{\times }$ denotes the unit group of 
$K$.
We will use the two parametrizations above to count cubic fields with bounded discriminants and skewed rings of integers in § 4.2 and then to count their 
$2$-Selmer groups in § 4.3 from which Theorem 4.1 follows.
4.2 The number of cubic fields with bounded discriminants and skewed rings of integers
The goal of this subsection is to prove the following result.
Proposition 4.4 Let 
$X$ and 
$Z$ be positive real numbers. There exists some constant 
$C$ such that 
${{\mathcal {R}}}^{\pm }_3(X,Z)$ is empty if 
$Z>CX^{1/6}$. Otherwise 
$|{{\mathcal {R}}}^{\pm }_3(X,Z)|=O(X/Z)$.
For any subset 
$S$ of 
$V_3({{\mathbb {R}}})$, let 
$S^{\pm }$ denote the set of elements 
$f$ such that 
$\pm \Delta (f)>0$. Then 
$V_3({{\mathbb {R}}})^{+}$ (respectively, 
$V_3({{\mathbb {R}}})^{-}$) consists of a single 
${{\rm GL}}_2({{\mathbb {R}}})$-orbit and corresponds to the cubic algebra 
${{\mathbb {R}}}^{3}$ (respectively, 
${{\mathbb {R}}}\oplus {{\mathbb {C}}}$). We denote this cubic 
${{\mathbb {R}}}$-algebra by 
$R^{\pm }$. Let 
${{\mathcal {F}}}_2$ denote Gauss’s fundamental domain for the action of 
${{\rm GL}}_2({{\mathbb {Z}}})$ on 
${{\rm GL}}_2({{\mathbb {R}}})$. We write elements of 
${{\rm GL}}_2({{\mathbb {R}}})$ in Iwasawa coordinates, in which case we have
\[ {{\mathcal{F}}}_2=\{n\alpha k\lambda:n\in N'(t),\alpha(t)\in A',k\in K,\lambda\in\Lambda\}, \]where
\begin{gather} \begin{array}{c}N'(t)=
\left\{\left(\begin{array}{@{}cc@{}} 1 & {} \\
{u} & 1
\end{array}\right): u\in\nu(t)
\right\},\quad A' = \left\{\left(\begin{array}{@{}cc@{}} t^{{-}1} & {} \\
{} & t \end{array}\right): t\geq \sqrt[4]3/\sqrt
2 \right\},\nonumber\\
\Lambda = \left\{\left(\begin{array}{@{}cc@{}} \lambda & {} \\ {}
& \lambda \end{array}\right): \lambda>0 \right\}, \end{array}
\end{gather}
and 
$K$ is as usual the (compact) real orthogonal group 
${\rm SO}_2({{\mathbb {R}}})$; here 
$\nu (t)$ is a union of one or two subintervals of 
$[-\frac 12,\frac 12]$ depending only on the value of 
$t$. Elements 
$n\alpha (t) k\lambda$ are expressed in their Iwasawa coordinates as 
$(n,t,\lambda ,k)$. Fix compact sets 
$B^{\pm }\subset V_3({{\mathbb {R}}})^{\pm }$ that are closures of open bounded sets. Then, for every point 
$v\in B^{\pm }$, the set 
${{\mathcal {F}}}_2\cdot v$, viewed as a multiset, is a cover of a fundamental domain for the action of 
${{\rm GL}}_2({{\mathbb {Z}}})$ on 
$V_3({{\mathbb {R}}})^{\pm }$ of absolutely bounded degree. Recall that for a cubic ring 
$R$, its skewness 
$\text {sk}(R)$ is defined to be the quotient 
$\ell _2(R)/\ell _1(R)$ where 
$1,\ell _1(R),\ell _2(R)$ are the successive minima of 
$R$, regarded as a lattice inside 
$R\otimes {{\mathbb {R}}}$. We have the following lemma.
Lemma 4.5 Let 
$v\in B^{\pm }$ be any binary cubic form. Let 
$\gamma =(n,t,\lambda ,k)\in {{\mathcal {F}}}_2$ be such that 
$f=\gamma \cdot v$ is an integral binary cubic form. Then we have
\[ {{\rm sk}}(R_f)\asymp t^{2}, \]
where 
$R_f$ denotes the cubic ring corresponding to 
$f$.
Proof. Every binary cubic form 
$v$ in 
$V_3({{\mathbb {R}}})^{\pm }$ gives rise to the cubic algebra 
$R^{\pm }$, where 
$R^{+}\cong {{\mathbb {R}}}^{3}$ and 
$R^{-}\cong {{\mathbb {C}}}\oplus {{\mathbb {R}}}$, along with elements 
$\alpha _v$ and 
$\beta _v$ such that 
$\langle 1,\alpha _v,\beta _v\rangle$ form a basis for 
$R^{\pm }$. Furthermore, the lattice spanned by 
$1$, 
$\alpha _v$, and 
$\beta _v$ has covolume 
$\sqrt {|\Delta (v)|}$. Since 
$B^{\pm }$ is compact, it follows that we have 
$|\alpha _v|\ll 1$ and 
$|\beta _v|\ll 1$ for 
$v\in B^{\pm }$. Additionally, the action of 
${{\rm GL}}_2({{\mathbb {R}}})$ on 
$V_3({{\mathbb {R}}})$ agrees with the action of 
${{\rm GL}}_2({{\mathbb {R}}})$ on pairs 
$(\alpha _v,\beta _v)$ by linear change of variables. That is, we have 
$(\alpha _{\gamma \cdot v},\beta _{\gamma \cdot v})=\gamma \cdot (\alpha _v,\beta _v)$.
Let 
$f=\gamma \cdot v$ be an integral binary cubic form as in the statement of the lemma. Since 
$\gamma \in {{\mathcal {F}}}_2$, it follows that 
$|\alpha _f|\asymp \lambda t^{-1}$ and 
$|\beta _f|\asymp \lambda t$. As a consequence, 
$|\alpha _f|\cdot |\beta _f|\asymp \sqrt {{{\rm Disc}}(f)}$. Hence the successive minima of 
$R_f$ are, up to a bounded multiplicative constant, 
$1$, 
$|\alpha _f|$, and 
$|\beta _f|$ (see, for example, [Reference Eskin and KatznelsonEK95, Lemma 2.2]). Therefore, we have 
$\ell _2(R_f)/\ell _1(R_f)\asymp |\beta _f|/|\alpha _f|\asymp t^{2}$ as necessary.
Next, we have the following lemma, due to Davenport [Reference DavenportDav51], that estimates the number of lattice points within regions of Euclidean space.
Proposition 4.6 [Reference DavenportDav51]
Let 
$\mathcal {R}$ be a bounded, semi-algebraic multiset in 
${{\mathbb {R}}}^{n}$ having maximum multiplicity 
$m$, and that is defined by at most 
$k$ polynomial inequalities each having degree at most 
$\ell$. Then the number of integral lattice points 
$($counted with multiplicity
$)$ contained in the region 
$\mathcal {R}$ is
\[ {{\rm Vol}}(\mathcal{R})+ O({{\rm max}}\{{{\rm Vol}}(\bar{\mathcal{R}}),1\}), \]
where 
${{\rm Vol}}(\bar {\mathcal {R}})$ denotes the greatest 
$d$-dimensional volume of any projection of 
$\mathcal {R}$ onto a coordinate subspace obtained by equating 
$n-d$ coordinates to zero, where 
$d$ takes all values from 
$1$ to 
$n-1$. The implied constant in the second summand depends only on 
$n$, 
$m$, 
$k$, and 
$\ell$.
We are now ready to prove Proposition 4.4.
Proof of Proposition 4.4 A general version of the first claim of the proposition, applying to number fields of all degrees, is obtained in [Reference Bhargava, Shankar, Taniguchi, Thorne, Tsimerman and ZhaoBST+17, Theorem 3.1], and further generalizations are proved in [Reference Chiche-LapierreChi19]. For completeness, we include a proof for our case below. Let 
$K$ be a cubic field whose ring of integers 
${{\mathcal {O}}}_K$ belongs to 
${{\mathcal {R}}}^{\pm }_3(X,Z)$, and let 
$\langle 1,\alpha ,\beta \rangle$ be a Minkowski basis for 
${{\mathcal {O}}}_K$ with 
$|\alpha |\leq |\beta |$. Consider the ring 
${{\mathbb {Z}}}[\alpha ]$ which is a suborder of 
${{\mathcal {O}}}_K$. We have
\[ X^{1/2}\asymp\sqrt{{{\rm Disc}}({{\mathcal{O}}}_K)}\ll\sqrt{{{\rm Disc}}({{\mathbb{Z}}}[\alpha])}\ll|\alpha|^{3}, \]
and it follows that 
$|\alpha |\gg X^{1/6}$. Since 
$|\alpha ||\beta |\asymp X^{1/2}$, we have 
$Z=|\beta |/|\alpha |\ll X^{1/2}/X^{2/6}=X^{1/6}$ and the first claim of the proposition follows.
We now estimate 
$|{{\mathcal {R}}}^{\pm }_3(X,Z)|$ under the assumption that 
$Z\ll X^{1/6}$ following the setup of [Reference Bhargava, Shankar and TsimermanBST13, § 5]. Let 
$v$ be an element of the compact set 
$B^{\pm }$. If 
$(n,t,\gamma ,k)\cdot v$ corresponds to a cubic ring 
$R$ with 
$X\leq {{\rm Disc}}(R)<2X$ and 
${{\rm sk}}(R)>Z$, then it follows that 
$\lambda \asymp X^{1/4}$ and 
$t\gg Z^{1/2}$, respectively, where the latter fact follows from Lemma 4.5. Hence we have
\begin{align*}
|{{\mathcal{R}}}^{{\pm}}_3(X,Z)|&\leq
\int_{\substack{g=(n,t,\lambda,k)\in{{\mathcal{F}}}_2\\\lambda\asymp
X^{1/4}\\t\gg Z^{1/2}}} \#\{g\cdot B^{{\pm}}\cap
V_3({{\mathbb{Z}}})^{{\rm irr}}\}\,dg\\ &\leq
\int_{\substack{g=(n,t,\lambda,k)\in{{\mathcal{F}}}_2\\
\lambda\asymp X^{1/4}\\Z^{1/2}\ll t\ll
X^{1/12}}} \#\{g\cdot B^{{\pm}}\cap
V_3({{\mathbb{Z}}})\}\,dg, \end{align*}
where the second inequality follows from the observation that if 
$t \gg X^{1/12}$, then every element 
$f(x,y)$ in 
$g\cdot B^{\pm }$ has 
$x^{3}$-coefficient less than 
$1$ in absolute value. Therefore no such integral element 
$f(x,y)$ can be irreducible since its 
$x^{3}$-coefficient must be 
$0$. Applying Proposition 4.6 on the set 
$g\cdot B^{\pm }$, we obtain
\begin{align*} |{{\mathcal{R}}}^{{\pm}}_3(X,Z)|&\ll \int_{\lambda\asymp X^{1/4}}\int_{Z^{1/2}\ll t\ll X^{1/12}} (\lambda^{4}+\lambda^{3}t^{3})t^{{-}2}\,d^{{\times}} t\,d^{{\times}}\lambda\\ &\ll \frac{X}{Z}+X^{5/6}\ll\frac{X}{Z}, \end{align*}
since 
$Z\ll X^{1/6}$. The proposition follows.
We end this subsection with a counting result on the number of primitive algebraic integers in a cubic field of bounded size to be used in § 6.1. We say an element 
$\alpha$ in a ring 
$R$ is primitive if 
$\alpha \neq n\beta$ for any 
$\beta \in R$ and any integer 
$n\geq 2$. We use the superscript 
${{\rm Tr}}=0$ to denote the subset of elements of trace 
$0$.
Lemma 4.7 Let 
$K$ be a cubic field with discriminant 
$D$. For any real number 
$Y>0$, let 
$N_K(Y)$ denote the number of primitive elements 
$\alpha \in {{\mathcal {O}}}_K^{{{\rm Tr}}=0}$ with 
$|\alpha | < Y$. Then
\begin{equation} N_K(Y)\leq\left\{
\begin{array}{@{}ll} 0 & \mbox{if }
Y<\ell_1(K),\\ 1 & \mbox{if }
\ell_1(K)\leq Y < \ell_2(K),\\
\displaystyle
\dfrac{Y^{2}}{\sqrt{D}}+O\bigg(\dfrac{Y}{\ell_1(K)}\bigg) &
\mbox{if } \ell_2(K)\leq Y. \end{array}\right.
\end{equation}
Note that if 
$\ell _2(K)\leq Y$, then 
${Y}/{\ell _1(K)}\ll {Y^{2}}/{\sqrt {D}}$ and so we simply have 
$N_K(Y)\ll {Y^{2}}/{\sqrt {D}}$, which is the best possible bound in this case.
Proof. The first two lines of (19) are clearly true. In fact, they are equalities. The final claim follows from Proposition 4.6 by replacing 
$N_K(Y)$ by the overcount where we count all (not merely primitive) traceless elements 
$\alpha \in {{\mathcal {O}}}_K$, since 
${{\mathcal {O}}}_K^{{{\rm Tr}}=0}$ considered as a lattice inside 
$(K\otimes {{\mathbb {R}}})^{{{\rm Tr}}=0}$ has covolume 
$\sqrt {D}$.
4.3 The 2-torsion subgroups in the class groups of cubic fields
Let 
$K$ be a cubic field, and let 
$f\in V_3({{\mathbb {Z}}})$ be the binary cubic form corresponding to 
${{\mathcal {O}}}_K$, the ring of integers of 
$K$. A consequence of Theorem 4.3 is that the set of 
$2$-torsion elements in the class group of 
$K$ injects into the set of 
${{\rm SL}}_3({{\mathbb {Z}}})$-orbits on the elements 
$(A,B)\in W({{\mathbb {Z}}})$ satisfying 
${{\rm Res}}(A,B)=f$.
Choose Iwasawa coordinates 
$(n,t,\lambda ,k_2)$ for 
${{\rm GL}}_2({{\mathbb {R}}})$ as in the previous subsection and 
$(u,s_1,s_2,k_3)$ for 
${{\rm SL}}_3({{\mathbb {R}}})$ as in [Reference BhargavaBha05, § 2.1]. A Haar measure for 
${{\rm SL}}_3({{\mathbb {R}}})$ in these coordinates is 
$s_1^{-6}s_2^{-6}dudk_3d^{\times } s_1d^{\times } s_2$. Let 
${{\mathcal {F}}}_3$ denote a fundamental domain for the action of 
${{\rm SL}}_3({{\mathbb {Z}}})$ on 
${{\rm SL}}_3({{\mathbb {R}}})$, such that 
${{\mathcal {F}}}_3$ is contained within a standard Seigel domain in 
${{\rm SL}}_3({{\mathbb {R}}})$. Then 
${{\mathcal {F}}}_{2,3}:={{\mathcal {F}}}_2\times {{\mathcal {F}}}_3$ is a fundamental domain for the action of 
$G_{2,3}({{\mathbb {Z}}})$ on 
$G_{2,3}({{\mathbb {R}}})$. There are four 
$G_{2,3}({{\mathbb {R}}})$-orbits having nonzero discriminant on 
$W({{\mathbb {R}}})$, and we denote them by 
$W({{\mathbb {R}}})^{(i)}$, 
$1\leq i\leq 4$. For each 
$i$, let 
${{\mathcal {B}}}_i\subset W({{\mathbb {R}}})^{(i)}$ be compact sets, which are closures of open sets, such that 
${{\rm Res}}({{\mathcal {B}}}_i)\subset B^{+}\cup B^{-}$, where 
$B^{+}$ and 
$B^{-}$ are as in the previous subsection. For each element 
$w\in {{\mathcal {B}}}_i$, the set 
${{\mathcal {F}}}_{2,3}\cdot w$ is a cover of a fundamental domain for the action of 
$G_{2,3}({{\mathbb {Z}}})$ on 
$W({{\mathbb {R}}})^{(i)}$. Let 
${{\mathcal {B}}}$ denote the union of the 
${{\mathcal {B}}}_i$.
Next, let 
$W({{\mathbb {Z}}})^{{\rm irr}}$ denote the set of elements 
$(A,B)\in W({{\mathbb {Z}}})$ such that the resolvent of 
$(A,B)$ corresponds to an integral domain, and such that 
$A$ and 
$B$ have no common root in 
${{\mathbb {P}}}^{2}({{\mathbb {Q}}})$. Elements in 
$W({{\mathbb {Z}}})$ that are not in 
$W({{\mathbb {Z}}})^{{\rm irr}}$ are said to be reducible. Given a reducible element 
$w$ with resolvent 
$f$, either 
$R_f$ is not an integral domain or 
$w$ corresponds to the identity element in the class group of 
$R_f$. We now have the following lemmas.
Lemma 4.8 Let 
$g=(g_2,g_3)$ be an element in 
${{\mathcal {F}}}_{2,3}$, where 
$g_2=(n,t,k_2,\lambda )\in {{\mathcal {F}}}_2$ and 
$g_3\in {{\mathcal {F}}}_3$. Let 
$(A,B)$ be an integral element in 
$g\cdot {{\mathcal {B}}}$ such that 
${{\rm Res}}(A,B)=f$. Then we have
\[ \Delta(f)\asymp \lambda^{12},\quad{{\rm sk}}(R_f)\asymp t^{2}. \]Proof. The lemma follows immediately from (17) in conjunction with Lemma 4.5 and the fact that 
$\Delta$ is a degree-
$4$ homogeneous polynomial in the coefficients of 
$V_3$.
Lemma 4.9 Let 
$(A,B)$ be an element in 
$W({{\mathbb {Z}}})$. Denote the coefficients of 
$A$ and 
$B$ by 
$a_{ij}$ and 
$b_{ij}$, respectively. If 
$\det (A)=0$ or 
$a_{11}=b_{11}=0$, then 
$(A,B)$ is reducible.
Proof. If 
$\det (A)=0$, then the cubic resolvent of 
$(A,B)$ has 
$x^{3}$-coefficient 
$0$, implying that 
$(A,B)$ is reducible. If 
$a_{11}=b_{11}=0$ then 
$A$ and 
$B$ have a common zero in 
${{\mathbb {P}}}^{2}({{\mathbb {Q}}})$, implying that 
$(A,B)$ corresponds to the identity element in the class group of 
$R_f$.
We are now ready to prove the second claim of Theorem 4.1.
Proof of Theorem 4.1 We follow the setup and methods of [Reference BhargavaBha05]. To begin with, averaging over 
$w\in {{\mathcal {B}}}$ as in [Reference BhargavaBha05, (6) and (8)], we obtain
\begin{align*} & \sum_{K\in{{\mathcal{R}}}^{{\pm}}_3(X,Z)}(|{{\rm Cl}}(K)[2]|-1)\\ &\quad \ll \int_{g\in{{\mathcal{F}}}_{2,3}} |\{w\in g\cdot{{\mathcal{B}}}\cap W({{\mathbb{Z}}})^{{\rm irr}}:K_{{{\rm Res}}(w)}\in{{\mathcal{R}}}_3(X,Z) \}|\,dg\\ &\quad \ll \int_{s_1,s_2,t\gg 1} |\{w\in ((\lambda,t),(s_1,s_2))\cdot{{\mathcal{B}}}\cap W({{\mathbb{Z}}})^{{\rm irr}}: K_{{{\rm Res}}(w)}\in {{\mathcal{R}}}_3(X,Z) \}|\frac{d^{{\times}}\lambda\, d^{{\times}} t\, d^{{\times}} s_1\, d^{{\times}} s_2}{t^{2} s_1^{6}s_2^{6}}, \end{align*}
where 
$K_f$ denotes the algebra 
${{\mathbb {Q}}}\otimes R_f$ for an integral binary cubic form 
$f$.
The action of an element 
$((\lambda ,t),(s_1,s_2))\in {{\mathcal {F}}}_{2,3}$ on 
$W({{\mathbb {R}}})$ multiplies each coordinate 
$c_{ij}$ of 
$W$ by a factor which we denote by 
$w(c_{ij})$. For example, we have 
$w(a_{11})=\lambda t^{-1}s_1^{-4}s_2^{-2}$. The volume of 
${{\mathcal {B}}}$ is some positive constant, and when 
${{\mathcal {B}}}$ is translated by an element 
$((\lambda ,t),(s_1,s_2))$, the volume is multiplied by a factor of 
$\lambda ^{12}$, the product of 
$w(c_{ij})$ over all the coordinates 
$c_{ij}$. Furthermore, the maximum of the volumes of the projections of 
$((\lambda ,t),(s_1,s_2))\cdot {{\mathcal {B}}}$ is
\[{\ll}\prod_{c_{ij}\in S}w(c_{ij})=\prod_{c_{ij}\not\in S}\lambda^{12}w(c_{ij}), \]
where 
$S$ denotes the set of coordinates 
$c_{ij}$ of 
$W({{\mathbb {R}}})$ such that the length of the projection of 
$((\lambda ,t),(s_1,s_2))\cdot {{\mathcal {B}}}$ onto the 
$c_{ij}$-coordinate is at least 
$\gg 1$.
For the set 
$((\lambda ,t),(s_1,s_2))\cdot {{\mathcal {B}}}\cap W({{\mathbb {Z}}})^{{\rm irr}}$ to be empty, it is necessary that the projection of 
${{\mathcal {B}}}':=((\lambda ,t),(s_1,s_2))\cdot {{\mathcal {B}}}$ onto the 
$b_{11}$-coordinate is 
$\gg 1$. Otherwise, every integral element of 
${{\mathcal {B}}}'$ has 
$a_{11}=b_{11}=0$, and is hence reducible by Lemma 4.9. Similarly, the projections of 
${{\mathcal {B}}}'$ onto the 
$a_{13}$- and 
$a_{22}$-coordinates are also 
$\gg 1$ (since otherwise every integral element 
$(A,B)$ of 
${{\mathcal {B}}}'$ satisfies 
$\det (A)=0$). Finally, for 
${{\mathcal {B}}}'\cap W({{\mathbb {Z}}})$ to contain an element whose resolvent cubic form corresponds to a field in 
${{\mathcal {R}}}_3(X,Z)$, we must have 
$\lambda \asymp X^{1/12}$ and 
$Z^{1/2}\ll t\ll X^{1/2}$ by Lemma 4.8.
Therefore, applying Proposition 4.6 to the sets 
$((\lambda ,t),(s_1,s_2))\cdot {{\mathcal {B}}}$, we obtain
\begin{align*}
&\sum_{K\in{{\mathcal{R}}}^{{\pm}}_3(X,Z)}(|{{\rm Cl}}(K)[2]|-1)\nonumber\\&
\quad\ll \int_{\substack{\lambda,t,s_1,s_2\\
\lambda\asymp X^{1/12}\\Z^{1/2}\ll t\ll
X^{1/12}\\ S_1,s_2\gg 1}}
(\lambda^{12}(1+w(a_{11})^{{-}1}+w(a_{11}a_{12})^{{-}1})
\frac{d^{{\times}}\lambda\, d^{{\times}} t\, d^{{\times}}
s_1\, d^{{\times}} s_2}{t^{2}s_1^{6}s_2^{6}} \\
&\quad\ll \int_{\substack{\lambda,t,s_1,s_2\\\lambda\asymp
X^{1/12}\\Z^{1/2}\ll t\ll X^{1/12}\\ S_1,s_2\gg 1}}
(\lambda^{12}+s_1^{4}s_2^{2}t\lambda^{11}+s_1^{4}s_2^{4}t^{2}\lambda^{10})
\frac{d^{{\times}}\lambda\, d^{{\times}} t\, d^{{\times}}
s_1\, d^{{\times}} s_2}{t^{2}s_1^{6}s_2^{6}}\\ &\quad\ll
X/Z+X^{11/12}/Z^{1/2}+X^{5/6+\epsilon},
\end{align*}
which is sufficient since 
$Z\ll X^{1/6}$. Theorem 4.1 now follows from this bound and Proposition 4.4.
5. Embedding into the space of binary quartic forms
Recall that 
$U_0({{\mathbb {Z}}})$ denotes the set of monic cubic polynomials with zero 
$x^{2}$-coefficient, and 
$U_0({{\mathbb {Z}}})^{{\rm sm}} = U_0({{\mathbb {Z}}}) \cap {\bigcap} _p U_0({{\mathbb {Z}}}_p)^{{\rm sm}}$ denotes the set of elements 
$f(x)\in U_0({{\mathbb {Z}}})$ such that the elliptic curve 
$y^{2}=f(x)$ has minimal discriminant among all its quadratic twists. We define the height function 
$H:U_0({{\mathbb {Z}}})\to {{\mathbb {R}}}_{\geq 0}$ by
\[ H(x^{3}+Ax+B)={{\rm max}}\{4|A|^{3},27B^{2}\}. \]
For 
$f(x)\in U_0({{\mathbb {Z}}})$, we write 
$K_f = {{\mathbb {Q}}}[x]/(f(x))$, 
$R_f = {{\mathbb {Z}}}[x]/(f(x))$, and let 
${{\mathcal {O}}}_f$ denote the maximal order in 
$K_f$. The 
$Q$-invariant 
$Q(f)$ of 
$f$ is defined as the index of 
$R_f$ in 
${{\mathcal {O}}}_f$, and 
$D(f)$ is defined to be the discriminant of 
$K_f$. Observe from Table 1 that for primes 
$p$ of type 
${{\mathrm I}}{{\mathrm I}}{{\mathrm I}}$, 
${{\mathrm I}}{{\mathrm V}}$ and 
${{\mathrm I}}_{2n+1}$, we have 
$p\mid Q(f)$ and 
$p\mid D(f)$. Note also 
$\gcd (Q(f),D(f))$ is squarefree.
In this section we obtain a bound on the number of elements 
$f\in U_0({{\mathbb {Z}}})^{{\rm sm}}$, having bounded height, such that both 
$Q(f)$ and 
$\gcd (Q(f),D(f))$ are large.
Theorem 5.1 Let 
$Q$ and 
$q$ be positive real numbers with 
$Q\geq q$. Let 
$N_{Q,q}(Y)$ denote the number of elements 
$f(x)\in U_0({{\mathbb {Z}}})^{{\rm sm}}$ such that 
$H(f) < Y$, 
$|Q(f)|>Q$, and 
$\gcd (Q(f),D(f))>q$. Then
\[ N_{Q,q}(Y)\ll_\epsilon \frac{Y^{5/6+\epsilon}}{qQ} + \frac{Y^{7/12+\epsilon}}{Q^{1/2}}, \]
where the implied constant is independent of 
$Q$, 
$q$, and 
$Y$.
This section is organized as follows. First, in § 5.1, we collect classical results on the invariant theory of the action of 
${{\rm PGL}}_2$ on the space 
$V_4$ of binary quartic forms, and summarize the reduction theory of binary quartics developed in [Reference Bhargava and ShankarBS15]. Next, in § 5.2, we restrict to the space 
$V_4({{\mathbb {Z}}})^{{\rm red}}$ of binary quartic forms with a linear factor. We develop the invariant theory for the action of 
${{\rm PGL}}_2$ on this space, and construct an embedding 
$U_0({{\mathbb {Z}}})^{{\rm sm}}\to V_4({{\mathbb {Z}}})^{{\rm red}}$.
In §§ 5.3–5.5, we estimate the number of 
${{\rm PGL}}_2({{\mathbb {Z}}})$-orbits on elements in 
$V_4({{\mathbb {Z}}})^{{\rm red}}$ with bounded height and large 
$Q$-invariant and whose 
$Q$- and 
$D$-invariants have a large common factor. We do this by fibering the space 
$V_4({{\mathbb {Z}}})^{{\rm red}}$ by their roots in 
${{\mathbb {P}}}^{1}({{\mathbb {Z}}})$. Given an element 
$r\in {{\mathbb {P}}}^{1}({{\mathbb {Z}}})$, the set of elements in 
$V_4({{\mathbb {Z}}})$ that vanish on 
$r$ is a lattice 
$\mathcal {L}_r$. We then count the number of elements in 
$\mathcal {L}_r$, using the Ekedahl sieve to exploit the condition that 
$\gcd (Q,D)$ is large.
5.1 The action of ${{\rm PGL}}_2$ on the space $V_4$ of binary quartic forms
Let 
$V_4$ denote the space of binary quartic forms. The group 
${{\rm PGL}}_2$ acts on 
$V_4$ as follows. Given 
$\gamma \in {{\rm GL}}_2$ and 
$g(x,y)\in V_4$, define
\[ (\gamma\cdot g)(x,y):=\frac{1}{(\det \gamma)^{2}}\,g((x,y)\cdot\gamma). \]
It is easy to check that the center of 
${{\rm GL}}_2$ acts trivially. Hence this action of 
${{\rm GL}}_2$ on 
$V_4$ descends to an action of 
${{\rm PGL}}_2$ on 
$V_4$.
The ring of invariants for the action of 
${{\rm PGL}}_2({{\mathbb {C}}})$ on 
$V_4({{\mathbb {C}}})$ is freely generated by two elements, traditionally denoted by 
$I$ and 
$J$. Explicitly, for 
$g(x,y)=ax^{4}+bx^{3}y+cx^{2}y^{2}+dxy^{3}+ey^{4}$, we have
\begin{align*} I(g) &= 12ae-3bd+c^{2},\\ J(g) &= 72ace+9bcd-27ad^{2}-27eb^{2}-2c^{3}. \end{align*}
The height 
$H(g)$ of a binary quartic form 
$g(x,y)$ is defined by 
$H(g) = {{\rm max}}\{4|I(g)|^{3},J(g)^{2}\}.$
We collect results from [Reference Bhargava and ShankarBS15, § 2.1] on the reduction theory of integral binary quartic forms. For 
$i=0,1,2$, we let 
$V_4({{\mathbb {R}}})^{(i)}$ be the set of elements in 
$V({{\mathbb {R}}})$ with nonzero discriminant, 
$i$ pairs of complex conjugate roots, and 
$4-2i$ real roots. Furthermore, we write 
$V_4({{\mathbb {R}}})^{(2)}=V_4({{\mathbb {R}}})^{(2+)}\cup V_4({{\mathbb {R}}})^{(2-)}$ as the union of forms that are positive definite and negative definite. The four sets 
$L^{(i)}$ for 
$i\in \{0,1,2+,2-\}$ constructed in [Reference Bhargava and ShankarBS15, Table 1] satisfy the following two properties: first, 
$L^{(i)}$ are fundamental sets for the action of 
${{\mathbb {R}}}_{>0}\cdot {{\rm PGL}}_2({{\mathbb {R}}})$ on 
$V_4({{\mathbb {R}}})^{(i)}$ where 
${{\mathbb {R}}}$ acts via scaling; and second, the sets 
$L^{(i)}$ are absolutely bounded. It follows that the sets 
$R^{(i)}:={{\mathbb {R}}}_{>0}\cdot L^{(i)}$ are fundamental sets for the action of 
${{\rm PGL}}_2({{\mathbb {R}}})$ on 
$V_4({{\mathbb {R}}})^{(i)}$, and that the coefficients of an element 
$f(x,y)\in R^{(i)}$ with 
$H(f)=Y$ are bounded by 
$O(Y^{1/6})$.
For 
$A'$, 
$N'(t)$, and 
$K$ defined in (18), set
\[ {{\mathcal{F}}}_0=\{n\alpha(t)k:n(u)\in N'(t),\alpha(t)\in A',k\in K\}. \]
Then 
${{\mathcal {F}}}_0$ is a fundamental domain for the left multiplication action of 
${{\rm PGL}}_2({{\mathbb {Z}}})$ on 
${{\rm PGL}}_2({{\mathbb {R}}})$; and the multisets 
${{\mathcal {F}}}_0\cdot R^{(i)}$ are 
$n_i$-fold fundamental domains for the action of 
${{\rm PGL}}_2({{\mathbb {Z}}})$ on 
$V({{\mathbb {R}}})^{(i)}$, where 
$n_0=n_{2\pm }=4$ and 
$n_1=2$. Let 
$S\subset V_4({{\mathbb {Z}}})^{(i)}=V_4({{\mathbb {Z}}})\cap V_4({{\mathbb {R}}})^{(i)}$ be any 
${{\rm PGL}}_2({{\mathbb {Z}}})$-invariant set. Let 
$N_4(S;X)$ denote the number of 
${{\rm PGL}}_2({{\mathbb {Z}}})$-orbits on 
$S$ with height bounded by 
$X$ such that each orbit 
${{\rm PGL}}_2({{\mathbb {Z}}})\cdot f$ is counted with weight 
$1/\#{{\rm Stab}}_{{{\rm PGL}}_2({{\mathbb {Z}}})}(f)$. Let 
$G_0\subset {{\rm PGL}}_2({{\mathbb {R}}})$ be a nonempty open bounded 
$K$-invariant set, and let 
$d\gamma =t^{-2}dnd^{\times } tdk$, for 
$\gamma =ntk$ in Iwasawa coordinates, be a Haar measure on 
${{\rm PGL}}_2({{\mathbb {R}}})$. Then, identically as in [Reference Bhargava and ShankarBS15, Theorem 2.5], we have the following result.
Theorem 5.2 We have
\[ N_4(S;X)=\frac{1}{n_i{{\rm Vol}}(G_0)} \int_{\gamma\in{{\mathcal{F}}}_0}\#\{S\cap \gamma G_0\cdot R^{(i)}_X\}\,d\gamma, \]
where 
$R^{(i)}_X$ denotes the set of elements in 
$R^{(i)}$ with height bounded by 
$X$, the volume of 
$G_0$ is computed with respect to 
$d\gamma$, and for any set 
$T\subset V({{\mathbb {R}}})$, the set of elements in 
$T$ with height less than 
$X$ is denoted by 
$T_X$.
Apart from its use in this section to obtain a bound on reducible binary quartic forms, Theorem 5.2 will also be used in § 7 to prove Theorem 1.2.
5.2 Embedding $U_0({{\mathbb {Z}}})^{{\rm sm}}$ into the space of reducible binary quartics
Let 
$f(x)=x^{3}+Ax+B$ be an element in 
$U_0({{\mathbb {Z}}})^{{\rm sm}}$ with 
$Q(f)=n$. From Theorem 1.6 it follows that there exists an integer 
$r$, defined uniquely modulo 
$n$, such that 
$f(x+r)$ is of the form
\[ f(x+r)=x^{3}+ax^{2}+bnx+cn^{2}. \]
The cubic ring 
$R_f={{\mathbb {Z}}}[x]/(f(x))$ is contained in the cubic ring corresponding to the binary cubic form
\[ h(x,y)=nx^{3}+ax^{2}y+bxy^{2}+cy^{3} \]
under the Delone–Faddeev correspondence, with index 
$n$. In other words, the cubic ring corresponding to 
$h(x,y)$ is the ring of integers 
${{\mathcal {O}}}_f$ in 
$K_f={{\mathbb {Q}}}[x]/(f(x))$.
Elements in 
$U_0({{\mathbb {Z}}})^{{\rm sm}}$ with 
$Q$-invariant 
$n$ thus correspond to integral binary cubic forms that represent 
$n$. However, the latter condition is difficult to detect, at least using geometry-of-numbers methods. Instead, we embed the space of binary cubic forms into the space 
$V_4({{\mathbb {Z}}})^{{\rm red}}$ of binary quartic forms with a linear factor over 
${{\mathbb {Q}}}$ by multiplying by 
$y$. In fact, we will replace 
$V_4({{\mathbb {Z}}})^{{\rm red}}$ with its (at most 
$4$ to 
$1$) cover 
$\tilde {V}_4({{\mathbb {Z}}})$ consisting of pairs 
$(g(x,y),[\alpha ,\beta ])$, where 
$g$ is a reducible binary quartic forms and 
$[\alpha ,\beta ]$ is a root of 
$f$. Explicitly,
\[ \tilde{V}_4({{\mathbb{Z}}}):=\{(g(x,y),[\alpha,\beta]):0\neq g(x,y)\in V_4({{\mathbb{Z}}})^{{\rm red}}, \alpha,\beta\in{{\mathbb{Z}}}, \gcd(\alpha,\beta)=1, g(\alpha,\beta)=0\}. \]
This gives us the following map 
$\tilde {\sigma }:U_0({{\mathbb {Z}}})^{{\rm sm}}\to \tilde {V}_4({{\mathbb {Z}}})$:
\begin{equation} \begin{aligned} \tilde{\sigma}:U_0({{\mathbb{Z}}})^{{\rm sm}} & \to V_3({{\mathbb{Z}}}) \to \tilde{V}_4({{\mathbb{Z}}})\\ f(x) & \mapsto h(x,y) \mapsto (yh(x,y),[1,0]). \end{aligned} \end{equation}
The group 
${{\rm PGL}}_2({{\mathbb {Z}}})$ acts on 
$\tilde {V}_4({{\mathbb {Z}}})$ via
\[ \gamma\cdot(g(x,y), [\alpha:\beta]) = ((\gamma\cdot g)(x,y),[\alpha:\beta]\gamma^{{-}1}); \]
this is an action since 
$(\gamma \cdot g)((\alpha ,\beta )\gamma ^{-1})=g(\alpha ,\beta )=0$. Aside from the classical invariants 
$I$ and 
$J$, this action has an extra invariant, which we denote by 
$Q$, defined as follows. Given 
$(g,[\alpha :\beta ])\in \tilde {V}_4({{\mathbb {Z}}})$, let 
$h(x,y) = g(x,y)/(\beta x-\alpha y)$ be the associated binary cubic form and we define
\begin{equation} Q(g,[\alpha:\beta]) = h(\alpha,\beta),\quad D(g,[\alpha:\beta])) = \Delta(h). \end{equation}
The 
$Q$- and 
$D$-invariants and the discriminant are related by
\[ \Delta(g) = Q(g,[\alpha:\beta])^{2}D(g,[\alpha:\beta]). \]We now have the following result.
Proposition 5.3 There is an injective map
\[ \sigma: U_0({{\mathbb{Z}}})^{{\rm sm}}\rightarrow \tilde{V}_4({{\mathbb{Z}}})\rightarrow {{\rm PGL}}_2({{\mathbb{Z}}}) \backslash \tilde{V}_4({{\mathbb{Z}}}), \]
such that for every 
$f(x) = x^{3} + Ax + B\in U_0({{\mathbb {Z}}})^{{\rm sm}}$, we have
\begin{equation} I(\sigma(f))={-}3A,\quad J(\sigma(f))={-}27B,\quad Q(f) = Q(\sigma(f)), \quad D(f)=D(\sigma(f)). \end{equation}Proof. The first two equalities of (22) can be checked by a direct computation. The injectivity of 
$\sigma$ then follows from the fact that 
$I(\sigma (f))$ and 
$J(\sigma (f))$ determine 
$f$. The third and fourth equalities of (22) are true by design: in the notation above, the ‘leading coefficient’ 
$h(1,0)$ of 
$h(x,y)$ is the 
$Q$-invariant of 
$f$ and the cubic ring corresponding to 
$h(x,y)$ is the ring of integers in 
$K_f$, which implies that 
$D(\sigma (f))=\Delta (h) = {{\rm Disc}}(K_f) = D(f)$.
Therefore, to prove Theorem 5.1, it suffices to count 
${{\rm PGL}}_2({{\mathbb {Z}}})$-orbits 
$(g,[\alpha :\beta ])$ in 
$\tilde {V}_4({{\mathbb {Z}}})$, such that both 
$Q(g,[\alpha :\beta ])$ and the radical 
${{\rm rad}}(\gcd (Q(g,[\alpha :\beta ]),D(g,[\alpha :\beta ])))$ are large.
5.3 Counting ${{\rm PGL}}_2({{\mathbb {Z}}})$-orbits on reducible binary quartic forms
We use the setup of [Reference Bhargava and ShankarBS15, § 2], which is recalled in § 5.1. Since the sets 
$L^{(i)}$ are absolutely bounded, the coefficients of any element in 
$R^{(i)}={{\mathbb {R}}}_{>0}\cdot L^{(i)}$ having height 
$Y$ are bounded by 
$O(Y^{1/6})$. Hence the same is true of every element in 
$G_0\cdot R^{(i)}_Y$, as 
$G_0$ is a bounded set. The set 
$V_4({{\mathbb {Z}}})^{{\rm red}}$ is not a lattice. To apply geometry-of-numbers methods, we fiber it over the set of possible linear factors. We write
\begin{equation} V_4({{\mathbb{Z}}})^{{\rm red}}=\bigcup_{r=[\alpha:\beta]}\mathcal{L}_r, \end{equation}
where 
$\alpha$ and 
$\beta$ are coprime integers and, for 
$r=[\alpha :\beta ]\in {{\mathbb {P}}}^{1}({{\mathbb {Z}}})$, we define 
$\mathcal {L}_r$ to be the set of all integral binary quartic forms 
$f$ such that 
$f(r)=0$. From Theorem 5.2, in conjunction with the injection 
$\sigma$ of § 5.2, we have
\begin{align} &N_{Q,q}(Y)\nonumber\\ & \ll \sum_{r\in{{\mathbb{P}}}^{1}({{\mathbb{Z}}})}\int_{(ntk)\in{{\mathcal{F}}}_0} \#\{g\in\mathcal{L}_r\cap (ntk)G_0R^{(i)}_Y: Q(g)>Q, {{\rm rad}}(\gcd(Q(g),D(g))) >q\} t^{{-}2}dn\,d^{{\times}} t\,dk. \end{align} As 
$\gamma$ varies over 
${{\mathcal {F}}}_0$, the set 
$\gamma G_0R^{(i)}_Y$ becomes skewed. More precisely, if 
$\gamma =ntk$ in Iwasawa coordinates, then the five coefficients 
$a$, 
$b$, 
$c$, 
$d$, and 
$e$ of any element of 
$\gamma G_0R^{(i)}(Y)$ satisfy
\begin{equation} a\ll \frac{Y^{1/6}}{t^{4}},\quad b\ll \frac{Y^{1/6}}{t^{2}},\quad c\ll Y^{1/6},\quad d\ll t^{2}Y^{1/6},\quad e\ll t^{4}Y^{1/6}. \end{equation}
Hence when 
$t\gg Y^{1/24}$, the 
$x^{4}$-coefficient of any integral binary quartic form in 
$\gamma G_0R^{(i)}(Y)$ is 
$0$, forcing a root at the point 
$[1,0]\in {{\mathbb {P}}}^{1}({{\mathbb {Z}}})$. Moreover, we expect it to be rare that such a binary quartic form has another integral root. In what follows, we first consider the lattice 
$\mathcal {L}_{[1,0]}$ in § 5.4, and consider the rest of the lattices in § 5.5.
5.4 The contribution from the root $r = [1:0]$
Let 
$g(x,y)=bx^{3}y+cx^{2}y^{2}+dxy^{3}+ey^{4}\in \mathcal {L}_{[1:0]}$ be an integral binary quartic form. We write 
$Q(g)$ for 
$Q(g,[1:0])$ and 
$D(g)$ for 
$D(g,[1:0])$. Then we have 
$Q(g)=b$ and 
$D(g)=\Delta (bx^{3}+cx^{2}y+dxy^{3}+ey^{3})$, the discriminant of the binary cubic form 
$g(x,y)/y$. Hence, if a fixed 
$t\geq 1$ contributes to the estimate 
$N_{Q,q}(Y)$ in (24), then we must have
\begin{equation} t\ll\frac{Y^{1/12}}{Q^{1/2}}. \end{equation}
We now fiber over the 
$O(Y^{1/6}/t^{2})$ choices for 
$b$. For each such choice, we have 
$O(Y^{\epsilon })$ possible squarefree divisors 
$m$ of 
$b$. Fix such a divisor 
$m>q$ such that 
${{\rm rad}}(\gcd (Q(g),D(g)))=m$. Then 
$m\mid D(g)$, which implies that
\[ 3c^{2}d^{2}-4c^{3}e\equiv 0\pmod{m}. \]
Thus, the residue class of 
$e$ modulo 
$m$ is determined by 
$c$ and 
$d$, unless 
$m\mid c$.
From (25), we see that the number of elements in 
$\mathcal {L}_{[1:0]}\cap (ntk)G_0R^{(i)}_Y$ with 
$b$ and 
$m$ fixed as above is bounded by
\[ O \biggl (\frac{t^{6}Y^{1/2}}{m}+t^{6}Y^{1/3} \biggr)= O \biggl (\frac{t^{6}Y^{1/2}}{q}+t^{6}Y^{1/3} \biggr), \]
where the second term deals with the case 
$q\gg Y^{1/6}$. It therefore follows that the contribution to 
$N_{Q,q}(Y)$ in (24) from the root 
$r=[1:0]$ is bounded by
\begin{equation} \int_{t=1}^{Y^{1/12}/Q^{1/2}}\frac{Y^{1/6+\epsilon}}{t^{2}} \biggl (\frac{t^{6}Y^{1/2}}{q}+t^{6}Y^{1/3} \biggr)t^{{-}2}\,d^{{\times}} t \ll_\epsilon \frac{Y^{5/6+\epsilon}}{qQ}+\frac{Y^{2/3+\epsilon}}{Q}, \end{equation}
which is sufficiently small. The contribution from the root 
$r=[0:1]$ can be identically bounded.
5.5 The contribution from a general root $r=[\alpha :\beta ]$ with $\alpha \beta \neq 0$
Write 
$r = [\alpha :\beta ]$ where 
$\alpha ,\beta$ are coprime integers and 
$\alpha \beta \neq 0$. Throughout this section, we denote the torus element in 
${{\mathcal {F}}}_0$ with entries 
$t^{-1}$ and 
$t$ by 
$a_t$. We have the bijection
\begin{equation} \begin{aligned} \theta_t:\{\mathcal{L}_r\cap a_tG_0\cdot R^{(i)}_Y\}& \longleftrightarrow \{a_t^{{-}1}\mathcal{L}_r \cap G_0 \cdot R^{(i)}_Y\}\\ ax^{4}+bx^{3}y+cx^{2}y^{2}+dxy^{3}+ey^{4} & \longmapsto t^{4}ax^{4}+t^{2}bx^{3}y+cx^{2}y^{2}+t^{{-}2}dxy^{3}+t^{{-}4}ey^{4}, \end{aligned} \end{equation}
which preserves the invariants 
$I$ and 
$J$. Define 
$\widetilde {V_4}({{\mathbb {R}}})$ to be the set of pairs 
$(g(x,y),r)$, where 
$g(x,y)\in V_4({{\mathbb {R}}})$ and 
$r\in {{\mathbb {R}}}^{2}$ such that 
$g(r)=0$. We extend the definitions of the 
$Q$- and 
$D$-invariants to the space 
$\widetilde {V_4}({{\mathbb {R}}})$ via (21). Set 
$r_t:=r\cdot a_t=[t^{-1}\alpha ,t\beta ]$. Then we have
\[ Q(g,r)=Q(\theta_t\cdot g,r_t),\quad D(g,r)=D(\theta_t\cdot g,r_t). \] Identifying the space of binary quartics with 
${{\mathbb {R}}}^{5}$ via the coefficients 
$(a,b,c,d,e)$, we write
\[ a_t^{{-}1}\mathcal{L}_r = \text{diag}(t^{4},t^{2},1,t^{{-}2},t^{{-}4}) \cdot ((\alpha^{4},\alpha^{3}\beta,\alpha^{2}\beta^{2},\alpha\beta^{3},\beta^{4})^{{\perp}}), \]
where 
$(\alpha ^{4},\alpha ^{3}\beta ,\alpha ^{2}\beta ^{2},\alpha \beta ^{3},\beta ^{4})^{\perp }$ is the sublattice of 
${{\mathbb {Z}}}^{5}$ perpendicular to 
$(\alpha ^{4},\alpha ^{3}\beta ,\alpha ^{2}\beta ^{2},\alpha \beta ^{3},\beta ^{4})$ with respect to the usual inner product on 
${{\mathbb {R}}}^{5}$. Since 
$\alpha$ and 
$\beta$ are coprime, the following vectors form an integral basis for 
$a_t^{-1}\mathcal {L}_r$:
\begin{gather*} w_1 = (t^{4}\beta,-t^{2}\alpha,0,0,0),\quad w_2 = (0,t^{2}\beta,-\alpha,0,0),\\ w_3 = (0,0,\beta,-t^{{-}2}\alpha,0),\quad w_4 = (0,0,0,t^{{-}2}\beta,-t^{{-}4}\alpha). \end{gather*}
Define the vector 
$v_t$ to be 
$v_t:=(t\beta ,-t^{-1}\alpha )\in {{\mathbb {R}}}^{2}$. Then it is easy to see that the lengths of 
$w_i$ are given by
\begin{equation} |w_1| = t^{3}|v_t|,\quad |w_2| = t|v_t|,\quad |w_3| = t^{{-}1}|v_t|,\quad |w_4| = t^{{-}3}|v_t|, \end{equation}
The next lemma proves that this basis is almost Minkowski. That is, the quotients 
$\langle w_i,w_j\rangle /(|w_i||w_j|)$, for 
$i\neq j$, are bounded from above by a constant 
$c<1$ independent of 
$t$ and 
$r$.
Lemma 5.4 For 
$i\neq j$, we have
\[ \langle w_i,w_{j}\rangle \leq \tfrac 12 |w_i||w_j|. \]Proof. The inner product 
$\langle w_i,w_j\rangle$ for 
$i<j$ is 
$0$ unless 
$j=i+1$. In those three cases, we have
\[ \frac{\langle w_i,w_{j}\rangle}{|w_i||w_j|}=\frac{|\alpha\beta|}{t^{{-}2}\alpha^{2}+t^{2}\beta^{2}}\leq \frac 12, \]by the inequality of arithmetic and geometric means.
We will represent elements in 
$a_t^{-1}\mathcal {L}_r$ by 
$4$-tuples 
$(a_1,a_2,a_3,a_4)\in {{\mathbb {Z}}}^{4}$, where such a tuple corresponds to the element 
$a_1w_1 + a_2w_2 + a_3w_3 + a_4w_4$. Then we have the following lemma.
Lemma 5.5 Let 
$g(x,y)$ be an element in 
$\mathcal {L}_r$, and let 
$a_t^{-1} g(x,y)$ correspond to the 
$4$-tuple 
$(a_1,a_2,a_3,a_4)$. Then we have
\begin{align*} g(x,y) &= (\beta x - \alpha y)(a_1 x^{3} + a_2 x^{2} y + a_3 x y^{2} + a_4 y ^{3}),\\ Q(g,r) &= a_1\alpha^{3} + a_2\alpha^{2}\beta + a_3\alpha\beta^{2} + a_4\beta^{3},\\ D(g,r) &= \Delta_3(a_1,a_2,a_3,a_4), \end{align*}
where 
$\Delta _3(a_1,a_2,a_3,a_4)$ denotes the discriminant of the binary cubic form with coefficients 
$a_i$.
The above lemma follows from a direct computation. Next, we determine when an element 
$(a_1,a_2,a_3,a_4)$ has small length.
Lemma 5.6 Suppose 
$g\in a_t^{-1}\mathcal {L}_r$, corresponding to 
$(a_1,a_2,a_3,a_4)$, belongs to 
$G_0\cdot R^{(i)}_Y$ for some 
$i$. Then
\begin{equation} a_1 \ll \frac{Y^{1/6}}{t^{3}|v_t|}, \quad a_2 \ll \frac{Y^{1/6}}{t|v_t|}, \quad a_3 \ll \frac{Y^{1/6}}{t^{{-}1}|v_t|}, \quad a_4 \ll \frac{Y^{1/6}}{t^{{-}3}|v_t|}. \end{equation}Proof. Let 
$|\cdot |$ denote the length of a binary quartic form, where 
$V_4({{\mathbb {R}}})$ has been identified with 
${{\mathbb {R}}}^{5}$ in the natural way. Then for 
$g$ to belong in 
$G_0\cdot R^{(i)}_Y$, it must satisfy 
$|g|\ll Y^{1/6}$. For any real numbers 
$a_1,a_2,a_3,a_4$, we compute
\begin{align*} |a_1w_1+a_2w_2+a_3w_3+a_4w_4|^{2} &\geq a_1^{2}|w_1|^{2} + a_2^{2}|w_2|^{2} + a_3^{2}|w_3|^{2} + a_4^{2}|w_4|^{2} \\ & \quad -|a_1||a_2||w_1||w_2|-|a_2||a_3||w_2||w_3| - |a_3||a_4||w_3||w_4|\\ &\geq \frac{3 - \sqrt{5}}{4}( a_1^{2}|w_1|^{2} + a_2^{2}|w_2|^{2} + a_3^{2}|w_3|^{2} + a_4^{2}|w_4|^{2}). \end{align*}
(Of course, the exact constant is not important.) Therefore in order that 
$|a_1w_1+a_2w_2+a_3w_3+a_4w_4|\ll Y^{1/6}$, (30) must be satisfied.
We now have the following proposition bounding the number of elements in 
$\mathcal {L}_r\cap a_tG_0\cdot R^{(i)}_Y$ whose 
$Q$- and 
$D$-invariants share a large common factor.
Proposition 5.7 For 
$t\gg 1$, we have
\begin{align}
&\#\{g(x,y)\in\mathcal{L}_r\cap a_tG_0\cdot
R^{(i)}_Y:{{\rm rad}}(\gcd(Q(g,r),D(g,r)))>q\}\nonumber\\
&\quad = \left\{ \begin{array}{@{}ll} 0 & \mbox{if } |v_t|\gg
Y^{1/6},\\ \displaystyle O \biggl
(\dfrac{Y^{2/3+\epsilon}}{q|v_t|^{4}}+\dfrac{Y^{1/2+\epsilon}}{t|v_t|^{3}}
\biggr) & \mbox{otherwise}, \end{array}
\right.\end{align}
where the implied constant is independent of 
$r$, 
$t$, and 
$Y$.
Proof. Using the bijection (28) in conjunction with Lemmas 5.5 and 5.6, we see that it is enough to prove that the number of 
$4$-tuples of integers 
$(a_1,a_2,a_3,a_4)$, satisfying (30) and
\[ {{\rm rad}}(\gcd(a_1\alpha^{3} + a_2\alpha^{2}\beta + a_3\alpha\beta^{2} + a_4\beta^{3},\Delta_3(a_1,a_2,a_3,a_4)))>q, \]
is bounded by the right-hand side of (31). Suppose first 
$|v_t|\gg Y^{1/6}$. Then any binary quartic form 
$g(x,y)$ represented by the 
$4$-tuple 
$(a_1,a_2,a_3,a_4)$ satisfying (30) must have 
$a_1=a_2=0$. From Lemma 5.5, it follows that 
$D(g,r)=0$ and hence 
$\Delta (g)=0$. Since 
$G_0\cdot R^{(i)}_Y$ contains no point with 
$\Delta =0$, it follows that the intersection is empty, proving the first part of the proposition.
The second part of the proposition is proved by using the Ekedahl sieve as developed in [Reference BhargavaBha14]. We carry out the sieve in detail so as to demonstrate that the implied constant in (31) is indeed independent of 
$r$ and 
$t$. Define
\[ T_{\alpha,\beta}(a_1,a_2,a_3) := \Delta_3(a_1\beta^{3}, a_2\beta^{3}, a_3\beta^{3}, -(a_1\alpha^{3} + a_2\alpha^{2}\beta + a_3\alpha\beta^{2})). \]
It is clear that if 
$m\mid Q(g,r)$ and 
$m\mid D(g,r)$ for any integer 
$m$, then 
$m\mid T_{\alpha ,\beta }(a_1,a_2,a_3)$.
First, we bound the number of triples 
$(a_1,a_2,a_3)$ satisfying (30) such that 
$T_{\alpha ,\beta }(a_1,a_2,a_3)=0$. For a fixed pair 
$(a_1,a_2)\neq (0,0)$, by explicitly writing out 
$T_{\alpha ,\beta }(a_1,a_2,a_3)$, we see that there are at most three possible values of 
$a_3$ with 
$T_{\alpha ,\beta }(a_1,a_2,a_3)=0$. This gives a bound of 
$O(Y^{1/3}/(t^{4}|v_t|^{2}))$ on the number of triples 
$(a_1,a_2,a_3)$ with 
$T_{\alpha ,\beta }(a_1,a_2,a_3)=0$. Multiplying with the number of all possibilities for 
$a_4$, we obtain the bound
\begin{equation} O \biggl(\frac{Y^{1/2}}{t|v_t|^{3}}\biggr) \end{equation}
on the number of 
$4$-tuples of integers 
$(a_1,a_2,a_3,a_4)$, satisfying (30) and 
$T_{\alpha ,\beta }(a_1,a_2,a_3)=0$.
Next, we fiber over triples 
$(a_1,a_2,a_3)$ with 
$T_{\alpha ,\beta }(a_1,a_2,a_3)\neq 0$ and satisfying (30). In this case, we have 
$(a_1,a_2)\neq (0,0)$. Hence by (30), we may assume 
$\alpha ,\beta ,t\ll Y^{1/6}$. Hence the value of 
$T_{\alpha ,\beta }(a_1,a_2,a_3)$ is bounded by a polynomial in 
$Y$ of fixed degree. It follows that the number of squarefree divisors of 
$T_{\alpha ,\beta }(a_1,a_2,a_3)$ is bounded by 
$O_\epsilon (Y^{\epsilon })$. Fix one such divisor 
$m>q$. We now fiber over a positive squarefree integer 
$\delta \ll Y^{1/6}/(t^{3}|v_t|)$ such that 
$\text {rad}(\gcd (a_1,a_2)) = \delta$. The number of such possible 
$(a_1,a_2)$ is
\[{\ll} \frac{1}{\delta^{2}}\frac{Y^{1/6}}{t^{3}|v_t|}\frac{Y^{1/6}}{t|v_t|}. \]
Fix any such pair. Let 
$a_3$ be any integer satisfying (30) such that 
$T_{\alpha ,\beta }(a_1,a_2,a_3)\neq 0$. Let 
$m_1 = \gcd (m,\delta )$ and let 
$m_2 = m/m_1 > q/\delta$. Then the polynomial 
$\Delta _3(a_1,a_2,a_3,a_4)$ is identically 
$0$ modulo 
$m_1$ and quadratic in 
$a_4$ modulo any prime factor of 
$m_2$. Hence the number of these quadruples with the extra condition that 
$m\mid \Delta _3(a_1,a_2,a_3,a_4)$ is
\[{\ll}\frac{1}{\delta^{2}}\frac{Y^{1/6}}{t^{3}|v_t|}\frac{Y^{1/6}}{t|v_t|} \frac{Y^{1/6}}{t^{{-}1}|v_t|}\bigg(\frac{1}{q/\delta}\frac{Y^{1/6}}{t^{{-}3}|v_t|}+1\bigg) \ll_\epsilon \frac{Y^{2/3}}{\delta q|v_t|^{4}} + \frac{Y^{1/2}}{\delta^{2}t^{3}|v_t|^{3}}. \]
Summing over 
$\delta$ and all possible divisors 
$m$ gives the bound
\begin{equation} O \biggl (\frac{Y^{2/3+\epsilon}}{q|v_t|^{4}} + \frac{Y^{1/2+\epsilon}}{t^{3}|v_t|^{3}} \biggr). \end{equation} We now impose the condition on the 
$Q$-invariant. From Lemma 5.5 and (30), we obtain
\[ Q< |Q(g,r)| = |a_1\alpha^{3} + a_2\alpha^{2}\beta + a_3\alpha\beta^{2} + a_4\beta^{3}| \ll Y^{1/6}|v_t|^{2}. \]In conjunction with (24) and the estimates of Proposition 5.7, this yields
\begin{align*} N_{Q,q}(Y)&\ll_\epsilon \sum_{k\ll\log Y}\int_{t\gg 1}\sum_{\substack{r=[\alpha:\beta]\\ 2^{k}<|v_t|\leq 2^{k+1}}} \biggl (\frac{Y^{2/3+\epsilon}}{q|v_t|^{4}}+\frac{Y^{1/2+\epsilon}}{t|v_t|^{3}} \biggr)t^{{-}2}\,d^{{\times}} t\\ &\ll_\epsilon \sum_{k\ll\log Y}\int_{t\gg 1} \biggl (\frac{Y^{5/6+\epsilon}}{qQ}+\frac{Y^{7/12+\epsilon}}{tQ^{1/2}} \biggr)t^{{-}2}\,d^{{\times}} t\\ &\ll_\epsilon \frac{Y^{5/6+\epsilon}}{qQ}+\frac{Y^{7/12+\epsilon}}{Q^{1/2}}. \end{align*}This concludes the proof of Theorem 5.1.
6. Uniformity estimates
In this section, we prove Theorems 1.4 and 1.5, the main uniformity estimates. First, in § 6.1, we use the results of § 4 to prove Theorem 1.4. Next, in § 6.2, we combine the results of §§ 3 and 5 in order to obtain Theorem 1.5. Finally, in § 6.3, we obtain uniformity estimates on the number of 
${{\rm PGL}}_2({{\mathbb {Z}}})$-orbits on integral binary quartic forms whose discriminants are divisible by the square of a large prime. This result significantly improves the previously best known estimates of [Reference Bhargava and ShankarBS15, Theorem 2.13].
6.1 The family of elliptic curves with squarefree index
Recall the family 
${{\mathcal {E}}}$ defined in the introduction. The assumption that elliptic curves 
$E\in {{\mathcal {E}}}$ satisfy 
$j(E)\ll \log (\Delta (E))$ implies the height bound 
$H(E)\ll \Delta (E)^{1+\epsilon }$. Given 
$E\in {{\mathcal {E}}}$, let 
$E:y^{2}=f(x)=x^{3}+Ax+B$ be the Weierstrass model for 
$E$ such that 
$p^{4}\nmid A$ or 
$p^{6}\nmid B$ for every prime 
$p$. Given an étale algebra 
$K$ over 
${{\mathbb {Q}}}$ with ring of integers 
${{\mathcal {O}}}_K$, let 
${{\mathcal {O}}}_K^{{{\rm Tr}}=0}$ denote the set of traceless integral elements in 
$K$. Consider the map
\[ {{\mathcal{E}}}\to \{(K,\alpha):K\mbox{ cubic algebra over }{{\mathbb{Q}}}, \alpha\in{{\mathcal{O}}}_K^{{{\rm Tr}}=0}\} \]
sending 
$E:y^{2}=f(x)$ to the pair 
$({{\mathbb {Q}}}[x]/(f(x)),x)$. This map is injective since if 
$E$ corresponds to the pair 
$(K,\alpha )$, then 
$y^{2}=N_{K/{{\mathbb {Q}}}}(x-\alpha )$ recovers 
$E$. Note we have 
$|a|\ll H(E)^{1/6}$ from, for example, Fujiwara's bound [Reference FujiwaraFuj16].
Suppose now 
$E\in {{\mathcal {E}}}$ corresponds to a pair 
$(K,\alpha )$. Let 
$\beta ={\rm Prim}(\alpha )$ denote the primitive part of 
$\alpha$, that is, the unique primitive integer in 
${{\mathcal {O}}}_K$ which is a positive rational multiple of 
$\alpha$. Note 
$\alpha = n\beta$ for some positive integer 
$n$. The cubic polynomial 
$g(x) = N_{K/{{\mathbb {Q}}}}(x - \beta ) = x^{3} + ax + b$ is integral and we have 
$A=n^{2}a$ and 
$B=n^{3}b.$ From Tables 3 and 4 in § 7 on the congruence conditions on 
$A,B$ in order for 
$E_{AB}$ to have good reduction at 
$A$ and 
$B$, we see that 
$g(x)$ uniquely determines the 
$2$- and the 
$3$-part of 
$n$. For primes 
$p\geq 5$, we see from Table 1 that if 
$E\in {{\mathcal {E}}}_{{\rm sf}}$, then the 
$p$-part of 
$n$ must be 
$1$. In other words, the modified map
\begin{equation} \begin{aligned} \sigma: {{\mathcal{E}}} & \to \{(K,\beta):K\mbox{ cubic \unicode{x00E9}tale algebra over }{{\mathbb{Q}}}, \beta\in{{\mathcal{O}}}_K^{{{\rm Tr}}=0}\}\\ E:y^{2}=f(x) & \mapsto ({{\mathbb{Q}}}[x]/(f(x)),{\rm Prim}(x)), \end{aligned} \end{equation}
is injective when restricted to 
${{\mathcal {E}}}_{{\rm sf}}$.
Elliptic curves 
$E_{AB}$ with good reduction at 
$2$.
Elliptic curves 
$E_{AB}$ with good reduction at 
$3$.
We start with the following lemma.
Lemma 6.1 Let 
$E$ be an elliptic curve and let 
$\sigma (E)=(K,\beta )$. Then 
$|{{\rm Sel}}_2(E)|\ll _\epsilon |{{\rm Cl}}(K)[2]|\cdot |\Delta (E)|^{\epsilon }$ and 
$|\beta |\ll H(E)^{1/6}$.
Proof. The first bound is a direct consequence of [Reference Brumer and KramerBK77, Proposition 7.1]. The second bound is already true without taking the primitive part as shown above.
We now prove the following result.
Proposition 6.2 For positive real numbers 
$X$ and 
$Q\leq X$, we have
\begin{equation} |\{(E,\eta):E\in{{\mathcal{E}}}_{{\rm sf}}, \eta\in{{\rm Sel}}_2(E), X < C(E)\leq 2X, QX < \Delta(E)\leq 2QX\}| \ll_\epsilon X^{5/6+\epsilon}/Q^{1/6}. \end{equation}
where the implied constant is independent of 
$X$ and 
$Q$.
Proof. Let 
$E\in {{\mathcal {E}}}_{{\rm sf}}$ be an elliptic curve satisfying the conductor and discriminant bounds of (35), and let 
$\sigma (E)=(K,\beta )$. It is easy to verify from Table 1 that 
$\Delta (K) = C(E)^{2}/\Delta (E)$. Therefore, it follows that 
$X/(2Q)<\Delta (K)\leq 4X/Q$, and that 
$|\beta |\ll H(E)^{1/6}\ll _\epsilon (QX)^{1/6+\epsilon }$.
Since the map 
$\sigma$ is injective, it follows that the left-hand side of (35) is
\begin{equation} \ll_\epsilon X^{\epsilon}\sum_{\substack{[K:{{\mathbb{Q}}}]=3\\ {X}/{2Q}<\Delta(K)\leq {4X}/{Q}}} N'_K((QX)^{1/6+\epsilon})\,|{{\rm Cl}}(K)[2]|, \end{equation}
where 
$N'_K(Y)$ denotes the number of primitive elements 
$\beta$ in 
${{\mathcal {O}}}_K^{{{\rm Tr}} =0}$ such that 
$|\beta | < Y$ and the pair 
$(K,\beta )$ is in the image of 
$\sigma$. We now split the above sum over cubic algebras 
$K$ into three parts, corresponding to the sizes 
$\ell _1(K)$ and 
$\ell _2(K)$ of the successive minima of 
${{\mathcal {O}}}_K^{{{\rm Tr}} = 0}$.
First, if 
$(QX)^{1/6+\epsilon }\ll \ell _1(K)$, then the contribution to (36) is 
$0$. Second, assume that 
$\ell _2(K)\ll (QX)^{1/6+\epsilon }$. Then Lemma 4.7 yields the bound
\[ N_K'((QX)^{1/6+\epsilon})\ll_\epsilon (QX)^{1/3+\epsilon}/\sqrt{X/Q}\ll \frac{Q^{5/6}}{X^{1/6-\epsilon}}. \]
Using Bhargava's result [Reference BhargavaBha05, Theorem 5] to bound the sum of 
$|{{\rm Cl}}(K)[2]|$ over cubic fields 
$K$ with the prescribed discriminant range, and using the well-known genus-theory bounds 
${{\rm Cl}}(K)[2]\ll |\Delta (K)|^{\epsilon }$, for each reducible cubic 
$K$, we obtain
\[ \sum_{\substack{[K:{{\mathbb{Q}}}]=3\\ {X}/{2Q}<\Delta(K)\leq {4X}/{Q}\\ \ell_2(K)\ll (QX)^{1/6+\epsilon}}} N'_K((QX)^{1/6+\epsilon})\,|{{\rm Cl}}(K)[2]|\ll_\epsilon X^{\epsilon}\cdot \frac{Q^{5/6}}{X^{1/6-\epsilon}}\cdot \frac{X}{Q}=\frac{X^{5/6+2\epsilon}}{Q^{1/6}}. \] Finally, we bound the contribution of cubic étale algebras 
$K$ such that 
$\ell _1(K)\ll (QX)^{1/6+\epsilon }\ll \ell _2(K)$. In this case, we have 
$N_K'((QX)^{1/6+\epsilon })\leq 1$ and
\[ {{\rm sk}}(K)=\ell_2(K)/\ell_1(K)\gg \sqrt{\Delta(K)}/\ell_1(K)^{2} \gg X^{1/6}/Q^{5/6}. \]
Suppose first 
$K = {{\mathbb {Q}}}\oplus L$ is reducible. Then 
${{\mathcal {O}}}_K^{{{\rm Tr}}=0}$ has an integral basis given by 
$\{(-2,1),(0,\sqrt {d})\}$ where 
$\Delta (K) = d$ or 
$4d$. When 
$d$ is small, say bounded by 
$100$, we get an 
$O(1)$ contribution to (36). When 
$d$ is large, the above basis is a Minkowski basis and 
$(-2,1)$ is the smallest, and hence unique, primitive traceless element. However, this point does not correspond to an elliptic curve since the corresponding cubic polynomial is 
$(x-2)(x+1)^{2}$ which has a double root. Hence, we get no contribution in this case. It remains to consider the case where 
$K$ is a cubic field. Applying Theorem 4.1, we obtain a bound of
\[ O_\epsilon(X^{\epsilon}(X/Q)/(X^{1/6}/Q^{5/6}))= O_\epsilon(X^{5/6+\epsilon}/Q^{1/6}), \]
on the contribution to (36) over cubic fields 
$K$ with 
$\ell _1(K)\ll (QX)^{1/6+\epsilon }\ll \ell _2(K)$, as desired.
Proof of Theorem 1.4 Note that if the conductor 
$C(E)$ is bounded by 
$X$ and the index 
$\Delta (E)/C(E)$ is squarefree, then the index is also bounded by 
$X$. Divide the conductor range 
$[1,X]$ into 
$\log X$ dyadic ranges, and for each such range divide the index range 
$[M,X]$ into 
$\log X$ dyadic ranges, and then apply Proposition 6.2 on each pair of dyadic ranges. Theorem 1.4 follows.
6.2 The family of elliptic curves with bounded index
As in § 3, let 
$\Sigma$ be a finite set of pairs 
$(p,T_p)$, where 
$p$ is a prime number and 
$T_p={{\mathrm I}}{{\mathrm I}}{{\mathrm I}}$, 
${{\mathrm I}}{{\mathrm V}}$, or 
${{\mathrm I}}_{\geq 2}$ is a Kodaira symbol. Recall the invariants 
$Q(\Sigma )$, 
$m_{{\rm odd}}(\Sigma )$ and 
$m_T(\Sigma )$ for Kodaira symbols 
$T$. We further define 
$m_{{\rm even}}(\Sigma )$ to be the product of 
$p$ over pairs 
$(p,{{\mathrm I}}_{2k})$ in 
$\Sigma$. We define 
${{\mathcal {E}}}(\Sigma )$ to be the set of elliptic curves 
$E\in {{\mathcal {E}}}$ such that the Kodaira symbol at 
$p$ of 
$E$ is 
$T_p$ for every pair 
$(p,T_p)\in \Sigma$. Given a set of five positive real numbers
\[ S=\{m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}},m_{{{\mathrm I}}{{\mathrm V}}},m_{{{\rm even}}},m_{{{\rm odd}}},Q\}, \]
we let 
${{\mathcal {E}}}(S)$ denote the set of elliptic curves 
$E$ such that the product 
$P$ of primes at which 
$E$ has Kodaira symbol 
${{\mathrm I}}{{\mathrm I}}{{\mathrm I}}$ (respectively, 
${{\mathrm I}}{{\mathrm V}}$, 
$I_{2(k\geq 1)}$, 
$I_{2(k\geq 1)+1}$) satisfies 
$m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}\leq P<2 m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}$ (respectively, 
$m_{{{\mathrm I}}{{\mathrm V}}}\leq P<2 m_{{{\mathrm I}}{{\mathrm V}}}$, 
$m_{{{\rm even}}}\leq P<2 m_{{{\rm even}}}$, 
$m_{{{\rm odd}}}\leq P<2 m_{{{\rm odd}}}$), and 
$Q\leq Q(E)<2Q$. The following result is a consequence of Theorems 3.1 and 5.1.
Proposition 6.3 Let 
$S=\{m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}, m_{{{\mathrm I}}{{\mathrm V}}}, m_{{{\rm even}}}, m_{{{\rm odd}}}, Q\}$ be as above and let 
$Y$ be a positive real number. Then
\begin{align} & \#\{E\in {{\mathcal{E}}}(S):|\Delta(E)| < Y\}\nonumber\\ &\quad \ll_\epsilon Y^{\epsilon} {{\rm min}}\biggl(\frac{Y^{5/6}m_{{{\rm even}}}}{Q^{2}m_{{{\mathrm I}}{{\mathrm V}}}}+\frac{Qm_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\rm even}}}m_{{{\rm odd}}}^{2}}{Y^{1/6}}, \frac{Y^{5/6}}{Qm_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\mathrm I}}{{\mathrm V}}}m_{{{\rm odd}}}}+\frac{Y^{7/12}}{Q^{1/2}}\biggr). \end{align}Proof. First note that if 
$E\in {{\mathcal {E}}}$, then 
$H(E)\ll \Delta (E)^{1+\epsilon }$ from the 
$j$-invariant bound. It is enough to prove that the left-hand side of (37) is bounded (up to a factor of 
$Y^{\epsilon }$) by both terms in the minimum. For the second term, this is a direct consequence of Theorem 5.1 and Table 1.
For the first term, note that the set of monic cubic polynomials corresponding to curves in 
${{\mathcal {E}}}(S)$ is clearly the union of 
$O_\epsilon (Y^{\epsilon } m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\mathrm I}}{{\mathrm V}}}m_{{{\rm even}}}m_{{{\rm odd}}})$ sets 
$U_0({{\mathbb {Z}}})_\Sigma$, where each such 
$\Sigma$ satisfies 
$m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}(\Sigma )\sim m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}$, 
$m_{{{\mathrm I}}{{\mathrm V}}}(\Sigma )\sim m_{{{\mathrm I}}{{\mathrm V}}}$, 
$m_{{{\rm even}}}(\Sigma )\sim m_{{{\rm even}}}$, 
$m_{{{\rm odd}}}(\Sigma )\sim m_{{{\rm odd}}}$, and 
$Q(\Sigma )\sim Q$. In § 3, we obtained bounds on the number of elements in 
$U({{\mathbb {Z}}})_\Sigma$ with height bounded by 
$Y$. Since the set 
$U({{\mathbb {Z}}})_\Sigma$ is invariant under the linear 
${{\mathbb {Z}}}$-action, we have
\[ |\{f\in U_0({{\mathbb{Z}}})_\Sigma:H(f) < Y\}|\ll Y^{{-}1/6}|\{f\in U({{\mathbb{Z}}})_\Sigma:H(f) < Y\}|. \]
Combining this with Theorem 3.1, and multiplying by the number of different 
$\Sigma$s required to cover the set 
${{\mathcal {E}}}(S)$, we obtain the result.
Proof of Theorem 1.5 Given real numbers 
$X$, 
$Y$, 
$m_0$, 
$m_1$ and 
$m_2$ at least 1, let 
${{\mathcal {E}}}(S;X,Y,m_0,m_1,m_2)$ denote the set of elliptic curves 
$E\in {{\mathcal {E}}}$ that satisfy 
$X\leq C(E)<2X$, 
$Y\leq \Delta (E)<2Y$, and such that the product 
$P$ of primes at which 
$E$ has Kodaira symbol 
${{\mathrm I}}_0^{*}$ (respectively, 
${{\mathrm I}}_{n\geq 1}^{*}$, 
${{\mathrm I}}{{\mathrm I}}^{*}$ or 
${{\mathrm I}}{{\mathrm I}}{{\mathrm I}}^{*}$ or 
${{\mathrm I}}{{\mathrm V}}^{*}$) satisfies 
$m_0\leq P<2 m_0$ (respectively, 
$m_1\leq P<2 m_1$, 
$m_2\leq P<2 m_2$), and that 
$E_0\in {{\mathcal {E}}}(S)$ where 
$E_0$ is the elliptic curve obtained from 
$E$ by applying a quadratic twist so that its defining cubic polynomial is small.
Fix constants 
$0<\kappa <7/4$ and 
$0<\delta$. We first obtain bounds on the sizes of the sets 
${{\mathcal {E}}}(S;X,Y,m_0,m_1,m_2)$. Let 
$P$ be the contribution to the conductor of 
$E_0$ that is prime to 
$m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\mathrm I}}{{\mathrm V}}}m_{{{\rm even}}}m_{{{\rm odd}}}$. We set
\[ X_0 = \frac{X}{m_0^{2}m_1},\quad Y_0 = \frac{Y}{m_0^{6}m_1^{6}m_2^{6}}. \]Then we have, by Table 1,
\begin{align*} X_0&\asymp C(E_0) \asymp m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}^{2}m_{{{\mathrm I}}{{\mathrm V}}}^{2}m_{{{\rm even}}}m_{{{\rm odd}}}P,\\ Y_0&\asymp \Delta(E_0) \asymp m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\mathrm I}}{{\mathrm V}}}^{2}m_{{{\rm odd}}}Q^{2}P. \end{align*}
Therefore, in order for 
${{\mathcal {E}}}(S;X,Y,m_0,m_1,m_2)$ to be nonempty, we must have
\begin{equation} \frac{Y_0}{Q^{2}}\asymp\frac{X_0}{m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\rm even}}}}. \end{equation}First note that we have
\begin{equation} \frac{Y_0^{5/6}m_{{{\rm even}}}}{Q^{2}m_{{{\mathrm I}}{{\mathrm V}}}}\ll \frac{X_0}{Y_0^{1/6}}. \end{equation}Moreover,
\begin{align} {{\rm min}} \biggl (\frac{Qm_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\rm even}}}m_{{{\rm odd}}}^{2}}{Y_0^{1/6}}, \frac{Y_0^{5/6}}{Qm_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\mathrm I}}{{\mathrm V}}}m_{{{\rm odd}}}} \biggr) &\leq \biggl(\frac{Y_0^{15/6-1/6}m_{{{\rm even}}}}{Q^{2}m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}^{2}m_{{{\mathrm I}}{{\mathrm V}}}^{3}m_{{{\rm odd}}}}\biggr)^{1/4}\notag\\ &\ll X_0^{1/4}Y_0^{1/3} \ll X^{1/4}Y^{1/3},\\ {{\rm min}} \biggl (\frac{Qm_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\rm even}}}m_{{{\rm odd}}}^{2}}{Y_0^{1/6}}, \frac{Y_0^{7/12}}{Q^{1/2}} \biggr) &\leq (Y_0m_{{{\mathrm I}}{{\mathrm I}}{{\mathrm I}}}m_{{{\rm even}}}m_{{{\rm odd}}}^{2})^{1/3}\ll\frac{Y_0^{2/3}}{X_0^{1/3}}\ll\frac{Y^{2/3}}{X^{1/3}}. \notag \end{align} Assume that 
$Y$ satisfies the bound 
$X^{1+\delta }\ll Y\ll X^{\kappa }$ for 
$\delta >0$ and 
$\kappa <7/4$. Proposition 6.3, (39), and (40) imply that we have
\begin{equation} |{{\mathcal{E}}}(S;X,Y,m_0,m_1,m_2)|\ll_\epsilon X^{5/6-\theta+\epsilon}, \end{equation}
for some positive constant 
$\theta$ depending only on 
$\delta$ and 
$\kappa$. It is clear that the set
\[ \{E\in{{\mathcal{E}}}_\kappa:C(E) < X,|\Delta(E)| > C(E)X^{\delta}\} \]
is the union of 
$O_\epsilon (X^{\epsilon })$ sets 
${{\mathcal {E}}}(S;X_1,Y_1,m_0,m_1,m_2)$, with 
$X_1\leq X$, 
$X_1^{1+\delta }\ll Y_1\ll X_1^{\kappa }$, and 
$m_0,m_1,m_2\ll X^{\kappa /6}$. Theorem 1.5 now follows from (41).
6.3 Additional uniformity estimates
Recall that 
$U_0$ denotes the space of monic traceless cubic polynomials. For a prime 
$p$, let 
${{\mathcal {U}}}_p$ denote the set of elements 
$f(x)\in U_0({{\mathbb {Z}}})$, such that 
$p^{2}\mid \Delta (f)$. We write 
${{\mathcal {U}}}_p$ as the disjoint union 
${{\mathcal {U}}}_p^{(1)}\cup {{\mathcal {U}}}_p^{(2)}$, where 
${{\mathcal {U}}}_p^{(1)}$ is the set of elements 
${{\mathcal {U}}}_p$ whose discriminants are strongly divisible (in the notation of [Reference BhargavaBha14, § 3.3]) by 
$p^{2}$ and 
${{\mathcal {U}}}_p^{(2)}$ is simply 
${{\mathcal {U}}}_p\backslash {{\mathcal {U}}}_p^{(1)}$. We begin by proving the following (averaged) improvement of [Reference Bhargava and ShankarBS15, Proposition 3.16].
Proposition 6.4 We have
\begin{align*} \#\biggl\{f\in \bigcup_{p > M}{{\mathcal{U}}}_p^{(1)}:H(f) < X\biggr\} &\ll_\epsilon \frac{X^{5/6}}{M^{1-\epsilon}}+X^{1/3},\\ \#\biggl\{f\in \bigcup_{p > M}{{\mathcal{U}}}_p^{(2)}:H(f) < X\biggr\} &\ll \frac{X^{5/6}}{M}. \end{align*}Proof. The first inequality of the proposition follows immediately from [Reference BhargavaBha14, Theorem 3.5]. We prove the second inequality as follows. If 
$f(x)\in {{\mathcal {U}}}_p^{(2)}$, then 
$R_f={{\mathbb {Z}}}[x]/f(x)$ is nonmaximal in 
$K_f={{\mathbb {Q}}}[x]/f(x)$ of index at least 
$p$. Therefore, we have the following injection:
\begin{equation} \biggl\{f\in \bigcup_{p>M}{{\mathcal{U}}}_p^{(2)}:H(f) < X \biggr \}\hookrightarrow \biggl \{(K,\alpha):\Delta(K) < \frac{X}{M^{2}}, \alpha\in{{\mathcal{O}}}_K^{{{\rm Tr}}=0}, |\alpha| < X^{1/6} \biggr\}, \end{equation}
where 
$K$ ranges over cubic étale algebras over 
${{\mathbb {Q}}}$. (The injection simply maps 
$f(x)$ to 
$({{\mathbb {Q}}}[x]/f(x),x)$.) To estimate the size of the right-hand side of (42), we simply use Theorem 4.1 in conjunction with Lemma 4.7 to obtain a bound of
\[ \frac{X}{M^{2}}\cdot\frac{X^{2/6}}{\sqrt{X/M^{2}}}+ \frac{X}{M^{2}} \biggl(\frac{X^{1/6}}{M}\biggr )^{{-}1}\ll \frac{X^{5/6}}{M}, \]
which is sufficient. Indeed, the first summand corresponds to the contribution from cubic fields 
$K$ with 
$\ell _2(K)\leq X^{1/6}$, while the second summand corresponds to the contribution of cubic fields with 
$\ell _1(K)\leq X^{1/6}<\ell _2(K)$ in which case 
${{\rm sk}}(K)>X^{1/6}/M$.
Next, we consider the space 
$V_4({{\mathbb {Z}}})$ of integral binary quartic forms. For a prime 
$p$, let 
${{\mathcal {V}}}_p$ denote the set of elements 
$f(x)\in V_4({{\mathbb {Z}}})$, such that 
$p^{2}\mid \Delta (f)$. As before, we write 
${{\mathcal {V}}}_p$ as the disjoint union 
${{\mathcal {V}}}_p^{(1)}\cup {{\mathcal {V}}}_p^{(2)}$, where 
${{\mathcal {V}}}_p^{(1)}$ is the set of elements 
${{\mathcal {V}}}_p$ whose discriminants are strongly divisible by 
$p^{2}$ and 
${{\mathcal {V}}}_p^{(2)}$ is 
${{\mathcal {V}}}_p\backslash {{\mathcal {V}}}_p^{(1)}$. We have the following estimate which is a significantly stronger version of [Reference Bhargava and ShankarBS15, Theorem 2.13].
Theorem 6.5 We have
\begin{align*} \# \biggl \{f\in {{\rm PGL}}_2({{\mathbb{Z}}})\bigg\backslash \bigcup_{p>M}{{\mathcal{V}}}_p^{(1)}:H(f) < X \biggr\} &\ll_\epsilon \frac{X^{5/6}}{M^{1-\epsilon}}+X^{19/24},\\ \# \biggl \{f\in {{\rm PGL}}_2({{\mathbb{Z}}})\bigg\backslash \bigcup_{p>M}{{\mathcal{V}}}_p^{(2)}:H(f) < X \biggr\} &\ll_\epsilon \frac{X^{5/6+\epsilon}}{M}. \end{align*}Proof. The first inequality follows from an application of [Reference BhargavaBha14, Theorem 3.5] in conjunction with the counting results of [Reference Bhargava and ShankarBS15, § 2.3]. We now prove the second estimate of the proposition. First note that to every integral binary quartic form 
$f$ we may associate its cubic resolvent 
$g(x)$ which is the unique monic traceless cubic polynomial with the same invariants as 
$f(x)$. We note the following two facts about the cubic resolvent. First, for every integral monic cubic polynomial 
$g(x)$, we have a bijection
\[ {{\rm PGL}}_2({{\mathbb{Z}}})\backslash{{\rm Res}}^{{-}1}(g)\longleftrightarrow {{\rm Sel}}_2(R_g). \]
Next, the cubic resolvent 
$g(x)$ of any element in 
${{\mathcal {V}}}^{(2)}_p$ belongs to 
${{\mathcal {U}}}^{(2)}_p$. Since we have
\[ |{{\rm Sel}}_2(R_g)|\leq 4|{{\rm Cl}}(R_g)[2]|\ll_\epsilon |\Delta(g)|^{\epsilon}|{{\rm Cl}}(K_g)[2]|, \]the required bound follows from Theorem 4.1 in conjunction with the proof of the second estimate in Proposition 6.4.
7. Asymptotics for families of elliptic curves
Let 
$p$ be a fixed prime. An elliptic curve 
$E$ over 
${{\mathbb {Q}}}$ has either good reduction, multiplicative reduction, or additive reduction at 
$p$. For every prime 
$p\geq 5$, let 
$\Sigma _p$ be a nonempty subset of possible reduction types. We say that 
$\Sigma =(\Sigma _p)_p$ is a collection of reduction types and that such a collection is large if, for all large enough primes 
$p$, the set 
$\Sigma _p$ contains at least the good and multiplicative reduction types.
For a large collection 
$\Sigma$, let 
${{\mathcal {E}}}_{{\rm sf}}(\Sigma )$ (respectively, 
${{\mathcal {E}}}_\kappa (\Sigma )$) denote the set of elliptic curves 
$E\in {{\mathcal {E}}}_{{\rm sf}}$ (respectively, 
$E\in {{\mathcal {E}}}_\kappa$) such that for all primes 
$p\geq 5$, the reduction type of 
$E$ at 
$p$ belongs to 
$\Sigma _p$. In this section we prove the following theorem, from which Theorem 1.2 and the first two asymptotics of Theorem 1.1 immediately follow.Footnote 3
Theorem 7.1 Let 
$\Sigma$ be a large collection of elliptic curves. Let 
$\kappa <7/4$ be a positive constant. Then we have
\begin{equation} \begin{aligned} &
\#\{E\in {{\mathcal{E}}}_{{\rm sf}}(\Sigma)^{{\pm}}:
C(E) < X\}\\ & \quad \sim \frac{\alpha^{{\pm}}}{60\sqrt{3}}
\frac{\Gamma(1/2)\Gamma(1/6)}{\Gamma(2/3)}\prod_p(c_g(p)e_g(p)+c_m(p)e_m(p)+c_a(p)e_a(p))\cdot
X^{5/6},\\ & \#\{E\in
{{\mathcal{E}}}_\kappa(\Sigma)^{{\pm}}:
C(E) < X\}\\ & \quad
\sim \frac{\alpha^{{\pm}}}{60\sqrt{3}}
\frac{\Gamma(1/2)\Gamma(1/6)}{\Gamma(2/3)}\prod_p(c_g(p)f_g(p)+c_m(p)f_m(p)+c_a(p)f_a(p))\cdot
X^{5/6}, \end{aligned} \end{equation}
where 
$\alpha ^{+}=1$, 
$\alpha ^{-}=\sqrt {3}$, 
$c_g(p)$ (respectively, 
$c_m(p)$, 
$c_a(p))$ is 
$1$ or 
$0$ depending on whether 
$\Sigma _p$ contains the good 
$($resp. multiplicative, additive) reduction type, and 
$e_*(p)$ and 
$f_*(p)$ are given by
\begin{gather*} e_g(p):= 1-\frac{1}{p}, \quad e_m(p):=\frac{1}{p} \biggl(1+\frac{1}{p^{1/6}}\biggr) \biggl(1-\frac{1}{p}\biggr)^{2},\quad e_a(p):= \frac{1}{p^{2}} \biggl (1+\frac{1}{p^{1/6}}\biggr) \biggl(1-\frac{1}{p}\biggr),\\ f_g(p):= 1-\frac{1}{p}, \quad f_m(p):=\frac{1}{p} \biggl(1-\frac{1}{p^{1/6}}\biggr)^{{-}1} \biggl(1-\frac{1}{p}\biggr)^{2},\\ f_a(p):= \frac{1}{p^{5/3}} \biggl(1-\frac{1}{p}\biggr) \biggl(1+\frac{1}{p^{1/6}}+\frac{1}{p^{7/6}}\biggr) + \frac{1}{p^{2}}\biggl(1-\frac{1}{p}\biggr) \biggl(1-\frac{1}{p^{1/6}}\biggr)^{{-}1} \biggl(3-\frac{2}{p^{1/2}} \biggr). \end{gather*}
Furthermore, when elliptic curves in 
${{\mathcal {E}}}_{{\rm sf}}(\Sigma )$ are ordered by conductor, the average size of their 
$2$-Selmer groups is 
$3$.
7.1 The family ${{\mathcal {E}}}$ ordered by discriminant
In this subsection we prove the final asymptotic in Theorem 1.1 on counting the family 
${{\mathcal {E}}}$ ordered by discriminant.
For a pair 
$(A,B)\in {{\mathbb {R}}}^{2}$, we write 
$\Delta (A,B)=-4A^{3} - 27B^{2}$ and write 
$j(A,B)=1728(4A^{3})/(4A^{3}+27B^{2}).$ Then 
$j(A,B)$ is the 
$j$-invariant of the elliptic curve 
$E_{AB}$, and when 
$(A,B)\in {{\mathbb {Z}}}^{2}$ is such that 
$E_{AB}$ has good reduction at 
$2$ and 
$3$, 
$\Delta (A,B)=2^{8}\Delta (E_{AB})$ if 
$3\nmid A$ and 
$\Delta (A,B)=2^{8}3^{12}\Delta (E_{AB})$ otherwise (see Tables 3 and 4). Hence we first count pairs 
$(A,B)\in {{\mathbb {Z}}}^{2}$ first with 
$j(A,B)\ll \log (\Delta (A,B)/c_0)$, where 
$c_0=2^{8}$ or 
$2^{8}3^{12}$.
The condition 
$j(A,B)\ll \log (\Delta (A,B)/c_0)$ is not a semialgebraic condition in 
$A$ and 
$B$. However it is clearly definable in the o-minimal structure where the distinguished functions are polynomials and the exponential function [Reference WilkieWil96]. Hence we can use the following result of Barroero and Widmer [Reference Barroero and WidmerBW14, Theorem 1.3].
Theorem 7.2 Let 
$m$ and 
$n$ be positive integers, let 
$\Lambda \subset {{\mathbb {R}}}^{n}$ be a lattice, and denote the successive minima of 
$\Lambda$ by 
$\lambda _i$. Let 
$Z\subset {{\mathbb {R}}}^{m+n}$ be a definable family in an o-minimal structure, and suppose the fibers 
$Z_T$ are bounded. Then there exists a constant 
$c_Z\in {{\mathbb {R}}}$, depending only on the family 
$Z$, such that
\[ \biggl |\# (Z_T\cap\Lambda)- \frac{{{\rm Vol}}(Z_T)}{\det(\Lambda)}\biggr | \leq c_Z\sum_{j=0}^{n-1}\frac{V_j(Z_T)}{\lambda_1\cdots\lambda_j}, \]
where 
$V_j(Z_T)$ is the sum of the 
$j$-dimensional volumes of the orthogonal projections of 
$Z_T$ on every 
$j$-dimensional coordinate subspace of 
${{\mathbb {R}}}^{n}$.
Suppose now we are given a set 
$S\subset {{\mathbb {Z}}}^{2}$ defined by congruence conditions modulo some positive integer 
$n$. Then we may break 
$S$ up into a union of 
$n^{2}\nu (S)$ translates of the lattice 
$n{{\mathbb {Z}}}\times n{{\mathbb {Z}}}$, where 
$\nu (S)$ denotes the volume of the closure of 
$S$ in 
$\hat {{{\mathbb {Z}}}}^{2}$. Applying Theorem 7.2 to each of these translates and summing gives the following immediate consequence of Theorem 7.2.
Proposition 7.3 Let 
$S\subset {{\mathbb {Z}}}^{2}$ denote a set of pairs 
$(A,B)$ defined by congruence conditions on 
$A$ and 
$B$ modulo some positive integer. Then we have
\begin{align*} &\#\{(A,B)\in\Lambda:
j(A,B)\ll\log(\Delta(A,B)/c_0), 0 < {\pm}\Delta(A,B) < X\}\\
&\quad=\nu(S)c_\infty^{{\pm}}(X)+O_\epsilon(n\nu(S)X^{1/2+\epsilon}),\end{align*}
where 
$c_\infty ^{\pm }(X)$ denotes the volume of the set
\[ C^{{\pm}}(X):=\{(A,B)\in{{\mathbb{R}}}^{2}: j(A,B)\ll\log(\Delta(A,B)/c_0), 0 < {\pm}\Delta(A,B) < X\} \]computed with respect to the usual Euclidean measure.
We now restate and prove the third asymptotic in Theorem 1.1.
Theorem 7.4 We have
\[ \#\{E\in{{\mathcal{E}}}:0 < {\pm}\Delta(E) < X\}\sim \frac{\alpha^{{\pm}}}{60\sqrt{3}}\cdot\frac{\Gamma(1/2)\Gamma(1/6)}{\Gamma(2/3)} \cdot\prod_{p\geq 5} \biggl (1-\frac{1}{p^{10}} \biggr)X^{5/6}, \]
where 
$\alpha ^{+}=1$ and 
$\alpha ^{-}=\sqrt {3}$.
Proof. First, we describe the set of elliptic curves 
$E_{AB}:y^{2}=x^{3}+Ax+B$ that have good reduction at 
$2$ and 
$3$ in Tables 3 and 4, respectively. In both tables, the first column describes the congruence conditions on 
$A$, the second describes congruence conditions at 
$B$, the third gives the 
$2$-part (respectively, the 
$3$-part) of the discriminant 
$\Delta (A,B)=4A^{3}+27B^{2}$, and the fourth column gives the density of these congruence conditions inside the space 
$(A,B)\in {{\mathbb {Z}}}_p^{2}$ for 
$p=2$ and 
$3$. Below, 
$\delta$ is either 
$0$ or 
$1$.
We now apply Proposition 7.3. Let 
$1\leq i\leq 9$ and 
$1\leq j\leq 11$ be integers, and consider the set of integers 
$(A,B)$ that satisfy line 
$i$ of Table 3 and line 
$j$ of Table 4. Let 
$\nu _{ij}=\nu _2(i)\cdot \nu _3(j)$ denote the density of this set of integers, and let 
$\Delta _{ij}=\Delta _2(i)\cdot \Delta _3(j)$ denote the product of the 
$2$- and 
$3$-parts of the discriminant 
$\Delta (A,B)$. It is necessary to count the number of pairs 
$(A,B)\in {{\mathbb {Z}}}^{2}$ that satisfy the following properties.
1. The pair
$(A,B)$ satisfies the $i$th (respectively, $j$th) condition of Table 3 (respectively, Table 4).2.
$0<\pm \Delta (A,B)<\Delta _{ij}X$.3.
$j(A,B)\ll \log (\Delta (A,B)/\Delta _{ij})$.4. For all primes
$p\geq 5$, either $p^{4}\nmid A$ or $p^{6}\nmid B$.
Counting the pairs 
$(A,B)$ which satisfy the first three properties is immediate from Proposition 7.3, and the fourth condition can be imposed by applying a simple inclusion exclusion sieve. For each 
$i,j$, let 
$S_{ij}$ denote the set of 
$(A,B)\in {{\mathbb {Z}}}^{2}$ satisfying the 
$i$th (respectively, 
$j$th) condition of Table 3 (respectively, Table 4). For every positive integer 
$m$, let 
$N_m^{\pm }(X)_{ij}$ denote the number of pairs 
$(A,B)\in S_{ij}$ satisfying the second and the third conditions above and 
$p^{4}\mid A$ and 
$p^{6}\mid B$ for each prime 
$p\mid m$ with 
$p\geq 5$. Let 
$\theta (p) = 1-p^{-10}$ for 
$p\geq 5$ and 
$\theta (2)=\theta (3)=1$. Set 
$\bar {\theta }(p) = 1 - \theta (p)$ and 
$\bar {\theta }(m)=\prod _{p\mid m}\bar {\theta }(p)$ for any squarefree 
$m$. Then, for any small 
$\alpha >0,$
\begin{align*} &\#\{(A,B)\in S_{ij}:0<{\pm}\Delta(A,B)<\Delta_{ij}X,j(A,B)\ll\log(\Delta(A,B)/\Delta_{ij})\}\\ &\quad = \sum_{m\geq 1}\mu(m)N_m^{{\pm}}(X)_{ij}\\ &\quad = \nu_{ij}\sum_{m=1}^{X^{\alpha}}\mu(m)\bar{\theta}(m) c_\infty^{{\pm}}(\Delta_{ij}\cdot X) + \sum_{m=1}^{X^{\alpha}}O(mX^{1/2+\epsilon}) + O\bigg(\sum_{\substack{m=X^{\alpha}\\m\text{ squarefree}}}N_m^{{\pm}}(X)\bigg)\\ &\quad = \nu_{ij}c_\infty^{{\pm}}(\Delta_{ij}\cdot X) \bigg(\prod_p\theta(p) + O(X^{{-}9\alpha})\bigg) + O(X^{1/2+2\alpha+\epsilon}) + O \bigg(\sum_{m=X^{\alpha}}^{X^{1/12+\epsilon}}\frac{X^{5/6+\epsilon}}{m^{10}}\bigg).\end{align*}
Here the very last term follows from a naive count as 
$A\ll X^{1/3+\epsilon }$ and 
$B\ll X^{1/2+\epsilon }$ and so 
$m$ is bounded by 
$X^{1/12+\epsilon }$ in order that 
$m^{4}\mid A$ and 
$m^{6}\mid B.$ We may now minimize the above with 
$\alpha = 1/33$ and sum over 
$i$ and 
$j$ to obtain
\[ \#\{E\in{{\mathcal{E}}}:0 < {\pm}\Delta(E) < X\} = \sum_{i,j}\nu_{ij}\cdot\prod_{p\geq 5}(1-p^{{-}10})\cdot c_\infty^{{\pm}}(\Delta_{ij}\cdot X) + O(X^{37/66 + \epsilon}). \] We next consider the volumes 
$c_\infty ^{\pm }(X)$. Let 
$d_\infty ^{\pm }(X)$ denote the volume of the same region but without the condition on 
$j(A,B)$. A direct integration over the region where 
$j(A,B)\gg \log (\Delta (A,B))$ gives
\[ c_\infty^{{\pm}}(X) = d_\infty^{{\pm}}(X) + O\bigg(\frac{X^{5/6}}{(\log X)^{1/6}}\bigg). \]
The volumes 
$d_\infty ^{\pm }(X)$ are computed in [Reference WatkinsWat08, § 2]:
\[ d_\infty^{{\pm}}(X)\sim X^{5/6}d_\infty^{{\pm}}(1), \]where
\[ d_\infty^{+}(1)=\frac{2}{4^{1/3}\cdot 27^{1/2}}\cdot\frac{1}{5}\cdot B(1/2,1/6),\quad d_\infty^{-}(1)=\frac{2}{4^{1/3}\cdot 27^{1/2}}\cdot \frac{3}{5}\cdot B(1/2,1/3)=\sqrt{3}\,d_\infty^{+}(1). \]
Above, 
$B(x,y)$ denotes the beta function given by
\[ B(x,y)=\int_{0}^{1} t^{x-1}(1-t)^{y-1}\,dt=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}. \]We therefore obtain
\begin{align*}
&\#\{E\in{{\mathcal{E}}}:0 < {\pm}\Delta(E) < X\}\\ &\quad
\sim \sum_{i,j}\nu_{ij}\Delta_{ij}^{5/6}\cdot\prod_{p\geq
5}(1-p^{{-}10})\cdot d_\infty^{{\pm}}(1)\cdot X^{5/6}\\ &\quad=
d_\infty^{{\pm}}(1) \biggl
(\sum_i\nu_2(i)\Delta_2(i)^{5/6}\biggr)
\biggl(\sum_i\nu_3(i)\Delta_3(i)^{5/6} \biggr) \prod_{p\geq
5}(1-p^{{-}10})\cdot X^{5/6}\\ &\quad=
\frac{2^{2/3}}{4}\frac{2\alpha^{{\pm}}}{4^{1/3}\cdot
3^{3/2}\cdot 5} \frac{\Gamma(1/2)\Gamma(1/6)}{\Gamma(2/3)}
\prod_{p\geq 5}(1-p^{{-}10})\cdot X^{5/6}\\ &\quad=
\frac{\alpha^{{\pm}}}{60\sqrt{3}}
\frac{\Gamma(1/2)\Gamma(1/6)}{\Gamma(2/3)} \prod_{p\geq
5}(1-p^{{-}10})\cdot X^{5/6}, \end{align*}as desired.
7.2 Ordering elliptic curves by conductor
Suppose that 
${{\mathcal {G}}}$ is equal to 
${{\mathcal {E}}}_*(\Sigma )$ for a large collection of reduction types 
$\Sigma$, where 
$*$ is either 
${{\rm sf}}$ or some positive 
$\kappa <7/4$. Pick a small positive constant 
$\delta <1/9$. Then there exists a positive constant 
$\theta$ such that
\begin{align} \#\{E\in {{\mathcal{G}}}^{{\pm}}:C(E) < X\}&= \sum_{n\geq 1} \#\{E\in {{\mathcal{G}}}^{{\pm}}:{{\rm ind}}(E)=n; \Delta(E) < nX\}\nonumber\\ &= \sum_{n,q\geq 1}\mu(q)\, \#\{E\in {{\mathcal{G}}}^{{\pm}}:nq\mid{{\rm ind}}(E); \Delta(E) < nX\}\nonumber\\ &= \sum_{\substack{n,q\geq 1\\nq < X^{\delta}}}\mu(q)\, \#\{E\in {{\mathcal{G}}}^{{\pm}}:nq\mid{{\rm ind}}(E); \Delta(E) < nX\}+O(X^{5/6-\theta}), \end{align}
where we bound the tail using the uniformity estimates in Theorems 1.4 and 1.5. We perform another inclusion–exclusion sieve to evaluate each summand of the right-hand side of the above equation. For each prime 
$p$, let 
$\chi _{\Sigma _p,nq}:{{\mathbb {Z}}}_p^{2}\to {{\mathbb {R}}}$ denote the characteristic function of the set of all 
$(A,B)\in {{\mathbb {Z}}}_p^{2}$ that satisfy the reduction type specified by 
$\Sigma _p$ and satisfy 
$nq\mid {{\rm ind}}(E_{AB})$. Let 
$\chi _p$ denote 
$1-\chi _{\Sigma _p,nq}$, and define 
$\chi _k:=\prod _{p\mid k}\chi _p$ for squarefree integers 
$k$. Then we have
\[ \prod_{p}\chi_{\Sigma_p,nq}(A,B)=\sum_k\mu(k)\chi_k(A,B) \]
for every 
$(A,B)\in {{\mathbb {Z}}}^{2}$. Set 
$\nu _*(nq,\Sigma )$ to be the product over all primes 
$p$ of the integral of 
$\chi _{\Sigma _p,nq}$. Therefore, for 
$nq < X^{\delta }$, we obtain
\begin{align*} \#\{E\in
{{\mathcal{G}}}^{{\pm}}:nq\mid{{\rm ind}}(E);
\Delta(E)<nX\}&=
\sum_{\substack{(A,B)\in{{\mathbb{Z}}}^{2}\\0 < {\pm}\Delta(E_{AB})<nX}}\sum_{k\geq
1}\mu(k)\chi_k(A,B)\\ &=
\sum_{\substack{(A,B)\in{{\mathbb{Z}}}^{2}\\0<{\pm}\Delta(E_{AB})<nX}}\sum_{k=1}^{X^{4\delta}}\mu(k)\chi_k(A,B)
+O \biggl(\frac{(nX)^{5/6}}{X^{2\delta}}\biggr)\\ &=
c_\infty^{{\pm}}(nX)\nu_*(nq,\Sigma)+O_\epsilon(X^{1/2+2\delta+\epsilon}+X^{5/6-7\delta/6}),
\end{align*}
where the second equality follows from the uniformity estimate in Proposition 6.4, and the third follows from Proposition 7.3 and adding up the volume terms by simply reversing the inclusion–exclusion sieve. Note that the constant 
$\delta$ has been specifically picked to be small enough so that Proposition 7.3 applies.
For each 
$n$, let 
$\lambda _*(n,\Sigma )$ denote the volume of the closure in 
$\hat {{{\mathbb {Z}}}}^{2}$ of the set of all 
$(A,B)\in {{\mathbb {Z}}}^{2}$ such that 
$E_{AB}$ belongs to 
${{\mathcal {G}}}={{\mathcal {E}}}_*(\Sigma )$ and 
$E_{AB}$ has index 
$n$. Returning to (44), we obtain
\begin{align*} \#\{E\in {{\mathcal{G}}}^{{\pm}}:C(E) < X\}&= c^{{\pm}}_\infty(1)X^{5/6} \sum_{\substack{n,q\geq 1\\nq < X^{\delta}}}\mu(q)n^{5/6}\nu_*(nq,\Sigma) +o(X^{5/6})\\ &= c_\infty^{{\pm}}(1)X^{5/6}\sum_{n\geq 1}n^{5/6}\lambda_*(n,\Sigma), \end{align*}where again, the final equality follows by reversing the inclusion–exclusion sieve of (44).
For each prime 
$p$ and 
$k\geq 0$, let 
$\bar {\nu }_*(p^{k},\Sigma )$ denote the 
$p$-adic density of the set of all 
$(A,B)\in {{\mathbb {Z}}}^{2}$ such that 
$E_{AB}\in {{\mathcal {E}}}_*(\Sigma )$ and 
${{\rm ind}}_p(E_{AB})=p^{k}$. The constant 
$\lambda _*(n,\Sigma )$ is a product over all 
$p$ of local densities:
\begin{align*} \lambda_*(n,\Sigma)&= \prod_{p\nmid n}\bar{\nu}_*(p^{0},\Sigma) \prod_{\substack{p^{k}\parallel n\\k\geq 1}}\bar{\nu}_*(p^{k},\Sigma)\\ &= \prod_p\bar{\nu}_*(p^{0},\Sigma) \prod_{\substack{p^{k}\parallel n\\k\geq 1}} \frac{\bar{\nu}_*(p^{k},\Sigma)}{\bar{\nu}_*(p^{0},\Sigma)}. \end{align*}
Hence 
$\lambda _*(n,\Sigma )$ is a multiplicative function in 
$n$, and we have
\begin{align*} \sum_{n\geq 1}n^{5/6}\lambda_*(n,\Sigma)&= \prod_p\bar{\nu}_*(p^{0},\Sigma) \prod_{p} \biggl (\sum_{k=0}^{\infty} p^{5k/6} \frac{\bar{\nu}_*(p^{k},\Sigma)}{\bar{\nu}_*(p^{0},\Sigma)} \biggr)\\ &= \prod_p \biggl (\sum_{k=0}^{\infty} p^{5k/6}\bar{\nu}_*(p^{k},\Sigma) \biggr). \end{align*} The values of 
$\bar {\nu }_*(p^{k},\Sigma )$ are easily computed from Table 2. We then have (43), proving the first and second asymptotics of Theorem 1.1.
7.3 The average size of the $2$-Selmer groups of elliptic curves in ${{\mathcal {E}}}_{{\rm sf}}(\Sigma )$
Let 
$\Sigma$ be a large collection of reduction types. For a positive integers 
$n$ and a positive real number 
$X$, let 
${{\mathcal {E}}}(\Sigma ,n,X)$ denote the set of elliptic curves 
$E\in {{\mathcal {E}}}_{{\rm sf}}(\Sigma )$, such that 
$n\mid {{\rm ind}}(E)$ and 
$|\Delta (E)| < X$. Then, as in the previous subsection, we have
\begin{align*}
\sum_{\substack{E\in{{\mathcal{E}}}(\Sigma)^{{\pm}}\\C(E) < X}}(|{{\rm
Sel}}_2(E)|-1)&= \sum_{n,q\geq
1}\mu(q)\sum_{E\in{{\mathcal{E}}}(\Sigma,nq,nX)^{{\pm}}}(|{{\rm
Sel}}_2(E)|-1)\\ &= \sum_{\substack{n,q\geq
1\\nq < X^{\theta}}}\mu(q)\sum_{E\in{{\mathcal{E}}}(\Sigma,nq,nX)^{{\pm}}}(|{{\rm
Sel}}_2(E)|-1)+O_\epsilon(X^{5/6-\theta/6+\epsilon}),
\end{align*}
for every 
$\theta >0$, where the second equality is a consequence of Theorem 1.4. Therefore, the final assertion of Theorem 7.1 follows immediately from the following result.
Proposition 7.5 There exist positive constants 
$\theta$ and 
$\theta _1$ such that
\[ \sum_{E\in{{\mathcal{E}}}(\Sigma,qn,nX)^{{\pm}}}(|{{\rm Sel}}_2(E)|-1)=2|{{\mathcal{E}}}(\Sigma,nq,nX)^{{\pm}}|+O(X^{5/6-\theta_1}), \]
for every 
$nq < X^{\theta }$.
Given the uniformity estimate (Theorem 6.5) that we have already proved, the proof of Proposition 7.5 very closely follows the proof of [Reference Bhargava and ShankarBS15, Theorem 3.1]. We briefly sketch the proof of Theorem 7.1, indicating the change needed at the places where it differs from [Reference Bhargava and ShankarBS15]. The starting point of the proof is the following parametrization of the 
$2$-Selmer groups of elliptic curves in terms of orbits on integral binary quartic forms. This correspondence is due to Birch and Swinnerton-Dyer, and we state it in the form of [Reference Bhargava and ShankarBS15, Theorem 3.5].
Theorem 7.6 Let 
$E:y^{2}=x^{3}+Ax+B$ be an elliptic curve over 
${{\mathbb {Q}}}$, and set 
$I=I(E):=-3A$ and 
$J=J(E):=-27B$. Then there is a bijection between 
${{\rm Sel}}_2(E)$ and the set of 
${{\rm PGL}}_2({{\mathbb {Q}}})$-equivalence classes of locally soluble integral binary quartic forms with invariants 
$2^{4}I$ and 
$2^{6}J$.
Moreover, the set of integral binary quartic forms that have a rational linear factor and invariants equal to 
$2^{4}I$ and 
$2^{6}J$ lie in one 
${{\rm PGL}}_2({{\mathbb {Q}}})$-equivalence class, and this class corresponds to the identity element in 
${{\rm Sel}}_2({{\mathbb {Q}}})$.
The second step in the proof is to obtain asymptotics for the number of 
${{\rm PGL}}_2({{\mathbb {Z}}})$-orbits on the set of integral binary quartic forms whose coefficients satisfy congruence conditions modulo some small number 
$n$, where these forms have bounded invariants. In [Reference Bhargava and ShankarBS15], the invariants were bounded by height. Here instead, we bound their discriminants and corresponding 
$j$-invariant. For an element 
$f\in V_4({{\mathbb {R}}})$ with 
$\Delta (f)\neq 0$, define 
$j(f)$ to be 
$j(E)$ with 
$E$ given by
\[ E:y^{2}=x^{3}-(I/3)x-J/27. \]
For any 
${{\rm PGL}}_2({{\mathbb {Z}}})$-invariant set 
$S\subset V_4({{\mathbb {Z}}})$, let 
$N^{(i)}_4(S;X)$ denote the number of 
${{\rm PGL}}_2({{\mathbb {Z}}})$-orbits on integral elements 
$f\in S\subset V_4^{(i)}({{\mathbb {Z}}})$, that do not have a linear factor over 
${{\mathbb {Q}}}$, and satisfy 
$\Delta (f) < X$ and 
$j(f) < \log \Delta (f)$.
In § 5 we defined the sets 
$R^{(i)}$ which are fundamental sets for the action of 
${{\rm PGL}}_2({{\mathbb {R}}})$ on 
$V({{\mathbb {R}}})^{(i)}$. Then 
$R^{(i)}$ contains one element 
$f\in V({{\mathbb {R}}})^{(i)}$ having invariants 
$I$ and 
$J$, for each 
$(I,J)\in {{\mathbb {R}}}^{2}$ with 
$4I^{3}-J^{2}\in {{\mathbb {R}}}_{>0}$ for 
$i=0,2\pm$ and 
$4I^{3}-J^{2}\in {{\mathbb {R}}}_{<0}$ for 
$i=1$. Furthermore, the coefficients of such an 
$f$ are bounded by 
$O(H(f)^{1/6})$. Define the sets
\[ R^{(i)}(X):=\{f\in R^{(i)}:0 < |\Delta(f)| < X;j(f) < \log\Delta(f)\}. \]
Clearly, if 
$f\in R^{(i)}(X)$ with 
$\Delta (f)=X$, then 
$H(f)\ll X^{1+\epsilon }$ and so the coefficients of 
$f$ are bounded by 
$O(X^{1/6+\epsilon })$.
Let 
$\delta =1/18$ be fixed. Let 
$L\subset V({{\mathbb {Z}}})$ be a lattice defined by congruence conditions modulo 
$n$, where 
$n < X^{\delta }$. Denote the set of elements in 
$L$ that have no linear factor by 
$L^{{\rm irr}}$ and define 
$\nu (L)$ to be the volume of the completion of 
$L$ in 
$V_4(\hat {{{\mathbb {Z}}}})$. Let 
$G_0\subset {{\rm PGL}}_2({{\mathbb {R}}})$ be a nonempty bounded open ball, and set 
$n_1=2$, 
$n_0=n_{2\pm }=4$. Identically to [Reference Bhargava and ShankarBS15, § 2.3], it follows that 
$N_4^{(i)}(L,X)$ is given by
\begin{align} N_4^{(i)}(L,X)&= \frac{1}{n_i{{\rm Vol}}(G_0)}\int_{\gamma\in{{\rm PGL}}_2({{\mathbb{Z}}})\backslash{{\rm PGL}}_2({{\mathbb{R}}})} \#\{\gamma G_0\cdot R^{(i)}(X)\cap L^{{\rm irr}}\}\,d\gamma \nonumber\\ &= \frac{1}{n_i}\int_{\gamma\in{{\rm PGL}}_2({{\mathbb{Z}}})\backslash{{\rm PGL}}_2({{\mathbb{R}}})} \nu(L){{\rm Vol}}(G_0\cdot R^{(i)}(X))\,d\gamma+O(X^{7/9}), \end{align}
where the error term is obtained in a similar manner to [Reference Bhargava and ShankarBS15, (18)–(20)]. There are two differences. First, we use Theorem 7.2 (instead of Davenport's result stated as [Reference Bhargava and ShankarBS15, Proposition 2.6]) to estimate the number of lattice points in 
$\gamma G_0\cdot R^{(i)}(X)$. Second, since we are imposing congruence conditions on 
$L$ modulo 
$n < X^{\delta }$ with 
$\delta =1/18$, we cut off the integral over 
$\gamma$ when the 
$t$-coefficient of 
$\gamma$ in its Iwasawa coordinate is 
$\gg X^{1/36}$. That way, the coefficients of the ball 
$\gamma G_0\cdot R^{(i)}(X)$ are always bigger than 
$n$. The precise values of 
$\delta =1/18$ and 
$7/9$, the exponent of the error term, are not important.
The third step in the proof is to introduce a bounded weight function 
$m:V_4({{\mathbb {Z}}})\to {{\mathbb {R}}}$, which is the product 
$m=\prod _p m_p$ of local weight functions 
$m_p:V_4({{\mathbb {Z}}}_p)\to {{\mathbb {R}}}$, such that, for all but negligibly many (
$\ll _\epsilon X^{3/4+\epsilon }$) elliptic curves 
$E_{AB}$, we have
\[ |{{\rm Sel}}_2(E_{AB})|-1=\sum_{f\in {V_4({{\mathbb{Z}}})_{A,B}}/{{{\rm PGL}}_2({{\mathbb{Z}}})}}m(f), \]
where 
$f$ is varying over 
${{\rm PGL}}_2({{\mathbb {Z}}})$-orbits on integral binary quartic forms with no linear factor and invariants 
$I(f)=-3\cdot 2^{4}I$ and 
$J(f)=-27\cdot 2^{6}J$. In our situation we do not need any changes to this part of the proof.
The fourth and final part of the proof is to perform a sieve so as to count 
${{\rm PGL}}_2({{\mathbb {Z}}})$-orbits on integral binary quartic forms with bounded invariants, so that each form 
$f$ is weighted by 
$m(f)$. Performing a standard inclusion–exclusion sieve using (45) together with the uniformity estimate Proposition 6.5 and the volume computations of [Reference Bhargava and ShankarBS15, §§ 3.3 and 3.6] yields Proposition 7.5. This concludes the proof of Theorem 7.1.
Acknowledgements
It is a pleasure to thank Manjul Bhargava, Étienne Fouvry, Benedict Gross, Hector Pasten, Peter Sarnak, and Jacob Tsimerman for many helpful conversations and comments. The second named author is supported an NSERC Discovery Grant and a Sloan Fellowship. The third named author is supported by an NSERC Discovery Grant.
