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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.10781 |
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| _version_ | 1866909957114298368 |
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| author | Freitas, Nuno Mocanu, Diana Sanchez-Rodriguez, Ignasi |
| author_facet | Freitas, Nuno Mocanu, Diana Sanchez-Rodriguez, Ignasi |
| contents | Let $E/\mathbb Q$ be an elliptic curve and $p \geq 3$ a prime. The modular curve $X_E^-(p)$ parameterizes elliptic curves with $p$-torsion modules anti-symplectically isomorphic to $E[p]$. The work of Freitas--Naskręcki--Stoll uses the modular method to show that all primitive non-trivial solutions of the Fermat-type equation $x^2 + y^3 = z^p$ give rise to rational points on $X_E^-(p)$ with $E \in \{27a1,54a1,96a1,288a1,864a1,864b1,864c1 \}$. Using a criterion classifying the existence of local points due to the first two authors, we show that, for $E$ any of the curves with conductor 864 and certain primes $p \equiv 19 \pmod{24}$, we have $X_E^-(p)(\mathbb Q_\ell) = \emptyset$. Furthermore, for each $E$ in the list and any $p$, we prove that either $X_E^-(p)$ can be discarded using the same criterion, or it cannot be discarded using purely local information. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_10781 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Revisiting the equation $x^2+y^3=z^p$ Freitas, Nuno Mocanu, Diana Sanchez-Rodriguez, Ignasi Number Theory Let $E/\mathbb Q$ be an elliptic curve and $p \geq 3$ a prime. The modular curve $X_E^-(p)$ parameterizes elliptic curves with $p$-torsion modules anti-symplectically isomorphic to $E[p]$. The work of Freitas--Naskręcki--Stoll uses the modular method to show that all primitive non-trivial solutions of the Fermat-type equation $x^2 + y^3 = z^p$ give rise to rational points on $X_E^-(p)$ with $E \in \{27a1,54a1,96a1,288a1,864a1,864b1,864c1 \}$. Using a criterion classifying the existence of local points due to the first two authors, we show that, for $E$ any of the curves with conductor 864 and certain primes $p \equiv 19 \pmod{24}$, we have $X_E^-(p)(\mathbb Q_\ell) = \emptyset$. Furthermore, for each $E$ in the list and any $p$, we prove that either $X_E^-(p)$ can be discarded using the same criterion, or it cannot be discarded using purely local information. |
| title | Revisiting the equation $x^2+y^3=z^p$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2512.10781 |