Statistics for Iwasawa invariants of elliptic curves

Let $\Gamma\mathrel{\mathop{\ordinarycolon}}=\operatorname{Gal}(\mathbb{Q}^{\operatorname{cyc}}/\mathbb{Q})\simeq\mathbb{Z}_{p}$. The Iwasawa algebra $\Lambda$ is the completed group algebra $\mathbb{Z}_{p}\llbracket\Gamma\rrbracket\mathrel{\mathop{\ordinarycolon}}=\varprojlim_{n}\mathbb{Z}_{p}[\Gamma/\Gamma^{p^{n}}]$. After fixing a topological generator $\gamma$ of $\Gamma$, there is an isomorphism of rings $\Lambda\cong\mathbb{Z}_{p}\llbracket T\rrbracket$, by sending $\gamma-1$ to the formal variable $T$.

Let M be a cofinitely generated cotorsion $\Lambda$-module. The Structure Theorem of $\Lambda$-modules asserts that the Pontryagin dual of M, denoted by $\rm{M}^{\vee}$, is pseudo-isomorphic to a finite direct sum of cyclic $\Lambda$-modules. In other words, there is a map of $\Lambda$-modules

with finite kernel and cokernel. Here, $m_{i}>0$ and $h_{j}(T)$ is a distinguished polynomial (i.e. a monic polynomial with non-leading coefficients divisible by $p$). The characteristic ideal of $\rm{M}^{\vee}$ is (up to a unit) generated by the characteristic element,

The $\mu$-invariant of M is defined as the power of $p$ in $f_{\rm{M}}^{(p)}(T)$. More precisely,

The $\lambda$-invariant of M is the degree of the characteristic element, i.e.

Let $E$ be an elliptic curve over $\mathbb{Q}$ with good reduction at $p$. It shall be assumed throughout that the prime $p$ is odd. Let $N$ denote the conductor of $E$ and set $S$ to denote the set of primes which divide $Np$. Let $\mathbb{Q}_{S}$ be the maximal algebraic extension of $\mathbb{Q}$ which is unramified at the primes $v\notin S$. Set $E[p^{\infty}]$ to be the Galois module of all $p$-power torsion points in $E({}\mkern 3.0mu\overline{\mkern-3.0mu\mathbb{Q}})$.

First, consider the case when $E$ has good (ordinary or supersingular) reduction at $p$. Let $v$ be a prime in $S$. For any finite extension $L/\mathbb{Q}$ contained in $\mathbb{Q}^{\operatorname{cyc}}$, write

where the direct sum is over all primes $w$ of $L$ lying above $v$. Then, the $p$-primary Selmer group over $\mathbb{Q}$ is defined as follows

This Selmer group fits into a short exact sequence

see [5]. Here, $\Sha(E/\mathbb{Q})$ is the Tate-Shafarevich group

Now, define

where $L$ ranges over the number fields contained in $\mathbb{Q}^{\operatorname{cyc}}$ and the inductive limit is taken with respect to the restriction maps. Taking direct limits, the $p$-primary Selmer group over $\mathbb{Q}^{\operatorname{cyc}}$ is defined as follows

As mentioned in the introduction, when $p$ is a prime of good supersingular reduction, the Pontryagin dual of $\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}^{\operatorname{cyc}})$ is not $\Lambda$-torsion. In this case, one studies the plus and minus Selmer groups which we describe below.

Let $E_{/\mathbb{Q}}$ be an elliptic curve with supersingular reduction at $p$. Denote by $\mathbb{Q}_{n}^{\operatorname{cyc}}$ the $n$-th layer in the cyclotomic $\mathbb{Z}_{p}$-extension, with $\mathbb{Q}_{0}^{\operatorname{cyc}}\mathrel{\mathop{\ordinarycolon}}=\mathbb{Q}$. Set $\widehat{E}$ to be the formal group of $E$ over $\mathbb{Z}_{p}$. Let $L$ be a finite extension of $\mathbb{Q}_{p}$ with valuation ring $\mathcal{O}_{L}$, let $\widehat{E}(L)$ denote $\widehat{E}(\mathfrak{m}_{L})$, where $\mathfrak{m}_{L}$ is the maximal ideal in $L$. Write $\mathfrak{p}$ for the (unique) prime above $p$ in $\mathbb{Q}^{\operatorname{cyc}}$, and for the prime above $p$ in every finite layer of the cyclotomic tower. Define the plus and minus norm groups as follows

where $\operatorname{tr}_{n/m+1}\mathrel{\mathop{\ordinarycolon}}\widehat{E}(\mathbb{Q}_{n,\mathfrak{p}}^{\operatorname{cyc}})\rightarrow\widehat{E}(\mathbb{Q}_{m+1,\mathfrak{p}}^{\operatorname{cyc}})$ denotes the trace map with respect to the formal group law on $\widehat{E}$. The completion $\mathbb{Q}^{\operatorname{cyc}}_{\mathfrak{p}}$ is the union of completions $\bigcup_{n\geq 1}\mathbb{Q}^{\operatorname{cyc}}_{n,\mathfrak{p}}$, and set $\widehat{E}^{\pm}(\mathbb{Q}_{\mathfrak{p}}^{\operatorname{cyc}})\mathrel{\mathop{\ordinarycolon}}=\bigcup_{n\geq 1}\widehat{E}^{\pm}(\mathbb{Q}_{n,\mathfrak{p}}^{\operatorname{cyc}})$. Define

where the inclusion

is induced via the Kummer map. For $v\in S\setminus\{p\}$, set $J_{v}^{\pm}(E/\mathbb{Q}^{\operatorname{cyc}})$ to be equal to $J_{v}(E/\mathbb{Q}^{\operatorname{cyc}})$. The plus and minus Selmer groups are defined as follows

For each choice of sign ${\ddagger}\in\{+,-\}$, set $f_{E}^{(p),{\ddagger}}(T)$ to denote the characteristic element of $\operatorname{Sel}_{p^{\infty}}^{{\ddagger}}(E/\mathbb{Q}^{\operatorname{cyc}})^{\vee}$, with $\mu_{p}^{{\ddagger}}(E)$, $\lambda_{p}^{{\ddagger}}(E)$ defined analogously.

Let $E$ be an elliptic curve with good (ordinary or supersingular) reduction at $p\geq 5$. In order to state results in both ordinary and supersingular case at once, we set for the remainder of the article the following notation,

Next, we introduce the fine Selmer group. Let $E$ be an elliptic curve over $\mathbb{Q}$ and $p$ be any odd prime. At each prime $v\in S$, set

The $p$-primary fine Selmer group of $E$ is defined as follows

By the result of Kato mentioned in the introduction, the Pontryagin dual of $\operatorname{Sel}^{0}_{p^{\infty}}(E/\mathbb{Q}^{\operatorname{cyc}})$ is known to be a cotorsion $\Lambda$-module independent of the reduction type at $p$. Further, it is conjectured that this fine Selmer group is a cotorsion $\mathbb{Z}_{p}$-module (i.e. the corresponding $\mu$-invariant vanishes), see [6, Conjecture A].

In what follows, M will be a cofinitely generated cotorsion $\Lambda$-module. Note that $H^{i}(\Gamma,\rm{M})$ is always zero for $i\geq 2$.

When the cohomology groups $H^{0}(\Gamma,\rm{M})$ and $H^{1}(\Gamma,\rm{M})$ are finite, the (classical) Euler characteristic $\chi(\Gamma,\rm{M})$ is defined as the alternating product

On the other hand, when the cohomology groups $H^{i}(\Gamma,\rm{M})$ are not finite, there is a generalized version denoted by $\chi_{t}(\Gamma,\rm{M})$. Since $H^{1}(\Gamma,\rm{M})$ is isomorphic to the group of coinvariants $H_{0}(\Gamma,\rm{M})=\rm{M}_{\Gamma}$, there is a natural map

sending $x\in\rm{M}^{\Gamma}$ to the residue class of $x$ in $\rm{M}_{\Gamma}$. We say that the truncated Euler characteristic is defined if the kernel and cokernel of $\Phi_{\rm{M}}$ are finite. In this case, the $\chi_{t}(\Gamma,\rm{M})$ is defined to be the following quotient,

It is easy to check that when $\chi(\Gamma,\rm{M})$ is defined, so is $\chi_{t}(\Gamma,\rm{M})$. In fact,

Express the characteristic element $f_{\rm{M}}^{(p)}(T)$ as a polynomial,

Let $r_{\rm{M}}$ denote the order of vanishing of $f_{\rm{M}}^{(p)}(T)$ at $T=0$. For $a,b\in\mathbb{Q}_{p}$, we write $a\sim b$ if there is a unit $u\in\mathbb{Z}_{p}^{\times}$ such that $a=bu$.

In particular, the classical Euler characteristic $\chi(\Gamma,\rm{M})$ is defined if and only if $r_{\rm{M}}=0$. When this happens, the constant coefficient $c_{0}\sim\chi(\Gamma,\rm{M})$.

We specialize the discussion on Euler characteristics to Selmer groups of elliptic curves. Let ${\ddagger}$ be a choice of sign and recall the definition of $\operatorname{Sel}_{p^{\infty}}^{{\ddagger}}(E/\mathbb{Q}^{\operatorname{cyc}})$. When $E$ has good ordinary reduction at $p$, the choice of ${\ddagger}$ is irrelevant. Denote by $\chi_{t}^{{\ddagger}}(\Gamma,E[p^{\infty}])$ (resp. $\chi^{{\ddagger}}(\Gamma,E[p^{\infty}])$) the truncated (resp. classical) Euler characteristic of the Selmer group $\operatorname{Sel}_{p^{\infty}}^{{\ddagger}}(E/\mathbb{Q}^{\operatorname{cyc}})$. The invariants $\mu_{p}^{{\ddagger}}(E)$ and $\lambda_{p}^{{\ddagger}}(E)$ simply refer to $\mu_{p}(E)$ and $\lambda_{p}(E)$ when $E$ has good ordinary reduction at $p$. When $E$ has good ordinary reduction at $p$, we shall drop the sign ${\ddagger}$ from the notation. It follows from Lemma 2.2 that the truncated Euler characteristic is always an integer.

In this subsection, we study the variation of the classical Euler characteristic as $p$ varies over primes of good reduction.

Let $E_{/\mathbb{Q}}$ be an elliptic curve. Denote by $S^{\operatorname{bad}}$ the finite set of primes at which $E$ has bad reduction. Denote by $S^{\operatorname{good}}$ the primes at which $E$ has good reduction (i.e., either ordinary reduction or supersingular reduction); write $S^{\operatorname{good}}=S^{\operatorname{ord}}\cup S^{\operatorname{ss}}$. When $E$ has good reduction at a fixed prime $p$, we set $\widetilde{E}$ to denote the reduced curve over $\mathbb{F}_{p}$.

The following result was initially proved by Greenberg for good ordinary primes.

Observe that $\Sigma^{\prime}$ is always a finite set. However, the set of primes $\Sigma$, referred to as the set of anomalous primes, is known to be a set of Dirichlet density zero. The Lang-Trotter Conjecture predicts that the proportion of anomalous primes $p<X$ is $C\cdot\frac{\sqrt{X}}{\log X}$ for some constant $C$. It has recently been shown that all but a density zero set of elliptic curves have infinitely many anomalous primes [1, Corollary 4.3]. However, in special cases one can in fact show that $C=0$ [25, §1.1]. By the Hasse inequality, a prime $p\geq 7$ is anomalous for an elliptic curve defined over $\mathbb{Q}$ if and only if $a_{p}=p+1-\#\widetilde{E}(\mathbb{F}_{p})=1$. When $p\geq 7$ and an elliptic curve has 2-torsion, $a_{p}$ is even and hence $C=0$. For more examples where the set $\Sigma$ is finite, and detailed discussion on this subject, we refer the reader to [18, 20, 23].

We prove a similar result for primes of supersingular reduction.

The following result gives evidence for the conjecture of Coates and Sujatha on the vanishing of the $\mu$-invariant of fine Selmer groups. In fact, more is true.

Next, we consider the case when the elliptic curve $E$ has positive rank.

When $E$ has good ordinary reduction at $p$, there is a notion of the $p$-adic height pairing, which is the $p$-adic analog of the usual height pairing, and was studied extensively in [27, 28]. This pairing is conjectured to be non-degenerate, and the $p$-adic regulator $R_{p}(E/\mathbb{Q})$ is defined to be the determinant of this pairing.

The above theorem implies that the right hand side of the equation is an integer since the left hand side is. This result does not assume that the truncated Euler characteristic is well-defined. The statement is proven in greater generality, in fact, for abelian varieties over number fields.

To analyze Iwasawa invariants on average, we shall apply Theorem 3.11. Let $E_{/\mathbb{Q}}$ be an elliptic curve. As before, we denote by $S^{\operatorname{ord}}$ the set of primes $p$ at which $E$ has good ordinary reduction. Let $\Sigma$ be the set of anomalous primes and $\Sigma^{\prime}$ the finite set of primes for which

1. (1)

   $p=2$,

2. (2)

   $p$ divides $\#\Sha(E/\mathbb{Q})$.

3. (3)

   $p$ divides the Tamagawa product $\prod_{l\in S^{\operatorname{bad}}}c_{l}(E)$.

Denote by $v_{p}$ the $p$-adic valuation on $\mathbb{Q}_{p}$ normalized by $v_{p}(p)=1$. Let $\Pi\subset S^{\operatorname{ord}}$ be the set of primes $p$ at which $v_{p}(R_{p}(E/\mathbb{Q}))\geq r_{E}$. In other words, it is the set of primes for which $p$ divides $\left(\frac{R_{p}(E/\mathbb{Q})}{p^{r_{E}}}\right)$. When $r_{E}=0$, the $p$-adic regulator $R_{p}(E/\mathbb{Q})=1$, therefore the set $\Pi$ is empty. On the other hand, the set of primes $\Pi$ need not be empty when $r_{E}\geq 1$. The following result generalizes Theorem 3.7 and is an easy consequence of Theorem 3.11.

While it is known that the set $\Sigma^{\prime}$ is finite and $\Sigma$ is a density zero set of primes, the same is not known for $\Pi$. For $N\geq 1$, let $\Pi^{\leq N}$ be the set of primes $p\in\Pi$ for which $5\leq p\leq N$. Calculations in [4] suggest that the set $\Pi$ is likely to have Dirichlet density zero, at least for elliptic curves of rank $1$ (see also [36]). We compute $\Pi^{\leq 1000}$ for the first 10 elliptic curves $E/\mathbb{Q}$ of rank $2$ (ordered by conductor). The calculations in the following table were done on sage.