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_{/\mathbb{Q}}^{(n)}) = 0 $, where $E^{(n)}\colon ny^2=x^3-x$ is a quadratic twist of the congruent number elliptic curve $E\colon y^2 = x^3 – x$.
