A $p$-Converse theorem for Real Quadratic Fields
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910946787590144 |
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| author | Bansal, Muskan Jha, Somnath Pal, Aprameyo Venkat, Guhan |
| author_facet | Bansal, Muskan Jha, Somnath Pal, Aprameyo Venkat, Guhan |
| contents | Let $E$ be an elliptic curve defined over a real quadratic field $F$. Let $p > 5$ be a rational prime that is inert in $F$ and assume that $E$ has split multiplicative reduction at the prime $\mathfrak{p}$ of $F$ dividing $p$. Let $\underline{III}(E/F)$ denote the Tate-Shafarevich group of $E$ over $F$ and $ L(E/F,s) $ be the Hasse-Weil complex $L$-function of $E$ over $F$. Under some technical assumptions, we show that when $rank_{\mathbb{Z}} \hspace{0.01mm} \hspace{1mm} E(F) = 1$ and $\#\Big(\underline{III}(E/F)_ {p^\infty}\Big) < \infty$, then $ord_{s=1} \ L(E/F,s) = 1$. Further, we give an applictaion to a $p$-converse theorem over $\mathbb{Q}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_21799 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A $p$-Converse theorem for Real Quadratic Fields Bansal, Muskan Jha, Somnath Pal, Aprameyo Venkat, Guhan Number Theory 11G40, 11G05, 11R23 Let $E$ be an elliptic curve defined over a real quadratic field $F$. Let $p > 5$ be a rational prime that is inert in $F$ and assume that $E$ has split multiplicative reduction at the prime $\mathfrak{p}$ of $F$ dividing $p$. Let $\underline{III}(E/F)$ denote the Tate-Shafarevich group of $E$ over $F$ and $ L(E/F,s) $ be the Hasse-Weil complex $L$-function of $E$ over $F$. Under some technical assumptions, we show that when $rank_{\mathbb{Z}} \hspace{0.01mm} \hspace{1mm} E(F) = 1$ and $\#\Big(\underline{III}(E/F)_ {p^\infty}\Big) < \infty$, then $ord_{s=1} \ L(E/F,s) = 1$. Further, we give an applictaion to a $p$-converse theorem over $\mathbb{Q}$. |
| title | A $p$-Converse theorem for Real Quadratic Fields |
| topic | Number Theory 11G40, 11G05, 11R23 |
| url | https://arxiv.org/abs/2504.21799 |