Imaginary quadratic fields $F$ with $X_0(15)(F)$ finite
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916247510188032 |
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| author | Evink, Tim |
| author_facet | Evink, Tim |
| contents | Caraiani and Newton have proven that if $F$ is an imaginary quadratic number field such that $X_0(15)$ has rank $0$ over $F$, then every elliptic curve over $F$ is modular. This paper is concerned with the quadratic fields $F=\mathbb{Q}(\sqrt{-p})$ for a prime number $p$. We give explicit conditions on $p$ under which the rank is $0$, and prove that these conditions are satisfied for $87,5\%$ of the primes for which the rank is expected to be even based on the parity conjecture. We also show these conditions are satisfied if and only if rank $0$ follows from a $4$-descent over $\mathbb{Q}$ on the quadratic twist $X_0(15)_{-p}$. To prove this, we perform two consecutive $2$-descents and prove this gives rank bounds equivalent to those obtained from a $4$-descent using visualisation techniques for $\mathrm{Sha}[2]$. In fact we prove a more general connection between higher descents for elliptic curves which seems interesting in its own right. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_09337 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Imaginary quadratic fields $F$ with $X_0(15)(F)$ finite Evink, Tim Number Theory Caraiani and Newton have proven that if $F$ is an imaginary quadratic number field such that $X_0(15)$ has rank $0$ over $F$, then every elliptic curve over $F$ is modular. This paper is concerned with the quadratic fields $F=\mathbb{Q}(\sqrt{-p})$ for a prime number $p$. We give explicit conditions on $p$ under which the rank is $0$, and prove that these conditions are satisfied for $87,5\%$ of the primes for which the rank is expected to be even based on the parity conjecture. We also show these conditions are satisfied if and only if rank $0$ follows from a $4$-descent over $\mathbb{Q}$ on the quadratic twist $X_0(15)_{-p}$. To prove this, we perform two consecutive $2$-descents and prove this gives rank bounds equivalent to those obtained from a $4$-descent using visualisation techniques for $\mathrm{Sha}[2]$. In fact we prove a more general connection between higher descents for elliptic curves which seems interesting in its own right. |
| title | Imaginary quadratic fields $F$ with $X_0(15)(F)$ finite |
| topic | Number Theory |
| url | https://arxiv.org/abs/2405.09337 |