Greenberg's $\mu = 0$ conjecture for elliptic curves over $\mathbb{Q}$: vanishing of the Iwasawa $\mu$-invariant of Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension

Greenberg's 1999 conjecture predicts that for every elliptic curve $E/\mathbb{Q}$ and every odd prime $p$ of good ordinary reduction with $E[p]$ irreducible as a Galois module, the Iwasawa $\mu$-invariant of the Pontryagin dual of the $p$-primary Selmer group over the cyclotomic $\mathbb{Z}_p$-extension $\mathbb{Q}^{\mathrm{cyc}}$ vanishes — equivalently, the Selmer group is cofinitely generated over $\mathbb{Z}_p$ and its order along the tower grows at most like $p^{\lambda n + O(1)}$ rather than $p^{\mu p^{n}}$. Under the cyclotomic Iwasawa main conjecture (proved by Rubin in the complex-multiplication case, and by Kato and Skinner–Urban under mild auxiliary hypotheses otherwise), the conjecture is equivalent to the Mazur–Swinnerton-Dyer $p$-adic $L$-function $L_p(E,T)$ not being divisible by $p$. The strongest general result toward it, by Chakravarthy (2024, corrected version), gives only $\mu \leq 1$ for all but finitely many primes of good ordinary reduction, ineffectively. The problem is to prove or refute the conjecture.

Background

Classical Iwasawa theory studies how arithmetic invariants grow in infinite towers of number fields. Given a $\mathbb{Z}_p$-extension $K_\infty/K$ — a Galois extension whose Galois group is isomorphic to the additive group $\mathbb{Z}_p$ of $p$-adic integers, with intermediate layers $K_n$ of degree $p^n$ over $K$ — Iwasawa proved that the exponent $e_n$ of $p$ in the class number of $K_n$ satisfies $e_n = \mu p^n + \lambda n + \nu$ for integers $\mu$, $\lambda$, $\nu$ and all sufficiently large $n$, and he conjectured that for the cyclotomic $\mathbb{Z}_p$-extension (the unique $\mathbb{Z}_p$-extension contained in $K(\zeta_{p^{\infty}})$, where $\zeta_{p^{\infty}}$ denotes a compatible system of $p$-power roots of unity) of any number field one always has $\mu = 0$. This is known only for abelian base fields, by Ferrero and Washington, and the source paper calls it the deepest open problem of classical Iwasawa theory.

The present problem is the elliptic-curve analogue of that conjecture. An elliptic curve $E/\mathbb{Q}$ is a smooth projective genus-one curve carrying a commutative group law on its points, given by a Weierstrass equation; its arithmetic is organized by the Mordell–Weil group $E(\mathbb{Q})$ (the finitely generated abelian group of rational points) and the Tate–Shafarevich group (the group measuring the failure of the local-to-global principle for rational points). An odd prime $p$ is a prime of good ordinary reduction for $E$ if $p$ does not divide the conductor $N_E$ (so the reduction modulo $p$ is again an elliptic curve) and the Frobenius trace $a_p(E)$ is not divisible by $p$. Let $\mathbb{Q}^{\mathrm{cyc}}$ be the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$, with finite layers $\mathbb{Q}_n$. The $p$-primary Selmer group $Sel(E/\mathbb{Q}^{\mathrm{cyc}})[p^\infty]$ — the subgroup of the Galois cohomology group $H^1(\mathbb{Q}^{\mathrm{cyc}}, E[p^\infty])$ defined by the usual unramified and local-divisibility conditions, which controls both the Mordell–Weil and Tate–Shafarevich groups along the tower — is the central object. A deep theorem of Kato shows that its Pontryagin dual $X(E/\mathbb{Q}^{\mathrm{cyc}})$ is finitely generated and torsion over the Iwasawa algebra $\Lambda = \mathbb{Z}_p[[T]]$; by the structure theorem for such modules it is pseudo-isomorphic to a finite direct sum of quotients $\Lambda/(f_i(T)^{a_i})$ and of quotients $\Lambda/(p^{m_j})$, together with a finite module. Its Iwasawa invariants are $\mu_E^{\mathrm{alg}}$, the largest power of $p$ dividing the characteristic ideal (equivalently, the number of $\Lambda/(p)$-type quotient factors), and $\lambda_E^{\mathrm{alg}}$, the number of characteristic-power zeros counted with multiplicity; they govern the growth law $|Sel(E/\mathbb{Q}_n)[p^\infty]| = p^{\mu_E^{\mathrm{alg}} p^n + \lambda_E^{\mathrm{alg}} n + O(1)}$. Thus $\mu_E^{\mathrm{alg}} = 0$ holds exactly when the Selmer group is cofinitely generated over $\mathbb{Z}_p$, that is, when its order grows at most like $p^{O(n)}$ instead of $p^{cp^n}$.

Greenberg conjectured in 1999 that $\mu_E^{\mathrm{alg}} = 0$ for every elliptic curve $E/\mathbb{Q}$ and every odd prime $p$ of good ordinary reduction for which $E[p]$ — the group of $p$-torsion points, a two-dimensional representation of the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ over the field with $p$ elements — is irreducible as a Galois module; irreducibility is equivalent to the absence of a rational $p$-isogeny and, by a theorem of Serre, holds for all but finitely many $p$ whenever $E$ does not have complex multiplication (i.e. its endomorphism ring is exactly $\mathbb{Z}$). The conjecture has an analytic counterpart. The Mazur–Swinnerton-Dyer $p$-adic $L$-function $L_p(E,T) \in \mathbb{Z}_p[[T]]$, constructed from modular symbols and interpolating the special values $L(E,\chi)$ of the complex $L$-function at Dirichlet characters $\chi$ of $p$-power conductor, has its own $\mu$-invariant $\mu_E^{\mathrm{an}}$. Kato proved one divisibility of the cyclotomic Iwasawa main conjecture — the statement identifying the characteristic ideal of $X(E/\mathbb{Q}^{\mathrm{cyc}})$ with the ideal generated by $L_p(E,T)$ — giving $\mu_E^{\mathrm{alg}} \leq \mu_E^{\mathrm{an}}$; Rubin proved the full main conjecture for curves with complex multiplication; and Skinner–Urban proved the converse divisibility under mild auxiliary conditions, so that under those conditions the full main conjecture holds and Greenberg's conjecture becomes equivalent to the single analytic statement that $L_p(E,T)$ is not divisible by $p$. In the complementary reducible case, Greenberg–Vatsal proved that $\mu = 0$ whenever $E[p]$ contains a Galois submodule that is unramified at $p$ and odd, or ramified at $p$ and even (the sign referring to the determinant character of the submodule), and Greenberg also formulated a generalized conjecture predicting that after a suitable $\mathbb{Q}$-rational isogeny of $p$-power degree the $\mu$-invariant always vanishes.

The strongest general result toward the conjecture is due to Chakravarthy. A first version (August 2024) claimed $\mu = 0$ for all but finitely many primes of good ordinary reduction; an error in that argument was found by Lei and Gajek-Leonard, and the corrected version (October 2024) proves only $\mu_E^{\mathrm{alg}} \leq 1$ for all but finitely many such primes $p$, for each fixed $E$. The proof expresses the coefficients of $L_p(E,T)$ through modular symbols — cohomology classes attached to paths between cusps on the modular curve $X_0(N_E)$ — and studies their $p$-divisibility using continued fractions; it is ineffective, giving no bound on and no algorithm for the finite exceptional set of primes where $\mu$ could be positive (or where $\mu = 1$). Companion results prove new cases of the conjecture for certain curves of analytic rank zero (analytic rank being the order of vanishing of $L(E,s)$ at $s = 1$), and a heuristic model of Müller and Ray predicts that $\mu = 0$ with probability one; no counterexample is known.

Problem Statement

Prove or refute Greenberg's $\mu = 0$ conjecture: for every elliptic curve $E/\mathbb{Q}$ and every odd prime $p$ of good ordinary reduction such that $E[p]$ is irreducible as a $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-module, determine whether the $\mu$-invariant of $X(E/\mathbb{Q}^{\mathrm{cyc}})$ — the Pontryagin dual of the $p$-primary Selmer group of $E$ over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ — is zero.

A proof must establish $\mu_E^{\mathrm{alg}} = 0$ for every such pair $(E, p)$; under the cyclotomic main conjecture (Kato's divisibility plus the Skinner–Urban converse divisibility under its auxiliary conditions, and Rubin's theorem in the complex-multiplication case) this is equivalent to proving that the Mazur–Swinnerton-Dyer $p$-adic $L$-function $L_p(E,T) \in \mathbb{Z}_p[[T]]$ is never divisible by $p$ for such $(E, p)$. A refutation must exhibit an explicit pair $(E, p)$ satisfying the hypotheses and certify $\mu_E^{\mathrm{alg}} \geq 1$.

References

1. Ralph Greenberg, Iwasawa theory for elliptic curves, Lecture Notes in Mathematics 1716, Springer, 1999, pp. 51–144, DOI: 10.1007/bfb0093453, https://doi.org/10.1007/bfb0093453 (also arXiv:math/9809206, https://arxiv.org/abs/math/9809206)
2. Adithya Chakravarthy, The Iwasawa μ-invariants of Elliptic Curves over ℚ, arXiv:2408.07826 (v2, 2024), DOI: 10.48550/arxiv.2408.07826, https://arxiv.org/abs/2408.07826
3. Adithya Chakravarthy, The Iwasawa μ-invariant of certain elliptic curves of analytic rank zero, arXiv:2405.05871 (2024), DOI: 10.48550/arxiv.2405.05871, https://arxiv.org/abs/2405.05871
4. Kazuya Kato, p-adic Hodge theory and values of zeta functions of modular forms, Astérisque 295 (2004), 117–290, DOI: 10.24033/ast.639, https://doi.org/10.24033/ast.639
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9. Anwesh Ray, Remarks on Greenberg's conjecture for Galois representations associated to elliptic curves, arXiv:2308.06673 (2023), DOI: 10.48550/arxiv.2308.06673, https://arxiv.org/abs/2308.06673
10. Katharina Müller and Anwesh Ray, A Heuristic approach to the Iwasawa theory of elliptic curves, arXiv:2409.15056 (2024), DOI: 10.48550/arxiv.2409.15056, https://arxiv.org/abs/2409.15056
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