$p$-adic Waldspurger Formula for Non-split Primes and Converse of Gross--Zagier and Kolyvagin Theorem
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910079900450816 |
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| author | Fan, Yangyu Wan, Xin |
| author_facet | Fan, Yangyu Wan, Xin |
| contents | 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 $ψ$ is 1, then the analytic rank of $L(ψ,s)$ 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_09806 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $p$-adic Waldspurger Formula for Non-split Primes and Converse of Gross--Zagier and Kolyvagin Theorem Fan, Yangyu Wan, Xin Number Theory 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 $ψ$ is 1, then the analytic rank of $L(ψ,s)$ 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). |
| title | $p$-adic Waldspurger Formula for Non-split Primes and Converse of Gross--Zagier and Kolyvagin Theorem |
| topic | Number Theory |
| url | https://arxiv.org/abs/2304.09806 |