Local points on twists of $X(p)$ with applications
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911267429548032 |
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| author | Freitas, Nuno Mocanu, Diana |
| author_facet | Freitas, Nuno Mocanu, Diana |
| contents | Let $E/\mathbb Q$ be an elliptic curve and $p \geq 3$ a prime. The modular curve $X_E^-(p)$ parametrizes elliptic curves with $p$-torsion modules anti-symplectically isomorphic to $E[p]$. We give a complete classification of when $X_E^-(p)(\mathbb Q_\ell)$ is non-empty, for all primes $\ell\neq p$; our result also includes $\ell=p$ in most cases when $E$ is semistable at $p$.
We give two different applications. First, we classify CM curves $E/\mathbb Q$ where the modular curve $X_E^-(p)$ is a counterexample to the Hasse principle for infinitely many $p$. Assuming the Frey--Mazur conjecture, we prove that for at least $60\%$ of rational elliptic curves $E$, the modular curve $X_E^-(p)$ is a counterexample to the Hasse principle for at least $50\%$ of primes $p$. Secondly, we introduce a new technique to the elimination stage of the modular method and apply it to show that $x^3+y^3=5^αz^p$ has no non-trivial primitive solutions for various primes $p$ satisfying $(α/p)=-1$. Moreover, as a by-product of our work, we simplify the assumptions of several local symplectic criteria due to the first author and Alain Kraus. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_04294 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local points on twists of $X(p)$ with applications Freitas, Nuno Mocanu, Diana Number Theory Primary 11G05, 11G07. Secondary 14H10, 14G12 Let $E/\mathbb Q$ be an elliptic curve and $p \geq 3$ a prime. The modular curve $X_E^-(p)$ parametrizes elliptic curves with $p$-torsion modules anti-symplectically isomorphic to $E[p]$. We give a complete classification of when $X_E^-(p)(\mathbb Q_\ell)$ is non-empty, for all primes $\ell\neq p$; our result also includes $\ell=p$ in most cases when $E$ is semistable at $p$. We give two different applications. First, we classify CM curves $E/\mathbb Q$ where the modular curve $X_E^-(p)$ is a counterexample to the Hasse principle for infinitely many $p$. Assuming the Frey--Mazur conjecture, we prove that for at least $60\%$ of rational elliptic curves $E$, the modular curve $X_E^-(p)$ is a counterexample to the Hasse principle for at least $50\%$ of primes $p$. Secondly, we introduce a new technique to the elimination stage of the modular method and apply it to show that $x^3+y^3=5^αz^p$ has no non-trivial primitive solutions for various primes $p$ satisfying $(α/p)=-1$. Moreover, as a by-product of our work, we simplify the assumptions of several local symplectic criteria due to the first author and Alain Kraus. |
| title | Local points on twists of $X(p)$ with applications |
| topic | Number Theory Primary 11G05, 11G07. Secondary 14H10, 14G12 |
| url | https://arxiv.org/abs/2509.04294 |