1 Introduction
Let
$E/\mathbb {Q}$
be an elliptic curve and fix a prime p of good reduction. Attached to E is the p-primary Selmer group
$\operatorname {Sel}(E/\mathbb {Q}_{\text {cyc}})$
, where
$\mathbb {Q}_{\text {cyc}}$
denotes the cyclotomic
$\mathbb {Z}_p$
-extension of
$\mathbb {Q}$
. This group fits into the exact sequence
$$ \begin{align} 0\rightarrow E(\mathbb{Q}_{\text{cyc}})\otimes \mathbb{Q}_p/\mathbb{Z}_p\rightarrow \operatorname{Sel}(E/\mathbb{Q}_{\text{cyc}})\rightarrow III(E/\mathbb{Q}_{\text{cyc}})\rightarrow 0, \end{align} $$
where
$III $
denotes the p-part of the Shafarevich–Tate group, and therefore encodes many arithmetic properties of E along the cyclotomic line.
If p is a prime of ordinary reduction then
$\operatorname {Sel}(E/\mathbb {Q}_{\text {cyc}})$
is cotorsion as an Iwasawa module (see [Reference Kato6, Theorem 17.4]) and its characteristic ideal is therefore generated by a polynomial
$L_p^{\text {alg}}\in \mathbb {Z}_p[T]$
. The algebraic Iwasawa invariants
$\lambda _p^{\text {alg}}$
and
$\mu _p^{\text {alg}}$
measure the degree and p-divisibility of
$L_p^{\text {alg}}$
, respectively. If
$E[p]$
is irreducible as a
$\operatorname {Gal}(\bar {\mathbb {Q}}/\mathbb {Q})$
-module, a well-known conjecture of Greenberg [Reference Greenberg5, Conjecture 1.11] asserts that
$\mu _p^{\text {alg}}=0$
.
If p is a prime of supersingular reduction then
$\operatorname {Sel}(E/\mathbb {Q}_{\text {cyc}})$
is no longer cotorsion, however Kobayashi [Reference Kobayashi8] introduced signed Selmer groups
$\operatorname {Sel}^\pm (E/\mathbb {Q}_{\text {cyc}})$
which are cotorsion and encode analogous arithmetic data. In particular, the characteristic ideals of
$\operatorname {Sel}^\pm (E/\mathbb {Q}_{\text {cyc}})$
are generated by polynomials
$L_{p,\text {alg}}^{\pm }\in \mathbb {Z}_p[T]$
which have associated pairs of Iwasawa invariants
$\mu _{p,\text {alg}}^\pm $
and
$\lambda _{p,\text {alg}}^{\pm }$
. In the supersingular setting,
$E[p]$
is automatically irreducible and it is similarly conjectured (see [Reference Pollack13, Conjecture 6.3] and [Reference Perrin-Riou12, Conjecture 7.1]) that
$\mu _{p,\text {alg}}^\pm =0$
.
Recently, Chakravarthy [Reference Chakravarthy2, Theorem 1.3] made progress toward Greenberg’s conjecture by proving that
$\mu _p^{\text {alg}}\leq 1$
for all but finitely many primes of good ordinary reduction. In this article, we prove a similar result in the supersingular setting.
Theorem 1.1 Let
$\ell \geq 0$
and
$*\in \{+,-\}$
. Then
$\mu _{p,\text {an}}^*\leq 1$
for all but finitely many good supersingular primes p with
$\lambda _{p,\text {an}}^{*}=\ell $
.
In the above theorem,
$\lambda _{p,\text {an}}^\pm $
and
$\mu _{p,\text {an}}^\pm $
denote the Iwasawa invariants attached to the analytic p-adic L-functions
$L_{p,\text {an}}^\pm \in \mathbb {Z}_p[\![ T]\!]$
defined by Pollack in [Reference Pollack13]. The construction of these p-adic L-functions requires
$a_p=0$
, which is automatically true when
$p> 3$
. The main conjecture of Iwasawa theory in this setting asserts that
$L_{p,\text {an}}^\pm $
and
$L_{p,\text {alg}}^\pm $
generate the same ideal in
$\mathbb {Z}_p[\![ T]\!]$
, and in particular that
$$ \begin{align} \lambda_{p,\text{alg}}^{\pm}=\lambda_{p,\text{an}}^{\pm}\qquad\text{and}\qquad \mu_{p,\text{alg}}^{\pm}=\mu_{p,\text{an}}^{\pm}. \end{align} $$
Thus, Theorem 1.1 provides support for the vanishing of
$\mu _{p,\text {alg}}^\pm $
.
Remark 1.2 The main conjecture is known to hold in many cases: the CM case was established by Pollack and Rubin [Reference Pollack and Rubin15], and Kobayashi [Reference Kobayashi8, Theorem 1.3] proved the containment
$(L_{p,\text {an}}^\pm )\subseteq (L_{p,\text {alg}}^\pm )$
for non CM curves. A proof of the full supersingular main conjecture was recently announced by Burungale, Skinner, Tian, and Wan [Reference Burungale, Skinner, Tian and Wan1, Theorem 1.2].
We henceforth assume (2) and write
$\lambda _p^\pm $
,
$\mu _p^\pm $
to mean either algebraic or analytic invariants. Letting
$r_E$
denote the Mordell–Weil rank of E, Kundu and Ray conjecture [Reference Kundu and Ray7, Conjecture 3.17] that
$\lambda _p^\pm =r_E$
on a density 1 set of good supersingular primes. Assuming this conjecture, the condition on
$\lambda $
-invariants in Theorem 1.1 could be removed and one would have the bound
$\mu _p^\pm \leq 1$
on a density 1 set of good supersingular primes for E.
We remark that if
$r_E=0$
then [Reference Kundu and Ray7, Theorem 3.8] implies that both
$\lambda _p^\pm $
and
$\mu _p^\pm $
vanish for all but finitely many primes p (in fact,
$\operatorname {Sel}^\pm (E/\mathbb {Q}_{\text {cyc}})=0$
for these primes). Thus, the primary contribution of Theorem 1.1 is in providing support for the vanishing of
$\mu _p^\pm $
in the positive rank case. We also note that, under some mild assumptions, it is known [Reference Kundu and Ray7, Lemma 3.3] that
$\lambda _p^\pm \geq r_E$
for all good supersingular primes
$p>2$
, thus the cases where
$\ell <r_E$
in Theorem 1.1 are mostly vacuous.
The crux of Chakravarthy’s proof in the ordinary setting is constructing a bound (which holds for all but finitely many p) on the size of the modular symbols defining the ordinary p-adic L-function. In the supersingular case, the signed p-adic L-functions are defined via a decomposition theorem of Pollack [Reference Pollack13, Theorem 5.6], and in particular they are not as immediately understood in terms of modular symbols. The approach taken here is to instead apply Chakravarthy’s bound to the sequence of Mazur–Tate elements
$\theta _n$
for E (which are defined using modular symbols), where one can show (see Proposition 3.8) that there exists an integer
$n_0$
such that for all but finitely many primes p,
$$ \begin{align} \mu(\theta_n)\leq 1, \quad \text{for all } n\geq n_0. \end{align} $$
The lower bound
$n_0$
depends only on the conductor of E (and not on p). We then relate the Iwasawa invariants of the Mazur–Tate elements to those of the signed p-adic L-functions in order to deduce Theorem 1.1.
The assumption on
$\lambda $
-invariants in Theorem 1.1 comes from the fact that, while one can show that
$\mu (\theta _n)=\mu _p^{*}$
for n large enough of fixed parity (see Proposition 3.1), in this case the lower bound on n in the asymptotic depends on both p and
$\lambda _p^\pm $
. The idea is that if we assume
$\lambda _p^\pm $
does not vary with p then it is possible to take
$p\gg 0$
so that
$\mu (\theta _n)=\mu _p^{*}$
holds for any fixed n, and in particular for the
$n_0$
appearing in (3).
2 Iwasawa invariants
Fix a prime p and let F be a nonzero power series in
$\Lambda =\mathbb {Z}_p[\![ T]\!]$
. By the Weierstrass preparation theorem [Reference Washington18, Theorem 7.3] there are unique nonnegative integers
$\lambda $
and
$\mu $
such that
$$ \begin{align} F=p^{\mu}DU, \end{align} $$
for some distinguished polynomial
$D\in \mathbb {Z}_p[T]$
of degree
$\lambda $
and some
$U\in \Lambda ^{\times }$
. (Recall that a polynomial
$D\in \mathbb {Z}_p[T]$
is called distinguished if
$D\equiv T^{\deg D}\ \mathrm {mod}\ p$
.) In terms of the coefficients of
$F=\sum _{i\geq 0} a_iT^i$
, we have
$$ \begin{align*} \mu &= \min\{\operatorname{ord}_p (a_i)\mid i\geq 0\},\\ \lambda &= \min \{\, i \, \mid \, \operatorname{ord}_p (a_i)=\mu\}. \end{align*} $$
2.1 Refined Iwasawa invariants
Let
$\Gamma \cong \mathbb {Z}_p$
be the Galois group of the cyclotomic
$\mathbb {Z}_p$
-extension of
$\mathbb {Q}$
and let
$\Gamma _n=\Gamma /\Gamma ^{p^n}\cong \mathbb {Z}/p^n\mathbb {Z}$
denote the Galois group of its nth-layer. Let
$\omega _n=(1+T)^{p^n}-1$
and
$\Lambda _n=\Lambda /(\omega _n)$
. Fixing a topological generator
$\gamma \in \Gamma $
, one has isomorphisms
$\Lambda \cong \mathbb {Z}_p[\![ \Gamma ]\!]$
and
$ \Lambda _n\cong \mathbb {Z}_p[\Gamma _n]$
induced by the map
$\gamma \mapsto 1+T$
. Refined Iwasawa invariants are those attached to elements of
$ \Lambda _n$
. We now give two definitions of refined Iwasawa invariants – both useful in different contexts – and then show that they are equivalent.
2.1.1 Definition via the division algorithm
Since
$\Lambda =\displaystyle \lim _{\leftarrow } \Lambda _n$
, for each
$n\geq 0$
there is a projection map
$\pi _n:\Lambda \twoheadrightarrow \Lambda _n,\, F\mapsto F\ \mathrm {mod}\ \omega _n,$
and we can define the Iwasawa invariants of
$\pi _n(F)$
as follows. Since
$\omega _n$
is a distinguished polynomial, the division algorithm for distinguished polynomials in
$\Lambda $
allows us to write
$$ \begin{align*}F=\omega_nQ_n+F_n, \end{align*} $$
for some unique
$Q_n\in \Lambda $
and a polynomial
$F_n\in \mathbb {Z}_p[T]$
of degree
$<p^n$
. Define
$$ \begin{align*} \lambda(\pi_n(F))&=\lambda(F_n),\\\mu(\pi_n(F))&=\mu(F_n). \end{align*} $$
2.1.2 Definition via augmentation ideals
Following [Reference Pollack14] and [Reference Pollack and Weston16], one can define the Iwasawa invariants of
$\theta \in \mathbb {Z}_p[\Gamma _n]$
as follows. For each
$n\geq 1$
, the element
$\gamma _n= \gamma \ \mathrm {mod}\ \Gamma ^{p^n}$
generates
$\Gamma _n$
and we define the
$\mu $
-invariant of
$\theta =\sum _{j=0}^{p^n-1} c_j\gamma _n^j$
by
$$ \begin{align*}\mu(\theta)=\min_{0\leq j\leq p^{n-1}} \operatorname{ord}_p(c_j). \end{align*} $$
For the
$\lambda $
-invariant, let
$\theta '=p^{-\mu (\theta )}\theta \in \mathbb {Z}_p[\Gamma _n]$
and let
$I_n$
be the augmentation ideal of
$\mathbb {F}_p[\Gamma _n]$
. (Thus,
$I_n$
is the ideal generated by the image of
$\gamma _n-1$
in
$\mathbb {F}_p[\Gamma _n]$
.) Since
$\theta '$
has nonzero image under the natural reduction map
$\overline {(\cdot )}:\mathbb {Z}_p[\Gamma _n]\rightarrow \mathbb {F}_p[\Gamma _n]$
and all ideals of
$\mathbb {F}_p[\Gamma _n]$
are powers of
$I_n$
, we can define
$$ \begin{align*}\lambda(\theta)=\operatorname{ord}_{I_n}\overline{\theta'}=\max\{j \mid \overline{\theta'}\in I_n^j\}\in \{0,1,\dots,p^n-1\}. \end{align*} $$
(If
$n=0$
then
$\theta \in \mathbb {Z}_p$
and we define
$\mu (\theta )=\operatorname {ord}_p(\theta )$
and
$\lambda (\theta )=0$
.)
2.1.3 Equivalence of definitions
We now show that the definitions of refined Iwasawa invariants given above agree.
Proposition 2.1 Let
$n\geq 0$
and
$\theta \in \mathbb {Z}_p[\Gamma _n]$
. If
$F\in \mathbb {Z}_p[T]$
is the unique polynomial of degree
$<p^n$
mapping to
$\theta $
under the composition
$$ \begin{align*}\Lambda \twoheadrightarrow \Lambda_n\xrightarrow{\cong}\mathbb{Z}_p[\Gamma_n], \quad T\mapsto \gamma_n-1, \end{align*} $$
then
$\lambda (\theta )=\lambda (F)$
and
$\mu (\theta )=\mu (F)$
. In particular, the Iwasawa invariants defined in Section 2.1.1 and Section 2.1.2 agree.
Proof The case
$n=0$
is clear, so suppose
$n\geq 1$
. Write
$\theta =p^{\mu (\theta )}\theta ' $
for some
$\theta '\in \mathbb {Z}_p[\Gamma _n]$
. Let
$F_{\theta '}\in \mathbb {Z}_p[T]$
be a representative of the image of
$\theta '$
in
$ \Lambda _n$
, so
$F\equiv p^{\mu (\theta )}F_{\theta '}\ \mathrm {mod}\ \omega _n$
. By the division algorithm, we can choose
$F_{\theta '}$
such that
$\deg F_{\theta '}<p^n$
, in which case the degree of F forces the equality
$F=p^{\mu (\theta )}F_{\theta '}$
in
$\mathbb {Z}_p[T]$
. Since
$\theta '\equiv (\gamma _n-1)^{\lambda (\theta )}\theta "\ \mathrm {mod}\ p$
for some
$\theta "\in \mathbb {Z}_p[\Gamma _n]$
, the commutativity of the diagram
where the horizontal maps are
$\gamma _n\mapsto 1+T$
and the vertical maps are reduction mod p, implies that
$F_{\theta '}\equiv T^{\lambda (\theta )}F_{\theta "}\mod p$
for some
$F_{\theta "}\in \mathcal {O}[T]$
of degree
$<p^n$
. Hensel’s lemma [Reference Neukirch11, Section II, Lemma 4.6] now gives a factorization
$F_{\theta '}=DU$
in
$\mathbb {Z}_p[T]$
, where
$D,U\in \mathcal {O}[T]$
are such that
$\deg D=\lambda (\theta )$
and
$D\equiv T^{\lambda (\theta )} \ \mathrm {mod}\ p$
(so D is distinguished), and
$U\equiv F_{\theta "}\mod p$
(so
$U\in \Lambda ^{\times }$
since the constant term of
$F_{\theta "}$
does not vanish mod p by maximality of
$\lambda (\theta )$
). It follows that
$F=p^{\mu (\theta )}DU$
and by uniqueness of the Weierstrass decomposition (4), we obtain
$\mu (\theta )=\mu (F)$
and
$\lambda (\theta )=\lambda (F)$
.
2.1.4 Relating invariants in
$\Lambda $
and
$\Lambda _n$
The following lemma is known in the literature (see [Reference Pollack14, Remark 4.3]), though we outline a proof for completeness.
Lemma 2.2 Let
$n\geq 0$
and
$F\in \Lambda $
. If
$\lambda (F)<p^n$
then the Iwasawa invariants of F and
$\pi _n(F)$
agree.
Proof We may assume
$\mu (F)=0$
. Use the division algorithm to write
$$ \begin{align} F=\omega_nQ_n+F_n. \end{align} $$
If
$\mu (F_n)$
were positive then (5) implies
$$ \begin{align*}F\equiv T^{p^n}Q_n \ \mathrm{mod}\ p, \end{align*} $$
which contradicts the fact that
$\lambda (F)<p^n$
. Hence we must have
$\mu (F_n)=0$
. From (4), we can write
$F=(T^{\lambda (F)}+ pF_0)U$
for some
$F_0\in \mathbb {Z}_p[T]$
of degree
$<\lambda (F)$
. Combining this decomposition with (5) yields
$$ \begin{align*}F_n\equiv T^{\lambda(F)}(U-T^{p^n-\lambda(F)}Q_n)\ \mathrm{mod}\ p, \end{align*} $$
but since U is a unit and
$\lambda (F)<p^n$
,
$U-T^{p^n-\lambda (F)}Q_n$
must also be a unit. It follows that
$\lambda (F_n)=\lambda (F)$
.
3 Bounding the
$\mu $
-invariant
Let
$E/\mathbb {Q}$
be an elliptic curve of conductor
$N_E$
and fix a prime p of good reduction such that
$a_p=0$
.
3.1 Mazur–Tate elements
Let
$L_p^\pm \in \Lambda $
and
$\theta _n\in \mathbb {Q}[T]$
denote the plus/minus p-adic L-functions and Mazur-Tate elements for E, as defined in [Reference Pollack13, Sections 2.9 and 6.15], respectively. The definition of both
$L_p^\pm $
and
$\theta _n$
depend on a choice of complex periods
$\Omega _E^\pm \in \mathbb {C}$
. We henceforth assume that
$\Omega _E^\pm $
are p-cohomological periods for E, in the sense of [Reference Pollack and Weston16, Section 2.2]. The choice of cohomological periods ensures that the coefficients of
$\theta _n$
are p-integral (cf. [Reference Pollack and Weston16, Remark 2.2]), thus we can view each
$\theta _n$
as an element of the localization
$\mathbb {Z}_{(p)}[T]\subseteq \mathbb {Q}[T]$
.
We now relate the Iwasawa invariants of
$\theta _n$
to those of
$L_p^\pm $
by showing that the even Mazur–Tate elements recover the minus invariants of the p-adic L-function, and vice-versa. Let
$q_1=q_0=0$
and define for
$n\geq 2$
the sequence
$$ \begin{align*}q_n=\begin{cases}p^{n-1}-p^{n-2}+\cdots +p-1 & \quad n \text{ even,}\\ p^{n-1}-p^{n-2}+\cdots +p^2-p & \quad n \text{ odd.} \end{cases} \end{align*} $$
Let
$\lambda _p^\pm $
and
$\mu _p^\pm $
denote the Iwasawa invariant of
$L_p^\pm $
.
Proposition 3.1 If
$n\geq 0$
is even (resp., odd) and
$\lambda _p^{-}<p^n-q_n$
(resp.,
$\lambda _p^{+}<p^n-q_n$
), then
$$ \begin{align*} \mu(\theta_n) &= \mu_p^\pm,\\ \lambda(\theta_n)&=\lambda_p^\pm +q_n, \end{align*} $$
where
$\pm $
is opposite the parity of n.
Proof This follows from the argument of [Reference Gajek-Leonard4, p. 3], which we reproduce in brief here. Let
$\varepsilon _n=\operatorname {sgn}(-1)^n$
denote the parity of n. By [Reference Pollack13, Proposition 6.18], we have
$$ \begin{align} \theta_n \equiv \omega_n^{-\varepsilon_n}L_p^{-\varepsilon_n}\ \mathrm{mod}\ \omega_n. \end{align} $$
Here
$$ \begin{align*}\omega_n^+=\prod_{\underset{i \text{ even}}{1\leq i\leq n}}\Phi_{p^i}(1+T)\qquad \text{and}\qquad \omega_n^-=\prod_{\underset{i \text{ odd}}{1\leq i\leq n}}\Phi_{p^i}(1+T), \end{align*} $$
where
$\Phi _{p^i}(T)$
is the
$p^i$
th cyclotomic polynomial. Since
$\lambda (\Phi _{p^n}(1+T))=p^n-p^{n-1}$
, we have
$\lambda \big (\omega _{n}^{-\varepsilon _n}L_p^{-\varepsilon _n}\big )=q_n+\lambda _p^{-\varepsilon _n}.$
As the sequence
$p^n-q_n$
tends to infinity, we may therefore take n large enough so that
$\lambda \big (\omega _{n}^{-\varepsilon _n}L_p^{-\varepsilon _n}\big )<p^n$
. The result now follows from Lemma 2.2.
Remark 3.2 The formula for
$\lambda $
-invariants in Proposition 3.1 can also be found in [Reference Perrin-Riou12, Section 5], [Reference Pollack and Weston16, Theorem 4.1], and [Reference Sprung17, Corollary 8.9], where it is instead deduced from a 3-term compatibility relation (see [Reference Pollack and Weston16, Proposition 2.5], for example) satisfied by Mazur–Tate elements.
We now fix an integer
$\ell \geq 0$
and let
$X^\pm (E,\ell )$
denote the set of all good supersingular primes
$p>3$
for which
$\lambda _{p}^\pm =\ell $
.
Remark 3.3 It is conjectured [Reference Kundu and Ray7, Conjecture 3.17] that
$\lambda _p^\pm $
coincides with the Mordell–Weil rank on a density 1 set of good supersingular primes, thus one expects
$X^\pm (E,\ell )$
to have density 0 except when
$\ell =r_E$
.
Corollary 3.4 Fix
$n\geq 1$
. For all but finitely many
$p\in X^\pm (E,\ell )$
, we have
$$ \begin{align*} \mu_p^\pm&=\mu(\theta_n) \end{align*} $$
where
$\pm $
is opposite the parity of n.
Proof Since
$\lambda _p^\pm = \ell $
, we can take p large enough so that
$\lambda _p^\pm <p^n-q_n$
. For such p, Proposition 3.1 yields the desired result.
3.2 Modular symbols
Let f be the cuspidal newform attached to E via modularity. For
$r\in \mathbb {Q}$
, recall the modular symbols of [Reference Mazur and Tate9] defined by
$$ \begin{align*} [r]^{\pm} = \frac{\pi i}{\Omega_E^{\pm}} \bigg(\int_{r}^{i\infty}f(z)dz \pm \int_{-r}^{i\infty}f(z)dz\bigg). \end{align*} $$
For odd primes p we have by definition (see [Reference Pollack13, Definition 6.15])
$$ \begin{align*} \theta_n = \sum_{a\in (\mathbb{Z}/p^{n+1}\mathbb{Z})^{\times}}[a/p^{n+1}]^{+}\gamma_n^{\log_{\gamma} a}\in \mathbb{Z}_p[\Gamma_n]. \end{align*} $$
Here
$0\leq \log _\gamma (a)\leq p^{n}-1$
is the unique integer for which
$a\equiv \omega (a)(1+p)^{ \log _\gamma (a)}\ \mathrm {mod}\ p^{n+1}$
, where
$\omega :(\mathbb {Z}/p^{n+1}\mathbb {Z})^{\times }\rightarrow \mathbb {Z}_p^{\times }$
is the mod p cyclotomic character (sending
$a\ \mathrm {mod}\ p^{n+1}$
to the
$(p-1)$
st root of unity
$\omega (a)\in \mathbb {Z}_p^{\times }$
with
$\omega (a)\equiv a\ \mathrm {mod}\ p$
.) Write
$\boldsymbol{\mu }_{p-1}$
for the set of
$(p-1)$
st roots of unity in
$\mathbb {Z}_p^{\times }$
, and if
$\alpha \in \mathbb {Z}_p$
define
$[\alpha /p^{n}]^\pm =[a/p^{n}]^\pm $
where
$\alpha \equiv a\ \mathrm {mod}\ p^n$
.
Lemma 3.5 Let p be an odd prime of good reduction. For any
$n\geq 0$
we have
$$ \begin{align*}\mu(\theta_n)=\displaystyle\min_{0\leq j\leq p^{n}-1} \operatorname{ord}_p\bigg(\sum_{\eta\in \boldsymbol{\mu}_{p-1}}\bigg[\frac{\eta (1+p)^{j}}{p^{n+1}}\bigg]^+\bigg). \end{align*} $$
Proof Since
$\log _\gamma (a)=\log _\gamma (b)$
if and only if
$a\omega (a)^{-1}=b\omega (b)^{-1}\ \mathrm {mod}\ p^{n+1}$
, we have
$$ \begin{align} \theta_n=\sum_{j=0}^{p^n-1} \sum_{\eta\in \boldsymbol{\mu}_{p-1}}\bigg[\frac{\eta (1+p)^{j}}{p^{n+1}}\bigg]^+\gamma_n^{j}. \end{align} $$
The result now follows from Definition 2.1.2.
Lemma 3.6 Let
$n\geq 0$
and
$C\in \mathbb {R}$
. For all but finitely many primes p, if
$a\in (\mathbb {Z}/p^{n+1}\mathbb {Z})^{\times }$
then
$$ \begin{align*}\bigg|C\bigg[\frac{a}{p^{n+1}}\bigg]^+\bigg|<p. \end{align*} $$
Proof By Chakravarthy’s bound [Reference Chakravarthy2, Proposition 4.1], there are constants
$c_1$
and
$c_2$
depending only on the conductor of E (and not p) such that for any
$x\in \mathbb {Q}$
,
$$ \begin{align*}|[x]^+|\leq c_1+c_2\log(\text{denominator}(x)). \end{align*} $$
The result now follows by letting
$x=a/p^{n+1}$
and taking p large enough so that
$c_1+c_2\log (p^{n+1})<\frac {p}{|C|}$
.
Lemma 3.7 Let p be an odd prime of good reduction. There is a constant
$n_0$
depending only on
$N_E$
such that if
$n\geq n_0$
then
$$ \begin{align*}\sum_{\eta\in \boldsymbol{\mu}_{p-1}}\bigg[\frac{\eta (1+p)^{j}}{p^{n+1}}\bigg]^+\neq0 \end{align*} $$
for some
$0\leq j\leq p^{n}-1$
.
Proof Since p is a good prime, a result of Chinta [Reference Chinta3, Theorem 2] guarantees the existence of an integer
$n_0$
(depending only on
$N_E$
and not on p) such that if
$n\geq n_0$
and
$\chi $
is a Dirichlet character of conductor
$p^n$
then
$L(E,\chi ,1)\neq 0$
. Let
$\chi $
be an even Dirichlet character of p-power order and conductor
$p^n$
with
$n\geq n_0$
. Setting
$v=1+p$
, we now have
$$ \begin{align*} \sum_{j=0}^{p^n-1} \sum_{\eta\in \boldsymbol{\mu}_{p-1}}\chi(\eta v^j)\bigg[\frac{\eta v^{j}}{p^{n+1}}\bigg]^+&=\sum_{a\in (\mathbb{Z}/p^{n+1}\mathbb{Z})^{\times}}\chi(a)\bigg[ \frac{a}{p^{n+1}} \bigg]^+\\ &=\tau(\bar \chi)\frac{L(E,\chi,1)}{\Omega_E^+}\\ &\neq 0. \end{align*} $$
Here
$\tau (\chi )$
is a Gauss sum and the middle equality above is due to [Reference Mazur, Tate and Teitelbaum10, (8.6)]. It follows that there is some
$0\leq j\leq p^{n}-1$
for which
$$ \begin{align*}\sum_{\eta\in \boldsymbol{\mu}_{p-1}}\chi(\eta v^j)\bigg[\frac{\eta v^{j}}{p^{n+1}}\bigg]^+=\chi(v^j)\sum_{\eta\in \boldsymbol{\mu}_{p-1}}\bigg[\frac{\eta v^{j}}{p^{n+1}}\bigg]^+\neq 0, \end{align*} $$
where the middle equality follows from the fact that
$\chi $
has p-power order and
$\eta $
is a
$(p-1)$
st root of unity. The result follows.
3.3 Main result
We now prove our main theorem. First, we give a bound on the
$\mu $
-invariants of Mazur–Tate elements.
Proposition 3.8 There is a constant
$n_0$
depending only on
$N_E$
such that if
$n\geq n_0$
then
$\mu (\theta _n)\leq 1$
for all but finitely many primes p.
Proof Let
$n_0$
be as in Lemma 3.7 and take
$n\geq n_0$
, so that
$$ \begin{align} \sum_{\eta\in \boldsymbol{\mu}_{p-1}}\bigg[\frac{\eta (1+p)^{j}}{p^{n+1}}\bigg]^+\neq0 \end{align} $$
holds for all good primes
$p>2$
and some
$0\leq j \leq p^{n}-1$
. Note the sum in (8) is the jth coefficient of
$\theta _n$
when written in the form (7). In particular, since
$\theta _n\in \mathbb {Z}_{(p)}[T]$
, this sum is a rational number whose denominator is
$d_n$
not divisible by p. Thus
$d_n \sum _{\eta \in \boldsymbol{\mu }_{p-1}}[\eta (1+p)^{j}/{p^{n+1}}]^+$
is a nonzero integer, and from Lemma 3.6 we can take p large enough so that
$$ \begin{align*}\bigg|d_n \sum_{\eta\in \boldsymbol{\mu}_{p-1}}\bigg[\frac{\eta (1+p)^{j}}{p^{n+1}}\bigg]^+\bigg|\leq \sum_{\eta\in \boldsymbol{\mu}_{p-1}}\bigg|d_n\bigg[\frac{\eta (1+p)^{j}}{p^{n+1}}\bigg]^+\bigg|<(p-1)p<p^2. \end{align*} $$
It now follows that
$$ \begin{align*}\operatorname{ord}_p\bigg(\sum_{\eta\in \boldsymbol{\mu}_{p-1}}\bigg[\frac{\eta (1+p)^{j}}{p^{n+1}}\bigg]^+\bigg)=\operatorname{ord}_p\bigg(d_n \sum_{\eta\in \boldsymbol{\mu}_{p-1}}\bigg[\frac{\eta (1+p)^{j}}{p^{n+1}}\bigg]^+\bigg)\leq 1. \end{align*} $$
From Lemma 3.5, we now have that
$\mu (\theta _{n})\leq 1$
.
Remark 3.9 It is interesting to note that the bound in Proposition 3.8 applies to both ordinary and supersingular primes. In particular, if p is an ordinary prime then by [Reference Pollack and Weston16, (4)] we have
$\mu _p=\mu (\theta _n(f_\alpha ))$
for
$n\gg 0$
, where
$\mu _p=\mu (L_{p}^{\text {an}})$
and
$f_\alpha $
is the p-stabilization of f to level
$pN$
at a root
$\alpha $
of the Hecke polynomial
$X^2-a_pX+p$
. It is therefore tempting to try to deduce Chakravarthy’s result [Reference Chakravarthy2, Theorem 1.3] that
$\mu _p\leq 1$
for all but finitely many p from Proposition 3.8, however this does not immediately follow since the Iwasawa invariants of
$\theta _n(E)$
and
$\theta _n(f_\alpha )$
need not always agree (see [Reference Pollack and Weston16, Example 3.4]).
Proof of Theorem 1.1
Fix
$\ell \geq 0$
. It suffices to show that for all but finitely many primes
$p\in X^\pm (E,\ell )$
, we have
$\mu _{p}^\pm \leq 1$
. By Proposition 3.8, there exists an odd integer
$n^+$
and an even integer
$n^-$
, neither of which depends on p, such that
$\mu (\theta _{n^\pm })\leq 1$
holds for all but finitely many p. But by Corollary 3.4, for either choice of sign
$*\in \{+,-\}$
we have
$\mu _p^{*}=\mu (\theta _{n^{*}})$
for all but finitely many
$p\in X^{*}(E,\ell )$
.
Acknowledgements
We are grateful to A. Chakravarthy, J. Hatley, A. Lei, and the anonymous referee for their helpful comments in the preparation of this article.
