A rank zero $p$-converse to a theorem of Gross--Zagier, Kolyvagin and Rubin
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915545208586240 |
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| author | Burungale, Ashay A. Tian, Ye |
| author_facet | Burungale, Ashay A. Tian, Ye |
| contents | 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 $E: y^{2}=x^{3}-x$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_03465 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A rank zero $p$-converse to a theorem of Gross--Zagier, Kolyvagin and Rubin Burungale, Ashay A. Tian, Ye Number Theory 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 $E: y^{2}=x^{3}-x$. |
| title | A rank zero $p$-converse to a theorem of Gross--Zagier, Kolyvagin and Rubin |
| topic | Number Theory |
| url | https://arxiv.org/abs/2506.03465 |