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

Abstract

Let $E$ be a CM elliptic curve defined over $\mathbb{Q}$ and $p$ a prime. We show that $${\mathrm{c}orank}_{{\mathbb{Z}}_p}{\mathrm{S}el}_{p^{\infty }}(E_{/\mathbb{Q}})=0⟹{\mathrm{o}rd}_{s=1}L(s,E_{/\mathbb{Q}})=0$$ for the $p^{\infty }$-Selmer group ${\mathrm{S}el}_{p^{\infty }}(E_{/\mathbb{Q}})$ 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 ${\mathrm{o}rd}_{s=1}L(s,E_{/\mathbb{Q}}^{(n)})=0,$ where $E^{(n)}:n{y}^2=x^3-x$ is a quadratic twist of the congruent number elliptic curve $E:y^2=x^3-x$.

Annals of Math., to appear