$p$-converse theorems for elliptic curves of potentially good ordinary reduction at Eisenstein primes

TL;DR

This paper establishes $p$-converse theorems for elliptic curves with potentially good ordinary reduction at Eisenstein primes, linking Selmer ranks to Iwasawa theory and improving rank distribution estimates in quadratic twist families.

Contribution

It proves $p$-converse theorems for certain elliptic curves using anticyclotomic Iwasawa Main Conjectures and descent methods, extending previous results to new cases.

Findings

Proved $p$-converse theorems for elliptic curves with reducible residual representation.

Established anticyclotomic Iwasawa Main Conjectures for specific imaginary quadratic fields.

Improved estimates for the proportion of elliptic curves with rank 0 or 1 in quadratic twist families.

Abstract

Let $E/\mathbf{Q}$ be an elliptic curve and $p\geq 3$ be a prime. We prove the $p$-converse theorems for elliptic curves of potentially good ordinary reduction at Eisenstein primes (i.e., such that the residual representation $E[p]$ is reducible) when the $p$-Selmer rank is $0$ or $1$. The key step is to obtain the anticyclotomic Iwasawa Main Conjectures for an auxiliary imaginary quadratic field $K$ where $E$ does not have CM similar to those in [CGLS22] and descent to $\mathbf{Q}$. As an application we get improved proportions for the number of elliptic curves in quadratic twist families having rank $0$ or $1$.