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arXiv · 2210.02316

A modular approach to Fermat equations of signature $(p,p,5)$ using Frey hyperelliptic curves

Imin Chen·Angelos Koutsianas
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Abstract

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.

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Imin Chen, Angelos Koutsianas. 2022-10-05. A modular approach to Fermat equations of signature $(p,p,5)$ using Frey hyperelliptic curves. https://arxiv.org/abs/2210.02316

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