A rank zero $p$-converse to a theorem of Gross--Zagier, Kolyvagin and Rubin

TL;DR

This paper establishes a new rank zero p-converse result for CM elliptic curves, linking Selmer group corank to the order of vanishing of the L-function, and confirms the even parity Goldfeld conjecture for a positive proportion of quadratic twists.

Contribution

It proves a rank zero p-converse theorem for CM elliptic curves and applies it to verify the even parity Goldfeld conjecture for half of the quadratic twists.

Findings

Corank zero Selmer groups imply L-function non-vanishing at s=1.

First proof of the even parity Goldfeld conjecture for a positive proportion of twists.

Connects Selmer group structure with analytic rank in the CM case.

Abstract

Let $E$ be a CM elliptic curve defined over $\mathbb{Q}$ and $p$ a prime. We show that $${\mathrm corank}_{\mathbb{Z}_{p}} {\mathrm Sel}_{p^{\infty}}(E_{/\mathbb{Q}})=0 \implies {\mathrm ord}_{s=1}L(s,E_{/\mathbb{Q}})=0 $$ for the $p^{\infty}$-Selmer group ${\mathrm Sel}_{p^{\infty}}(E_{/\mathbb{Q}})$ and the complex $L$-function $L(s,E_{/\mathbb{Q}})$. Along with Smith's work on the distribution of $2^\infty$-Selmer groups, this leads to the first instance of the even parity Goldfeld conjecture: For $50\%$ of the positive square-free integers $n$, we have $ {\mathrm ord}_{s=1}L(s,E^{(n)}_{/\mathbb{Q}})=0, $ where $E^{(n)}: ny^{2}=x^{3}-x $ is a quadratic twist of the congruent number elliptic curve $E: y^{2}=x^{3}-x$.