Non-vanishing of central $L$-values of the Gross family of elliptic curve
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912741274419200 |
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| author | Kezuka, Yukako Li, Yong-Xiong |
| author_facet | Kezuka, Yukako Li, Yong-Xiong |
| contents | We prove non-vanishing theorems for the central values of $L$-series of quadratic twists of the Gross elliptic curve with complex multiplication by the imaginary quadratic field $\mathbb{Q}(\sqrt{-q})$, where $q$ is any prime congruent to $7$ modulo $8$. This completes the non-vanishing theorems proven by Coates and the second author in which the primes $q$ were taken to be congruent to $7$ modulo $16$. From this, we obtain the finiteness of the Mordell-Weil group and the Tate-Shafarevich group for these curves. For a prime $\mathfrak{P}$ lying above the prime $2$, we also prove a converse theorem in the rank $0$ case and the $\mathfrak{P}$-part of the Birch-Swinnerton-Dyer conjecture for the higher-dimensional abelian varieties obtained by restriction of scalars. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_08689 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Non-vanishing of central $L$-values of the Gross family of elliptic curve Kezuka, Yukako Li, Yong-Xiong Number Theory 11R23, 11G05, 11G40 We prove non-vanishing theorems for the central values of $L$-series of quadratic twists of the Gross elliptic curve with complex multiplication by the imaginary quadratic field $\mathbb{Q}(\sqrt{-q})$, where $q$ is any prime congruent to $7$ modulo $8$. This completes the non-vanishing theorems proven by Coates and the second author in which the primes $q$ were taken to be congruent to $7$ modulo $16$. From this, we obtain the finiteness of the Mordell-Weil group and the Tate-Shafarevich group for these curves. For a prime $\mathfrak{P}$ lying above the prime $2$, we also prove a converse theorem in the rank $0$ case and the $\mathfrak{P}$-part of the Birch-Swinnerton-Dyer conjecture for the higher-dimensional abelian varieties obtained by restriction of scalars. |
| title | Non-vanishing of central $L$-values of the Gross family of elliptic curve |
| topic | Number Theory 11R23, 11G05, 11G40 |
| url | https://arxiv.org/abs/2305.08689 |