1- https://doi.org/10.4007/annals.2026.203.1.1
A rank zero $p$-converse to a theorem of Gross--Zagier, Kolyvagin and Rubin
This paper proves that for a CM elliptic curve over Q, a zero corank of the p-infinity Selmer group implies the L-function has rank zero, establishing a rank zero p-converse to a classical theorem. Combined with Smith's work, it confirms that 50% of quadratic twists of the congruent number curve have L-value zero at s=1, supporting the even parity Goldfeld conjecture.
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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 $E: y^2=x^3-x$.
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