On the conductor of a family of Frey hyperelliptic curves
Pedro-José Cazorla García, Lucas Villagra Torcomian
TL;DR
The paper develops cluster-picture techniques to compute the odd conductor exponents of a broad biparametric family of Frey hyperelliptic curves $C(z,s)$, extending Darmon's GL$_2$-type program for Generalized Fermat Equations. It provides explicit root descriptions, discriminants, and cluster pictures, and then derives the tame and wild contributions to the conductor at odd primes, both over $\mathbb{Q}$ and over the totally real field $K=\mathbb{Q}(\omega_1)$. The results hinge on the irreducibility of the associated polynomial $F(x)$ and the $q$-adic valuations of $\Delta=s^2-4z^r$, yielding concrete conductor-exponent tables and ramification data, with applications to signatures $(p,p,r)$, $(r,r,p)$, and $(2,r,p)$ in the Generalized Fermat setting. By linking to hypergeometric motives and the approach of Golfieri and Pacetti, the work enables a broader application of the modular method to GFEs with coefficients and paves the way for constructing new Frey curves with computable conductors.
Abstract
In his breakthrough article, Darmon presented a program to study Generalized Fermat Equations (GFE) via abelian varieties of $\text{GL}_2$-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 $z_0$ and $s_0$, and we find the conductor exponent at odd places of the natural Frey hyperelliptic curves attached to $Ax^p+By^p=Cz^r$ and $Ax^r+By^r=Cz^p$, generalizing the results due to Azon, Curcó-Iranzo, Khawaja, Maistret and Mocanu, and opening the door for future research in GFE with coefficients. Moreover, we show how a new Frey hyperelliptic curve for $Ax^2+By^r=Cz^p$ can be constructed in this way, giving new results on the conductor exponents for this equation. Finally, following the recent approach by Golfieri and Pacetti, 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 $p$.