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On the $\mathfrak{M}_H(G)$-property for Selmer groups at supersingular reduction

Sören Kleine, Ahmed Matar, Sujatha Ramdorai

TL;DR

The paper extends the M_H(G) framework to the supersingular setting for elliptic curves, introducing and analyzing signed Selmer groups over Z_p^2-extensions inside the two-dimensional extension 𝕂_∞ of an imaginary quadratic field. It establishes a suite of equivalent criteria for the M_H(G) property, proves a robust control theorem connecting signed Selmer groups to their Λ(H)–reductions, and develops two-variable Iwasawa invariant growth formulas (μ and λ) in Greenberg neighborhoods, including both upper and lower bounds and pseudo-null considerations. The work ties these algebraic results to conjectures of Mazur and Coates–Sujatha, providing conditions under which the M_H(G) property holds or fails and constructing explicit examples, especially in the challenging supersingular environment. Overall, the article delivers new cohomological and algebraic tools to study non-cyclotomic Iwasawa theory for supersingular elliptic curves, with implications for refined Selmer groups and related invariants. The results sharpen our understanding of how μ- and λ-invariants behave in non-cyclotomic Z_p-extensions and illuminate connections between signed Selmer groups, the classical Selmer group, and the fine Selmer group in this delicate setting.

Abstract

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ which has good supersingular reduction at the odd prime $p$. We study the variation of Iwasawa invariants and the $\mathfrak{M}_H(G)$-property for signed Selmer groups over $\mathbb{Z}_p$-extensions of an imaginary quadratic number field $K$ that lie inside the $\mathbb{Z}_p^2$-extension $\mathbb{K}_\infty$ of $K$ and are not necessarily cyclotomic. We prove several equivalent criteria for the validity of the $\mathfrak{M}_H(G)$-property which involve the growth of $μ$-invariants of the signed Selmer groups over intermediate shifted $\mathbb{Z}_p$-extensions in $\mathbb{K}_\infty$, and the boundedness of $λ$-invariants as one runs over $\mathbb{Z}_p$-extensions of $K$ inside $\mathbb{K}_\infty$. We give examples where the $\mathfrak{M}_H(G)$-property holds, and also examples where we can prove that it does not hold. It is striking that although the case of supersingular reduction is much more difficult than the case of ordinary reduction, we get finer results here; moreover, we are able to derive analogous criteria for the validity of the $\mathfrak{M}_H(G)$-property of the classical Selmer group, as well as the fine Selmer group. Many of the properties that we investigate have not been studied before in this non-torsion setting. Further, we study various implications between the $\mathfrak{M}_H(G)$-properties for Selmer groups, signed Selmer groups and fine Selmer groups. We apply our results to a conjecture of Mazur, and prove implications between the $\mathfrak{M}_H(G)$-property and Conjectures A and B of Coates and Sujatha.

On the $\mathfrak{M}_H(G)$-property for Selmer groups at supersingular reduction

TL;DR

The paper extends the M_H(G) framework to the supersingular setting for elliptic curves, introducing and analyzing signed Selmer groups over Z_p^2-extensions inside the two-dimensional extension 𝕂_∞ of an imaginary quadratic field. It establishes a suite of equivalent criteria for the M_H(G) property, proves a robust control theorem connecting signed Selmer groups to their Λ(H)–reductions, and develops two-variable Iwasawa invariant growth formulas (μ and λ) in Greenberg neighborhoods, including both upper and lower bounds and pseudo-null considerations. The work ties these algebraic results to conjectures of Mazur and Coates–Sujatha, providing conditions under which the M_H(G) property holds or fails and constructing explicit examples, especially in the challenging supersingular environment. Overall, the article delivers new cohomological and algebraic tools to study non-cyclotomic Iwasawa theory for supersingular elliptic curves, with implications for refined Selmer groups and related invariants. The results sharpen our understanding of how μ- and λ-invariants behave in non-cyclotomic Z_p-extensions and illuminate connections between signed Selmer groups, the classical Selmer group, and the fine Selmer group in this delicate setting.

Abstract

Let be an elliptic curve defined over which has good supersingular reduction at the odd prime . We study the variation of Iwasawa invariants and the -property for signed Selmer groups over -extensions of an imaginary quadratic number field that lie inside the -extension of and are not necessarily cyclotomic. We prove several equivalent criteria for the validity of the -property which involve the growth of -invariants of the signed Selmer groups over intermediate shifted -extensions in , and the boundedness of -invariants as one runs over -extensions of inside . We give examples where the -property holds, and also examples where we can prove that it does not hold. It is striking that although the case of supersingular reduction is much more difficult than the case of ordinary reduction, we get finer results here; moreover, we are able to derive analogous criteria for the validity of the -property of the classical Selmer group, as well as the fine Selmer group. Many of the properties that we investigate have not been studied before in this non-torsion setting. Further, we study various implications between the -properties for Selmer groups, signed Selmer groups and fine Selmer groups. We apply our results to a conjecture of Mazur, and prove implications between the -property and Conjectures A and B of Coates and Sujatha.
Paper Structure (36 sections, 120 theorems, 387 equations)

This paper contains 36 sections, 120 theorems, 387 equations.

Table of Contents

  1. Introduction and overview of results
  2. Introduction
  3. Basic definitions and statement of results
  4. Definitions and important notation
  5. Main results
  6. Relations between $X(E/\mathds{K}_{\infty})$ and $Y(E/\mathds{K}_{\infty})$
  7. Results related to Conjectures A and B
  8. On the non-triviality of torsion submodules
  9. Mazur's growth number conjecture
  10. Key results used for the main theorems
  11. Further definitions and the control theorem for signed Selmer groups
  12. Results on $\mathop{\mathrm{Sel}}\nolimits^{\bullet\star}_{p^\infty}(E/L)$ for $L=\mathds{K}_{\infty}$ and $L \in \mathcal{E}$
  13. The control theorem for the Selmer group and results on the Iwasawa invariants of $T_{\Lambda_{H,n}}([F_{\Lambda_2}(X(E/\mathds{K}_{\infty}))]_{H_n})$
  14. Surjectivity of the global-to-local map for signed Selmer groups
  15. Pseudo-null submodules and a comparison theorem
  16. ...and 21 more sections

Key Result

Theorem 2.4

With notation as above, let $\bullet, \star \in \{+,-\}$. We assume that $\mathcal{H}^{\bullet\star}$ is non-empty and we choose some ${H=\mkern 1.5mu\overline{\mkern-1.5mu\langle \sigma^a\tau^b \rangle\mkern-1.5mu}\mkern 1.5mu\in \mathcal{H}^{\bullet\star}}$. Then $X^{\bullet\star}(E/\mathds{K}_{\i Moreover, if ${\star = \bullet}$, i.e. either ${H \in \mathcal{H}^{++}}$ or ${H \in \mathcal{H}^{--

Theorems & Definitions (255)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Theorem 2.10
  • ...and 245 more