Galois cohomology of elliptic curves over anticyclotomic extensions

TL;DR

This paper investigates the Iwasawa theory and Galois cohomology of elliptic curves over anticyclotomic extensions of imaginary quadratic fields, providing a unified framework for these complex arithmetic structures.

Contribution

It introduces a unifying approach to study the Iwasawa theory and Galois cohomology of elliptic curves over anticyclotomic extensions of imaginary quadratic fields.

Findings

Analysis of the Galois cohomology of the dual Selmer group over $Z_p^2$-extensions.

Results on the structure of Selmer groups over anticyclotomic extensions.

Insights into the behavior of elliptic curves in anticyclotomic Iwasawa theory.

Abstract

Let $K$ be an imaginary quadratic field and $p$ be an odd prime number. Let $E/\mathbb{Q}$ be an elliptic curve with good ordinary reduction at $p$. We study the Iwasawa theory of $E$ over the anticyclotomic $\mathbb{Z}_p$-extension of $K$ by adopting a unifying framework. We also study the Galois cohomology of the dual Selmer group of $E$ over the unique $\mathbb{Z}_p^2$-extension of $K$ as well as over the anticyclotomic extension of $K$.