A rank zero $p$-converse to a theorem of Gross--Zagier, Kolyvagin and Rubin
Ashay A. Burungale, Ye Tian
TL;DR
The paper proves a rank-zero $p$-converse for CM elliptic curves over imaginary quadratic fields, establishing that ${\mathrm corank}_{\mathbb{Z}_p}{\mathrm{Sel}}_{p^{\infty}}(E_{/K})=0$ implies ${\mathrm ord}_{s=1}L(s,E_{/K})=0$, and deduces the corresponding result over ${\mathbb Q}$ when $E$ descends. The method hinges on the CM case of Kato's main conjecture, via an equivariant main conjecture for imaginary quadratic fields and a tight link between Beilinson--Kato elements and elliptic units, yielding a uniform $p$-adic framework across primes. As a significant corollary, the authors obtain the first instance of the even parity Goldfeld conjecture for the congruent number curve, combining a rank-zero $p$-converse with Smith’s distribution of $2^{\infty}$-Selmer ranks. The work integrates Iwasawa theory of zeta elements, CM modular forms, and Beilinson--Kato theory to relate arithmetical invariants to the analytic behavior of $L$-functions, with potential implications for the equivariant Tamagawa number conjecture and broader BSD-type conjectures.
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 $E: y^{2}=x^{3}-x$.