M(K,ℓ)
1. Introduction
Let K be a number field of degree
$d\gt1$
and denote by
$D_K$
the absolute value of its discriminant. For arbitrary
$\ell\in\mathbb{N}=\{1,2,3,\ldots\}$
let
$Cl_K[\ell]=\{[\mathfrak{a}]\in Cl_K\: ;\: [\mathfrak{a}]^\ell=[\mathcal{O}_K]\}$
be the
$\ell$
-torsion of the ideal class group of the number field K. The most important tool to upper bound
$\#Cl_K[\ell]$
in terms of
$D_K,d,\ell$
is Ellenberg and Venkatesh’s “Key-Lemma” [
Reference Ellenberg and Venkatesh5
, lemma 2·3] which yields
\begin{equation} \#Cl_K[\ell] \ll_{d,\ell,\varepsilon} D_K^{1/2-1/(2\ell(d-1))+\varepsilon}, \end{equation}
provided there are sufficiently many small suitable (e.g., splitting) primes, which is guaranteed under GRH.
Let
$H_K(\!\cdot\!)$
be the (relative) Weil height on K (see (2·1) for the definition). Let
$N'_K(X)$
be the number of primitive elements in K of Weil height less than X. Following Ellenberg [
Reference Ellenberg4
] we define
\begin{align*}M(K,\ell)&\;:\!=\;\inf_{X\gt0}X^{-1/\ell}(1+N'_K(X)), \text{ and }\\f(\ell,d)&\;:\!=\;\liminf_{K\atop [K:\mathbb{Q}]=d}\frac{-\log M(K,\ell)}{\log D_K}.\end{align*}
It follows from a well-known result of Silverman that
$f(\ell,d) \geq {1}/{(2\ell(d-1))}$
(see Section 2 for details). Ellenberg [
Reference Ellenberg4
] pointed out that the proof of (1·1) actually yields the stronger conclusion
$$\#Cl_K[\ell] \ll_{d,\ell,\varepsilon} D_K^{1/2-f(\ell,d)+\varepsilon},$$
and he proposed study the function
$f(\ell,d)$
. More specifically he wrote: “But at present it is not at all clear how to bound
$M(K,\ell)$
or
$f(\ell, d)$
. As far as we know it might be possible for
$N_K'(X)$
to start growing quite quickly once it becomes nonzero; in this case we would have
$f(\ell,d) = 1/(2\ell(d-1))$
and no improvement would be made on the results of [
Reference Ellenberg and Venkatesh5
]. In fact, Lecoanet [
Reference Lecoanet12
] has carried out experiments for several dozen cubic fields K which seem to show just this kind of behavior. It would be very interesting to understand more fully the situation for cubic fields.”
In this short paper we compute
$f(\ell,3)$
and more generally
$f(\ell,d)$
whenever
$\ell\geq d/2$
.
Proposition 1·1. Let
$d\gt1$
and
$\ell\geq d/2$
be integers. Then
$f(\ell,d)=1/(2\ell(d-1))$
.
Therefore, at least for
$\ell\geq d/2$
, a direct implementation of Ellenberg’s idea does not work. However, the topic has much evolved since 2008. The Key-Lemma has seen various refinements ([
Reference Heath-Brown and Pierce9
, proposition 2·1], [
Reference Frei and Widmer7
, proposition 2·1] and [
Reference Koymans and Thorner11
, theorem 3·3]), leading to more suitable quantities than the above
$N_K'(X)$
. For some of these refined quantities sufficiently good upper bounds can be established, so that an adapted version of Ellenberg’s idea does work, see, e.g., Chan and Koymans’ new bound [
Reference Chan and Koymans2
, theorem 1·1] on the 3-torsion part of the class group of quadratic number fields.
There are rather few other instances where the hypothesis of the Key-Lemma can be established unconditionally, hence giving unconditional pointwise upper bounds for the
$\ell$
-torsion. Wang [
Reference Wang16
] handled non-cyclic elementary abelian extensions (and later extended her result to include many additional nilpotent extensions [
Reference Wang15
]). Heath-Brown [
Reference Heath-Brown10
, theorem 10 and 11] established (1·1) for quadratic and cubic fields with smooth discriminant. Assuming the minimal polynomial of a generator of
$K/\mathbb{Q}$
has a particular shape also allows for pointwise bounds. Heath-Brown [
Reference Heath-Brown10
, theorem 1] pointed out that (1·1) holds for pure cubic number fields, i.e., one has the unconditional upper bound
\begin{equation}\#Cl_K[\ell]\ll_{\varepsilon,\ell}D_K^{1/2-1/(4\ell)+\varepsilon}\end{equation}
for
$\ell$
prime and every
$K=\mathbb{Q}(\sqrt[3]{a})$
where
$a\gt1$
is a cube-free integer. Using ideas of Dubickas we can sharpen (1·2).
Proposition 1·2. Let
$K=\mathbb{Q}(a^{1/3})$
be a pure cubic field, and
$a=A_1A_2^2$
with positive integers
$A_1,A_2$
both squarefree and coprime. Then we have
\begin{align*}\#Cl_K[\ell]\ll_{\varepsilon,\ell}D_K^{1/2-1/(3\ell)+\varepsilon}A_2^{1/(3\ell)}\end{align*}
for any
$\varepsilon\gt0$
, and any positive integer
$\ell$
.
We have
$(A_1A_2)^2\ll D_K\ll (A_1A_2)^2$
(see (1·5) below). By replacing a with
$a^2/A_2^3$
we can swap the roles of
$A_1$
and
$A_2$
. Hence we can assume
$A_2\leq A_1$
, and therefore
$A_2\ll D_K^{1/4}$
. If a is squarefree (or squarefull) then we get
\begin{align*}\#Cl_K[\ell]\ll_{\varepsilon,\ell}D_K^{1/2-1/(3\ell)+\varepsilon}.\end{align*}
In the worst case when
$A_1\asymp A_2$
we recover (1·2). It should also be mentioned that for
$\ell=3$
one has
\begin{align*}\#Cl_K[3]\ll_{\varepsilon}D_K^{\varepsilon}\end{align*}
by work of Gerth [ Reference Gerth III8 ] (see [ Reference Heath-Brown10 , (1·4)] for more details).
Our results apply for pure fields of any odd degree. We say K is a pure field of degree d if
$K=\mathbb{Q}(\theta)$
, where
$\theta$
is a root of some irreducible
$f(x)\in \mathbb{Z}[x]$
of the form
$f(x)=x^d-a$
. Although stated only for cubic fields, Heath-Brown’s observation holds for pure fields of any odd degree, giving
\begin{equation}\#Cl_K[\ell]\ll_{\varepsilon,\ell,d}D_K^{1/2-1/(2(d-1)\ell)+\varepsilon}\end{equation}
for any
$\varepsilon\gt0$
, and any positive integer
$\ell$
(not just primes).
If
$K=\mathbb{Q}(a^{1/d})$
is a pure field of degree d then we can assume
\begin{equation}a=\prod_{i=1}^{d-1}A_i^i\end{equation}
where the positive integers
$A_1,\ldots,A_{d-1}$
are squarefree and pairwise coprime. If k and d are coprime then
$(a^k)^{1/d}$
is also a generator of
$\mathbb{Q}(a^{1/d})$
. Therefore, one can replaceFootnote 1
$A_1$
by
$A_k$
as we did above whenever helpful.
It is clear that each prime p dividing
$A_1$
must occur with exponent at least
$d-1$
in
$D_K$
. This shows that
$\prod_{(k,d)=1} A_k^{d-1}$
divides
$D_K$
.
Each prime
$p\gt d$
divides
$D_K$
at most with exponent
$d-1$
. For primes
$p\leq d$
we note that the discriminant of the polynomial
$x^d-a$
is
$\pm d^da^{d-1}$
. Hence, we get
\begin{equation}\left(\prod_{(k,d)=1} A_k\right)^{d-1}\leq D_K \ll_d (A_1\cdots A_{d-1})^{d-1}.\end{equation}
Proposition 1·2 is an immediate consequence of the following result.
Proposition 1·3. Let
$d\gt1$
be odd, and let
$K=\mathbb{Q}(a^{1/d})$
be a pure field of degree d. Then we have
\begin{align*}\#Cl_K[\ell]\ll_{\varepsilon,\ell,d}D_K^{1/2+\varepsilon}\left(\min_{\frac{d+1}{2}\leq m\leq d-1}\prod_{i=1}^{d-1}A_i^{\frac{i\cdot m}{d}-\lfloor \frac{i\cdot m}{d}\rfloor}\right)^{-1/\ell}\end{align*}
for any
$\varepsilon\gt0$
, and any positive integer
$\ell$
.
Let us consider two examples. If a is squarefree (i.e.,
$a=A_1$
) then we get
\begin{align*}\#Cl_K[\ell]\ll_{\varepsilon,\ell,d}D_K^{1/2-1/(2(d-1)\ell)-1/(2d(d-1)\ell)+\varepsilon},\end{align*}
giving a small but significant improvement upon (1·3). If a is cubefree (i.e.,
$a=A_1A_2^2$
), then we get
\begin{align*}\#Cl_K[\ell]\ll_{\varepsilon,\ell,d}D_K^{1/2-(d+1)/(2(d-1)\ell)+\varepsilon}A_2^{\frac{d-1}{2d\ell}},\end{align*}
or equivalently
\begin{align*}\#Cl_K[\ell]\ll_{\varepsilon,\ell,d}D_K^{1/2-1/(2(d-1)\ell)+\varepsilon}\left(\frac{A_2^{d-2}}{A_1}\right)^{1/(2d\ell)},\end{align*}
giving an improvement upon (1·3) provided
$A_1$
is sufficiantly large in terms of
$A_2$
and d.
Throughout this article the implied constants in Vinogradov’s notation
$\ll$
and
$\gg$
are absolute unless dependence on further parameters is explicitly mentioned by adding subscripts.
2. Background and definitions
Let
\begin{equation}H_K(\alpha)=\prod_{v\in M_K}\max\{1,|\alpha|_v\}^{d_v}\end{equation}
be the relative multiplicative Weil height of
$\alpha\in K$
. Here
$M_K$
denotes the set of places of K, and for each place v we choose the unique representative
$| \cdot |_v$
that either extends the usual Archimedean absolute value on
$\mathbb{Q}$
or a usual p-adic absolute value on
$\mathbb{Q}$
, and
$d_v = [K_v \;:\; \mathbb{Q}_v]$
denotes the local degree at v. Note that this is exactly the height in [
Reference Ellenberg and Venkatesh5
, (2·2)] for the principal divisor
$(\alpha, (\alpha))$
associated to
$\alpha\in K^\times$
.
Since
$\max\{1,|\alpha\beta|_v\}\leq \max\{1,|\alpha|_v\}\max\{1,|\beta|_v\}$
we conclude that
$$H_K(\alpha\beta)\leq H_K(\alpha)H_K(\beta)$$
for all
$\alpha,\beta \in K$
. We write
$\eta(K)$
for the minimal height of a primitive elementFootnote
2
\begin{align*}\eta(K)=\inf\{H_K(\alpha);\; K=\mathbb{Q}(\alpha)\}.\end{align*}
First we recall Ellenberg and Venkatesh’s Key-Lemma [
Reference Ellenberg and Venkatesh5
, lemma 2·3]. Recall from [
Reference Ellenberg and Venkatesh5
] that a prime ideal
$\mathfrak{p}$
of
$\mathcal{O}_K$
is said to be an extension of a prime ideal from a subfield
$K_0\subsetneq K$
if there exists a prime ideal
$\mathfrak{p}_0$
of
$\mathcal{O}_{K_0}$
such that
$\mathfrak{p}=\mathfrak{p}_0\mathcal{O}_K$
. For such a prime ideal
$\mathfrak{p}$
the residue degree
$f(\mathfrak{p}/p)$
is necessarily larger than 1. We say
$\mathfrak{p}$
is unramified in
$K/K_0$
if the ramification index
$e(\mathfrak{p}/(\mathfrak{p}\cap K_0))$
is one.
Lemma 2·1 (Ellenberg and Venkatesh). Suppose K is a number field of degree
$d\gt1$
,
$\delta\lt$
$1/(2\ell(d-1))$
, and
$\varepsilon\gt0$
. Moreover, suppose
$\mathfrak{p}_1,\ldots,\mathfrak{p}_M$
are M prime ideals in
$\mathcal{O}_K$
of norm
$N(\mathfrak{p}_i)\leq D_K^\delta$
that are unramified in
$K/\mathbb{Q}$
and are not extensions of prime ideals from any proper subfield of K. Then we have
$$\#Cl_K[\ell] \ll_{d,\ell,\gamma,\varepsilon} D_K^{1/2+\varepsilon}M^{-1}.$$
Here the hypothesis
$\delta\lt1/(2\ell(d-1))$
can be replaced by
$\delta\lt\gamma/\ell$
as long as
$\eta(K)\gt D_K^\gamma$
. This fact will be used for the proof of Proposition 1·2.
It turns out that for a “typical” K one expects
$\eta(K)$
to be much larger than
$D_K^{1/(2(d-1))}$
and this led to improvements for the average
$\#Cl_K[\ell]$
over various families (see [
Reference Frei and Widmer6, Reference Frei and Widmer7, Reference Widmer17
].
Recall that
$N'_K(X)$
is the number of primitive elements in K of (relative) height less than X, Ellenberg [
Reference Ellenberg4
, proposition 1] pointed out that the proof of [
Reference Ellenberg and Venkatesh5
, lemma 2·3] even provides the stronger conclusion
\begin{equation}\#Cl_K[\ell] \ll_{d,\ell,\varepsilon} D_K^{1/2+\varepsilon}X^{-1/\ell+\varepsilon}(1+N'_K(X)),\end{equation}
provided there are
$\gg_{d,\ell,\varepsilon} X^{1/\ell-\varepsilon}$
prime ideals
$\mathfrak{p}$
in
$\mathcal{O}_K$
that are unramified in
$K/\mathbb{Q}$
and are not extensions of prime ideals from any proper subfield of K. It is well known (cf. [
Reference Silverman14
, theorem 2] or [
Reference Roy and Thunder13
, lemma 2]) that
$\eta(K)\gt(1/2)D_K^{1/(2(d-1))}$
, so that
$N'_K(X)=0$
whenever
$X\leq (1/2)D_K^{1/(2(d-1))}$
. Hence,
$$M(K,\ell)=\inf_{X\geq (1/2)D_K^{1/(2(d-1))}}(X^{-1/\ell}(1+N'_K(X)))\leq 2D_K^{-1/(2\ell(d-1))},$$
and thus
\begin{align*}f(\ell,d)\;:\!=\;\liminf_{K\atop [K:\mathbb{Q}]=d}\frac{-\log M(K,\ell)}{\log D_K}\geq \frac{1}{2\ell(d-1)}.\end{align*}
3. A lower bound for
$M(K,\ell)$
To prove Proposition 1·1 we need a sufficiently good lower bound for
$M(K,\ell)$
. And for this in turn we need to have a good lower bound on the number of small generators
$$N'_K(X)=\#\{\gamma\in K;\; K=\mathbb{Q}(\gamma), H_K(\gamma)\lt X\}.$$
Surprisingly, it suffices to consider rational multiples of a minimal generator, and this gives us the next lemma.
Lemma 3·1. Let K be a number field of degree
$d\gt1$
, and let
$\delta\gt0$
. Then
$N'_K(\eta(K) D_K^\delta)\gg D_K^{2\delta/d}$
.
Proof. Let
$\alpha\in K$
with
$H_K(\alpha)=\eta(K)$
and
$\mathbb{Q}(\alpha)=K$
. Now consider the primitive elements
$\alpha\beta$
with nonzero
$\beta\in \mathbb{Q}$
and
$H_K(\beta)\lt D_K^{\delta/d}$
. Note that
$H_K(\beta)=H_{\mathbb{Q}}(\beta)^d$
. Writing
$\beta=b_1/b_0$
with coprime integers
$b_0\gt0$
and
$b_1$
it follows from the product formula that
$H_{\mathbb{Q}}(\beta)=\max\{b_0,|b_1|\}$
. Hence, there are
$\gg T^2$
non-zero elements
$\beta\in \mathbb{Q}$
with
$H_{\mathbb{Q}}(\beta)\lt T$
if
$T\gt1$
. Applying this with
$T=D_K^{\delta/d}\gt1$
we conclude that there are
$\gg D_K^{2\delta/d}$
of these elements
$\beta\in \mathbb{Q}$
. Using the trivial estimate
$H_K(\alpha\beta)\leq H_K(\alpha)H_K(\beta)$
we conclude that
$N'_K(\eta(K) D_K^\delta)\gg D_K^{2\delta/d}$
.
Now we can prove a lower bound for
$M(K,\ell)$
.
Lemma 3·2. Let K be a number field of degree
$d\gt1$
and suppose
$\ell\geq d/2$
. Then
$$M(K,\ell)\gg \eta(K)^{-1/\ell}.$$
Proof. If
$X\leq \eta(K)$
then
$N'_K(X)=0$
. Hence,
\begin{equation}\inf_{X\leq \eta(K)}(X^{-1/\ell}(1+N'_K(X)))=\eta(K)^{-1/\ell}.\end{equation}
If
$X\gt\eta(K)$
then we can write
$X=\eta(K)D_K^\delta$
for some
$\delta\gt0$
, and thus
\begin{equation}\inf_{X\gt \eta(K)}(X^{-1/\ell}(1+N'_K(X)))=\inf_{\delta\gt0}\left((\eta(K)D_K^\delta)^{-1/\ell}(1+N'_K(\eta(K)D_K^\delta))\right).\end{equation}
Plugging the bound from Lemma 3·1 into (3·2) shows that
\begin{align*}\inf_{X\gt \eta(K)}(X^{-1/\ell}(1+N'_K(X)))\gg \inf_{\delta\gt0}((\eta(K)D_K^\delta)^{-1/\ell}D_K^{2\delta/d})=\eta(K)^{-1/\ell}\inf_{\delta\gt0}D_K^{\delta(2/d-1/\ell)}.\end{align*}
Since
$\ell \geq d/2$
we have
$\inf_{\delta\gt0}D_K^{\delta(2/d-1/\ell)}=1$
, and thus
\begin{equation}\inf_{X\gt \eta(K)}(X^{-1/\ell}(1+N'_K(X)))\gg \eta(K)^{-1/\ell}.\end{equation}
4. Proof of Proposition 1·1
We have already seen that
$f(\ell,d)\geq 1/(2\ell(d-1))$
. Let us now suppose that
$\ell\geq d/2$
. From Lemma 3·2 we know that
$-\log M(K,\ell)\leq ({1}/{\ell})\log \eta(K) -\log C$
for some absolute constant
$C\gt0$
. Hence,
\begin{equation}f(\ell,d)=\liminf_{K\atop [K:\mathbb{Q}]=d}\frac{-\log M(K,\ell)}{\log D_K}\leq \liminf_{K\atop [K:\mathbb{Q}]=d}\frac{\log \eta(K)}{\ell \log D_K}.\end{equation}
The family of degree d-fields
$K=\mathbb{Q}(a^{1/d})$
as in (1·5) with
$a=A_1 A_{d-1}^{d-1}$
and
$A_{d-1}\leq A_1\leq 2 A_{d-1}$
have a generator
$\alpha=(A_1/A_{d-1})^{1/d}$
. Now
$H_K(\alpha)=H_K(\alpha^d)^{1/d}=H_\mathbb{Q}(\alpha^d)=A_1$
, where we used that
$A_1$
and
$A_{d-1}$
are coprime. Hence,
$$\eta(K) \leq H_K(\alpha)=A_1\leq \sqrt{2A_1A_{d-1}}\leq \sqrt{2}D_K^{1/(2(d-1))}.$$
Plugging the above estimate into (4·1) for this infinite family of fields shows that
$f(\ell,d)\leq 1/(2\ell(d-1))$
. This completes the proof of Proposition 1·1.
5. Constructing suitable primes
The following lemma was observed by Heath-Brown (and possibly others including Jiuya Wang) but stated only for
$d=3$
. For the convenience of the reader we give all details for general odd d. Recall that K is a pure field of degree d if
$K=\mathbb{Q}(\theta)$
for a root
$\theta$
of some irreducible
$f(x)\in \mathbb{Z}[x]$
of the form
$f(x)=x^d-a$
.
Lemma 5·1 Let K be a pure field of degree
$d\gt1$
, and let
$\delta\gt\varepsilon\gt0$
. Then there are
$\gg_{\varepsilon,d} D_K^{\delta-\varepsilon}$
many prime ideals
$\mathfrak{p}|p$
in
$\mathcal{O}_K$
of degree
$f(\mathfrak{p}/p)=1$
, ramification index
$e(\mathfrak{p}/p)=1$
, and norm
$N(\mathfrak{p})\lt D_K^{\delta}$
.
Proof. Let K be a pure field of odd degree d. Hence there is a non-zero integer m, free of dth powers, such that
$f(x)=x^d-m$
is irreducible in
$\mathbb{Z}[x]$
,
$K=\mathbb{Q}(\theta)$
and
$f(\theta)=0$
.
Note that the discriminant of f has modulus
$|\Delta_f|=d^dm^{d-1}$
. Let
$p|m$
be a prime divisor and
$p^a$
the maximal power dividing m. Thus
$1\leq a\leq d-1$
. Let
$\mathfrak{p}\subset \mathcal{O}_K$
be a prime ideal above p. Since
$m=\theta^d$
it follows that
$\mathfrak{p}|(\theta)$
and therefore
$\mathfrak{p}^d|p^a$
. Hence,
$d\leq a\cdot e(\mathfrak{p}/p)$
. This shows that p ramifies in K.
The total number of primes p that ramify in K is
$\omega(D_K)\ll_\varepsilon D_K^\varepsilon$
. Next we show that for any unramified prime
$p\equiv 2 \pmod d$
there exists a prime ideal
$\mathfrak{p}|p$
in
$\mathcal{O}_K$
of degree
$f(\mathfrak{p}/p)=1$
, so that the claim follows from Dirichlet’s prime number theorem.
So let p be a prime with
$p\equiv 2 \pmod d$
and
$p\nmid m$
. Let g be a generator of the cyclic group
$\mathbb{F}_p^\times$
. Reducing the polynomial f modulo p gives
$\overline{f}(x)=x^d-g^t$
for some integer t. Hence
$\overline{f}$
has a root
$g^s$
in
$\mathbb{F}_p^\times$
if and only if
$sd\equiv t \pmod {p-1}$
. The latter has a solution s if and only if
$(d,p-1)|t$
which is true as
$p\equiv 2 \pmod d$
. Noting that
$p\nmid [\mathcal{O}_K\;:\;\mathbb{Z}[\theta]]$
and applying Dedekind’s factorisation theorem completes the proof.
6. A height lower bound for generators of pure fields and proof of Proposition 1·3
Ellenberg and Venkatesh used Silverman’s classical lower bound for generators
$\alpha$
of K
\begin{equation}H_K(\alpha)\gg_d D_K^{1/(2(d-1))}\end{equation}
to get (1·1). Dubickas [
Reference Dubickas3
, theorem 1] has improved (6·1) for pure fields
$K=\mathbb{Q}(a^{1/d})$
of odd degree d, provided a is prime. Following his proof and implementing the obvious minor modifications provides a lower bound for general a that improves (6·1) in some new cases, e.g., when a is squarefree.
For the next result let
$K=\mathbb{Q}(a^{1/d})$
be a pure field of odd degree
$d\geq 3$
with
$a\gt1$
. Recall that we can assume
$$a=\prod_{i=1}^{d-1}A_i^i,$$
where the positive integers
$A_1,\ldots,A_{d-1}$
are squarefree and pairwise coprime.
Proposition 6·1 (Dubickas, 2023). Suppose
$\alpha\in K$
and
$K=\mathbb{Q}(\alpha)$
. Then
\begin{equation}H_K(\alpha)\gt C_d \min_{\frac{d+1}{2}\leq m\leq d-1}\left(\prod_{i=1}^{d-1}A_i^{\frac{i\cdot m}{d}-\lfloor \frac{i\cdot m}{d}\rfloor}\right).\end{equation}
One can take
$C_d=d^{-(2d-1)}$
.
Proof. For the convenience of the reader we reproduce Dubickas’ proof [ Reference Dubickas3 , section 4, ‘proof of theorem 1’] with the necessary slight modification in the final step. Let
$$\alpha=b_0+b_1 a^{1/d}+\cdots +b_{m}a^{m/d},$$
where
$m\in \{1,2,\ldots,d-1\}$
,
$b_0,\ldots, b_m\in \mathbb{Q}$
and
$b_m\neq 0$
. Replacing
$\alpha$
by
$\alpha^{-1}$
(which does not change the height), we can assume that
$m\geq (d+1)/2$
(see [
Reference Dubickas3
, line after (4·3)]). Let
$\zeta=e^{2\pi i/d}$
so that the d conjugates
$\alpha_j$
(
$1\leq j\leq d$
) satisfy
$$\alpha_j=\sum_{k=0}^m b_ka^{k/d}\zeta^{(j-1)k}.$$
By [
Reference Dubickas3
, lemma 7] there are
$X_1,\ldots,X_{m+1}\in F=\mathbb{Q}(\zeta)$
with
$d^mX_j\in \mathcal{O}_F$
, and
\begin{equation}X_1\alpha_1+\cdots + X_{m+1}\alpha_{m+1}=b_ma^{m/d}\end{equation}
and moreover,
\begin{equation}|X_j|\leq \frac{1}{(2\sin(\pi/d))^m} \quad (1\leq j\leq m+1).\end{equation}
We consider the
$X_j$
and the conjugates
$\alpha_i$
as complex numbers and
$|\cdot|$
denotes the standard absolute value on
$\mathbb{C}$
. Let T be the leading coefficient of the minimal polynomial of
$\alpha$
in
$\mathbb{Z}[x]$
. Note that
$T\alpha_j$
and
$d^m X_j$
are algebraic integers and therefore also
$$d^mT(X_1\alpha_1+\cdots + X_{m+1}\alpha_{m+1})=d^mTb_ma^{m/d}$$
is an algebraic integer. Now
$d^mTb_m$
is a non-zero rational number, say
$D_0/D$
with
$(D_0,D)=1$
and
$D\geq 1$
.
Combining (6·3) and (6·4) gives
$$a^{m/d}/D\leq d^mT|b_m|a^{m/d}\leq \frac{(m+1)d^mT\max_{1\leq j\leq m+1}|\alpha_j|}{(2\sin(\pi/d))^m}.$$
Since the height
$H_K(\alpha)$
is equal to the Mahler measure of the minimal polynomial of
$\alpha$
(see [
Reference Bombieri and Gubler1
, proposition 1·6·6]) we get
$$H_K(\alpha)=T\prod_{j=1}^d\max\{1,|\alpha_j|\}\geq T\max_{1\leq j\leq m+1}{|\alpha_j|}\geq\frac{(2\sin(\pi/d))^m}{(m+1)d^m}\cdot \frac{a^{m/d}}{D}\gt C_d\cdot \frac{a^{m/d}}{D}.$$
Next we claim that
$$D\leq \prod_{p|a}p^{\lfloor\frac{m\cdot \text{ord}_p(a)}{d}\rfloor}.$$
Now
$D_0 a^{m/d}/D$
and
$a^{(d-m)/d}$
are both algebraic integers. Therefore, also their product
$aD_0/D$
is an algebraic integer. Since
$D_0$
and D are coprime it follows that each prime divisor p of D must also divide a. Taking dth powers and recalling that
$D_0a^{m/d}/D$
is an algebraic integer implies
$d \cdot \text{ord}_p(D)\leq m\cdot \text{ord}_p(a)$
, proving the claim.
Finally, we note that for
$p|A_i$
we have
$\text{ord}_p(A_i)=i$
, and hence,
$$\frac{a^{m/d}}{D}\geq\prod_{p|a}p^{\frac{m\cdot \text{ord}_p(a)}{d}-\lfloor\frac{m\cdot \text{ord}_p(a)}{d}\rfloor}\geq \prod_{i=1}^{d-1}A_i^{\frac{i\cdot m}{d}-\lfloor \frac{i\cdot m}{d}\rfloor}.$$
Recalling that
$(d+1)/2\leq m\leq d-1$
this completes the proof of the lemma.
First we recall that (1·3) is an immediate consequence of Lemma 2·1 and Lemma 5·1.
To prove Proposition 1·2 we use the stronger form of the Key-Lemma (Lemma 2·1), using the invariant
$\eta(K)$
, so that we can replace the hypothesis
$\delta\lt1/(2\ell(d-1))$
by
$\delta\lt\gamma/\ell$
as long as
$\eta(K)\gt D_K^\gamma$
.
We set
$$\mathcal{A}\;:\!=\;C_d \min_{\frac{d+1}{2}\leq m\leq d-1}\left(\prod_{i=1}^{d-1}A_i^{\frac{i\cdot m}{d}-\lfloor \frac{i\cdot m}{d}\rfloor}\right).$$
Define
$\gamma$
by
$\mathcal{A}=D_K^\gamma$
so that
$\eta(K)\gt D_K^\gamma$
by Proposition 6·1. If
$\gamma\leq 1/(2(d-1))$
then the inequality claimed in Proposition 1·3 follows from (1·3). Hence, we can assume
$\gamma\gt1/(2(d-1))$
. Next let
$0\lt\varepsilon\lt\gamma/\ell$
and set
$\delta=\gamma/\ell-\varepsilon$
. Applying the Key-Lemma (Lemma 2·1) and using Lemma 5·1 gives
\begin{align*}\#Cl_K[\ell]\ll_{\varepsilon,\ell,d}D_K^{1/2+\varepsilon}D_K^{-(\delta-\varepsilon)}=D_K^{1/2+3\varepsilon}\mathcal{A}^{-1/\ell}.\end{align*}
This concludes the proof.
Acknowledgements
We thank the referee for their careful reading of the manuscript, and for providing useful comments that improved the exposition of this paper.
