Equivariant Iwasawa theory for elliptic curves

TL;DR

This paper develops an equivariant Iwasawa theory framework for elliptic curves over rationals, constructing Coleman maps and proving divisibility results, advancing understanding of $p$-adic $L$-functions and related conjectures.

Contribution

It introduces equivariant Coleman maps for elliptic curves and proves a divisibility in the main conjecture, extending Iwasawa theory to an equivariant setting.

Findings

Constructed equivariant Coleman maps using Kobayashi's method.

Proved a divisibility in the equivariant main conjecture under certain assumptions.

Confirmed a case of the Mazur-Tate conjecture for elliptic curves.

Abstract

We discuss abelian equivariant Iwasawa theory for elliptic curves over $\mathbb{Q}$ at good supersingular primes and non-anomalous good ordinary primes. Using Kobayashi's method, we construct equivariant Coleman maps, which send the Beilinson-Kato element to the equivariant $p$-adic $L$-functions. Then we propose equivariant main conjectures and, under certain assumptions, prove one divisibility via Euler system machinery. As an application, we prove a case of a conjecture of Mazur-Tate.