Galois cohomology of elliptic curves over anticyclotomic extensions

TL;DR

This work develops a unified, algebraic framework for the Iwasawa theory of elliptic curves over the anticyclotomic $\mathbb{Z}_p$-extension of an imaginary quadratic field. It analyzes the dual Selmer group $X(E/K_{\text{ac}})$ in both definite (torsion) and indefinite (rank $1$) settings, connecting Galois cohomology, Heegner points, and existing results from Bertolini– Darmon, Howard, and Perrin-Riou. The paper also investigates the vanishing of the $\mu$-invariant in both settings, providing criteria under which $\mu=0$ without relying on the Iwasawa main conjecture and presenting concrete numerical examples. Overall, the results offer streamlined proofs and a cohesive approach to the structure of Selmer groups over anticyclotomic towers, with implications for the arithmetic of $E$ and the behavior of $p$-adic $L$-functions.

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 $K$.

Theorems & Definitions (48)

• Theorem 3.2
• Proposition 3.3
• Definition 3.4
• Definition 3.5
• Proposition 3.6
• proof
• Theorem 3.7
• Theorem 3.8
• Remark 3.9
• Theorem 3.11