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On the cohomology of plus/minus Selmer groups of supersingular elliptic curves in weakly ramified base fields

Ben Forrás, Katharina Müller

TL;DR

This work extends the signed Iwasawa theory for supersingular elliptic curves to weakly ramified base fields, establishing a Kida-type relation for Iwasawa invariants and an integrality result for characteristic elements in a ramified setting. The authors develop local cohomology and Iwasawa-theoretic tools without relying on norm-coherent points, then globalize to define signed Selmer groups and prove cotorsion, absence of finite submodules, and behavior under non-primitive variants. They formulate a ramified Kida formula, prove the integrality of characteristic elements in graduated orders, and analyze how Iwasawa invariants behave under congruences of p-torsion modules, offering a robust framework for non-ordinary primes in ramified Iwasawa theory. The results extend Lim’s unramified theory and have implications for main conjectures and congruence phenomena in the setting of supersingular elliptic curves over number fields.

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve and let $p\ge 5$ be a prime of good supersingular reduction. We generalize results due to Meng Fai Lim proving Kida's formula and integrality results for characteristic elements of signed Selmer groups along the cyclotomic $\mathbb{Z}_p$-extension of weakly ramified base fields $K/\mathbb{Q}_p$.

On the cohomology of plus/minus Selmer groups of supersingular elliptic curves in weakly ramified base fields

TL;DR

This work extends the signed Iwasawa theory for supersingular elliptic curves to weakly ramified base fields, establishing a Kida-type relation for Iwasawa invariants and an integrality result for characteristic elements in a ramified setting. The authors develop local cohomology and Iwasawa-theoretic tools without relying on norm-coherent points, then globalize to define signed Selmer groups and prove cotorsion, absence of finite submodules, and behavior under non-primitive variants. They formulate a ramified Kida formula, prove the integrality of characteristic elements in graduated orders, and analyze how Iwasawa invariants behave under congruences of p-torsion modules, offering a robust framework for non-ordinary primes in ramified Iwasawa theory. The results extend Lim’s unramified theory and have implications for main conjectures and congruence phenomena in the setting of supersingular elliptic curves over number fields.

Abstract

Let be an elliptic curve and let be a prime of good supersingular reduction. We generalize results due to Meng Fai Lim proving Kida's formula and integrality results for characteristic elements of signed Selmer groups along the cyclotomic -extension of weakly ramified base fields .
Paper Structure (16 sections, 48 theorems, 133 equations)

This paper contains 16 sections, 48 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Galois cohomology
  3. Local considerations
  4. Corank computations
  5. Freeness of Iwasawa cohomology
  6. Computation of cohomology groups
  7. p-primary part
  8. Global considerations
  9. Definition of signed Selmer groups
  10. Torsion properties of Xŝ(E/K_infty)
  11. Finite submodules of Xŝ(E/K_infty)
  12. Non-primitive Selmer groups
  13. Projectivity results
  14. Kida's formula
  15. Integrality of characteristic elements
  16. ...and 1 more sections

Key Result

Theorem 1

Assume that $\textup{Sel}^{\vec{s}}(E/K_\infty)$ is $\Lambda$-cotorsion and that $\theta(X^{\vec{s}}(E/K_\infty))\le 1$. Let $P_1\subset \Sigma'$ be the primes where $E$ has split multiplicative reduction and let $P_2$ be the set of primes in $\Sigma'$ where $E$ has good reduction and $E(K)[p]\neq 0

Theorems & Definitions (100)

  • Theorem : Kida formula, \ref{['thm:Kida']}
  • Theorem : Integrality of characteristic elements, \ref{['thm:integrality']}
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 90 more