On the conductor of a family of Frey hyperelliptic curves

Abstract

El artículo estudia el conductor de una familia biparamétrica de curvas hiperbólicas de Frey, denotada C(z, s), utilizada en el enfoque modular para las ecuaciones de Fermat generalizadas. Los autores muestran que muchas de las curvas conocidas son casos particulares de esta familia y aplican la metodología de “cluster pictures” para calcular los exponentes del conductor en todos los lugares impares. Posteriormente, se analizan las aplicaciones a las ecuaciones de Fermat generalizadas con firmas (p, p, r), (r, r, p) y (2, r, p), obteniendo fórmulas explícitas para los exponentes de los conductores. Además, se vinculan los resultados con las representaciones de Galois asociadas a motivos hipergeométricos, extendiendo la aplicabilidad del método modular a nuevas firmas.

In the breakthrough article [Dar00], Darmon presented a program to study Generalized Fermat Equations (GFE) via abelian varieties of GL2-type over totally real fields. So far, only Jacobians of some Frey hyperelliptic curves have been used with that purpose. In the present article, we show how most of the known Frey hyperelliptic curves are particular instances of a more general biparametric family of hyperelliptic curves C(z, s). Then, we apply the cluster picture methodology to compute the conductor of C(z, s) at all odd places. As a Diophantine application, we specialize C(z, s) in some particular values z0 and s0, and we find the conductor exponent at odd places of the natural Frey hyperelliptic curves attached to Axp + Byp = Czr and Axr + Byr = Czp, generalizing the results on [ACIK+24] and opening the door for future research in GFE with coefficients. Moreover, we show how a new Frey hyperelliptic curve for Ax2 + Byr = Czp can be constructed in this way, giving new results on the conductor exponents for this equation. Finally, following the recent approach in [GP24], we consider the Frey representations attached to a general signature (q, r, p) via hypergeometric motives and, using C(z, s), we compute the wild part of the conductor exponent at primes above q and r of the residual representation modulo a prime above p