A note on conductors of Frey representations at $2$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908855006396416 |
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| author | Chen, Imin Torcomian, Lucas Villagra |
| author_facet | Chen, Imin Torcomian, Lucas Villagra |
| contents | In 2000, Darmon introduced the notion of Frey representations within the framework of the modular method for studying the generalized Fermat equation. A central step in this program is the computation of their conductors, with the case at the prime $2$ presenting particular challenges. In this article we study the conductor exponent at $2$ for Frey representations of signatures $(p,p,r)$, $(r,r,p)$, $(2,r,p)$, and $(3,5,p)$, all of which have hyperelliptic realizations. In particular we are able to determine the conductor at $2$ for even degree Frey representations of signature $(p,p,r)$ and $(3,5,p)$ and all rational parameters. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_23540 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on conductors of Frey representations at $2$ Chen, Imin Torcomian, Lucas Villagra Number Theory 11D41, 11D61, 11G30, 11G20, 11F80 In 2000, Darmon introduced the notion of Frey representations within the framework of the modular method for studying the generalized Fermat equation. A central step in this program is the computation of their conductors, with the case at the prime $2$ presenting particular challenges. In this article we study the conductor exponent at $2$ for Frey representations of signatures $(p,p,r)$, $(r,r,p)$, $(2,r,p)$, and $(3,5,p)$, all of which have hyperelliptic realizations. In particular we are able to determine the conductor at $2$ for even degree Frey representations of signature $(p,p,r)$ and $(3,5,p)$ and all rational parameters. |
| title | A note on conductors of Frey representations at $2$ |
| topic | Number Theory 11D41, 11D61, 11G30, 11G20, 11F80 |
| url | https://arxiv.org/abs/2509.23540 |