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 $\text{corank}_{\mathbb{Z}_{p}}\mathrm{Sel}_{p^\infty}(E_{/\mathbb{Q}}) = 0 \implies \mathrm{ord}_{s=1}L(s,E_{/\mathbb{Q}}) = 0$ and the 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 $E: y^2 = x^3 – x$.