Residual supersingular Iwasawa theory and signed Iwasawa invariants

TL;DR

This paper develops a residual Iwasawa theory framework for supersingular elliptic curves, introducing a signed residual Selmer group that captures key invariants and facilitates the study of congruences modulo p.

Contribution

It introduces a new residual signed Selmer group framework that links residual Galois representations to Iwasawa invariants in supersingular settings.

Findings

Defines a fine signed residual Selmer group dependent on residual representation.

Establishes connections between residual invariants and classical Iwasawa invariants.

Provides a natural setting for studying congruences modulo p in supersingular Iwasawa theory.

Abstract

For an odd prime $p$ and a supersingular elliptic curve over a number field, this article introduces a fine signed residual Selmer group, under certain hypotheses on the base field. This group depends purely on the residual representation at $p$, yet captures information about the Iwasawa theoretic invariants of the signed $p^\infty$-Selmer group that arise in supersingular Iwasawa theory. Working in this residual setting provides a natural framework for studying congruences modulo $p$ in Iwasawa theory.