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

Published 4 Jun 2025 in math.NT | (2506.03465v1)

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 $E: y^{{2}=x^{3}-x$.}