A modular approach to Fermat equations of signature $(p,p,5)$ using Frey hyperelliptic curves
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arXiv
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| Format: | Preprint |
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2022
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| author | Chen, Imin Koutsianas, Angelos |
| author_facet | Chen, Imin Koutsianas, Angelos |
| contents | In this paper we carry out the steps of Darmon's program for the generalized Fermat equation $$ x^n + y^n = z^5. $$ In particular, we develop the machinery necessary to prove an optimal bound on the exponent $n$ for solutions satisfying certain $2$-adic and $5$-adic conditions which are natural from the point of view of the method. We also reduce the problem of resolving this equation to a `big image conjecture', completing a line of ideas suggested in his original program.
The above equation is an example of a generalized Fermat equation for which the predicted Frey abelian varieties have dimension $ > 1$ and thus it represents an interesting test case for Darmon's program. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_02316 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A modular approach to Fermat equations of signature $(p,p,5)$ using Frey hyperelliptic curves Chen, Imin Koutsianas, Angelos Number Theory 11D41 In this paper we carry out the steps of Darmon's program for the generalized Fermat equation $$ x^n + y^n = z^5. $$ In particular, we develop the machinery necessary to prove an optimal bound on the exponent $n$ for solutions satisfying certain $2$-adic and $5$-adic conditions which are natural from the point of view of the method. We also reduce the problem of resolving this equation to a `big image conjecture', completing a line of ideas suggested in his original program. The above equation is an example of a generalized Fermat equation for which the predicted Frey abelian varieties have dimension $ > 1$ and thus it represents an interesting test case for Darmon's program. |
| title | A modular approach to Fermat equations of signature $(p,p,5)$ using Frey hyperelliptic curves |
| topic | Number Theory 11D41 |
| url | https://arxiv.org/abs/2210.02316 |