A Heuristic approach to the Iwasawa theory of elliptic curves

Problem Statement

The paper addresses Greenberg's $\mu = 0$ conjecture within the Iwasawa theory of elliptic curves. Let $E/\mathbb{Q}$ be an elliptic curve and $p$ an odd prime where $E$ has good ordinary reduction. The conjecture posits that the $p$-primary Selmer group $S_{p^\infty}(E/\mathbb{Q}^{\text{cyc}})$ over the cyclotomic $\mathbb{Z}_p$-extension is cofinitely generated as a $\mathbb{Z}_p$-module. This condition is equivalent to the vanishing of the $\mu$-invariant of the Selmer group. While proven for abelian extensions of $\mathbb{Q}$ by Ferrero and Washington, the conjecture remains open for elliptic curves.

The authors investigate the conjecture from a statistical perspective, specifically testing Conjecture 1.2, which asserts that for a fixed odd prime $p$, "most" elliptic curves with good ordinary reduction at $p$ satisfy $\mu_p(E) = 0$. The central challenge is to provide heuristic evidence for this average vanishing, extending the probabilistic frameworks established by Cohen-Lenstra for class groups and Poonen-Rains for Selmer groups.

Methodology

The authors extend the heuristics of Poonen and Rains [PR12], who modeled the distribution of $p$-Selmer groups via the intersection of maximal isotropic subspaces in an inner product space. The methodology proceeds through the following steps:

1. Reduction to Residual Modules: The vanishing of the $\mu$-invariant for the full Selmer group is linked to the finiteness of the residual Greenberg Selmer group, $\text{Sel}^{Gr}(E[p]/\mathbb{Q}^{\text{cyc}})$. This group is analyzed via the quotient $\Pi_E := \text{Sel}^{Gr}(E[p]/\mathbb{Q}^{\text{cyc}}) / R(E[p]/\mathbb{Q}^{\text{cyc}})$, where $R$ denotes the fine Selmer group.
2. Geometric Interpretation: The authors define a space $V(E)$ equipped with a non-degenerate, symmetric, bilinear pairing over the ring $\Omega = \mathbb{F}_p[[T]]$. They identify $\Pi_E$ with the intersection of two specific $\Omega$-submodules, $V_1(E)$ and $V_2(E)$, within $V(E)$.
   • $V_1(E)$ is the image of the restriction map from global cohomology, shown to be a maximal isotropic submodule.
   • $V_2(E)$ is defined by local conditions at $p$ (related to the formal group), shown to be isomorphic to $\Omega^\vee$ (the Pontryagin dual of $\Omega$) plus a finite torsion part.
3. Projection to Cyclic Submodules: The problem of the finiteness of $V_1(E) \cap V_2(E)$ is reduced to studying the intersection of two submodules $M_1$ and $M_2$ of $(\Omega^\vee)^2$, where $M_1, M_2 \cong \Omega^\vee$. This is further dualized to studying the intersection of two maximal cyclic submodules $N_1, N_2$ of $\Omega^2$.
4. Probabilistic Model: The authors construct a probability space $\mathcal{M}$ of pairs of maximal cyclic submodules $(N_1, N_2)$ in $\Omega^2$. This space is defined as the inverse limit of spaces $\mathcal{M}_n$ of pairs of maximal submodules in $\Omega_n^2 = (\Omega/(T^n))^2$, equipped with uniform probability measures $P_n$.
5. Heuristic Assumption: The core heuristic (Heuristic 1.3) assumes that as elliptic curves vary over those with good ordinary reduction at $p$, the associated pairs $(M_1, M_2)$ are equidistributed with respect to the natural probability measure $P$ on $\mathcal{M}$.

Key Contributions and Results

• Theoretical Framework: The paper establishes a precise link between the arithmetic of the $\mu$-invariant and the geometric intersection of submodules in a specific inner product space over the Iwasawa algebra modulo $p$.
• Probabilistic Calculation: The authors prove Theorem 4.8, which states that for a pair of maximal cyclic submodules $(N_1, N_2)$ chosen uniformly at random from the space $\mathcal{M}$, the probability that their intersection is trivial (i.e., $N_1 \cap N_2 = 0$) is 1.
  • The proof relies on showing that $N_1 \cap N_2 \neq 0$ implies $N_1 = N_2$ (Lemma 4.6).
  • The probability that two randomly chosen maximal submodules in the finite quotient $\Omega_n^2$ are identical is calculated as $1/((p+1)p^{n-1})$, which tends to 0 as $n \to \infty$.
• Main Theorem: Under the assumption of Heuristic 1.3 and Coates and Sujatha's Conjecture A (which predicts the vanishing of the $\mu$-invariant for the fine Selmer group), the authors prove Theorem A:
  $$\lim_{x \to \infty} \frac{\#S_p(x)}{\#C(x)} = 0$$
  where $C(x)$ is the set of minimal Weierstrass models of height $\le x$, and $S_p(x)$ is the subset of curves with good ordinary reduction at $p$ for which the $\mu$-invariant of the Selmer group is non-zero.

Significance and Claims

The paper claims to provide further heuristic evidence for Greenberg's conjecture in the "average" sense. By demonstrating that the event of the $\mu$-invariant being non-zero corresponds to a probability zero event in the constructed random model, the authors support the conjecture that $\mu_p(E) = 0$ for almost all elliptic curves with good ordinary reduction at $p$.

The authors explicitly state that their results suggest a "broader validity of the conjecture on the average vanishing of the $\mu$-invariant." They position this work as an enrichment of the interplay between arithmetic statistics and Iwasawa theory, noting that such investigations remain a "fertile ground for future research." The paper does not claim to prove the conjecture for any specific curve or to resolve the general case without the heuristic assumption; rather, it offers a statistical justification for the conjecture's truth on average, consistent with the approach of Poonen and Rains.