$p$-adic Waldspurger Formula for Non-split Primes and Converse of Gross--Zagier and Kolyvagin Theorem

Abstract

Let $p$ be a prime and $\mathcal{K}$ be an imaginary quadratic field. In this paper we generalize a recent construction of a new type of $p$-adic $L$-function and $p$-adic Waldspurger formula by Andreatta-Iovita for $p$ non-split in $\mathcal{K}$, as long as the ${\mathrm{GL}}_2$ automorphic representation is principal series at $p$. Then we develop a new kind of anticyclotomic local $\pm$-Iwasawa theory at $p$ for self-dual Hecke characters over imaginary quadratic fields (including elliptic curves $E/\mathbb{Q}$ with complex multiplication) which is valid for all ramification types of $p$ (split, inert and ramified, and allowing $p=2$). As the main consequence, we prove the converse of the Gross--Zagier--Kolyvagin theorem for self-dual CM characters: if the Selmer rank of $\psi$ is 1, then the analytic rank of $L(\psi ,s)$ at $s=1$ is also 1. As corollaries, we prove Sylvester's conjecture (1879) on sums of two rational cubes, and Goldfeld's conjecture for CM elliptic curves over $\mathbb{Q}$ (conditional on work of A. Smith).

68 pages, with full generality