Fermat-type equations of signature $$(13,13,p)$$ via Hilbert cuspforms

Abstract

In this paper we prove that equations of the form x 13 +y 13 = Cz p have no non-trivial primitive solutions (a, b, c) such that 13 ∤ c if p > 4992539 for an infinite family of values for C. Our method consists in relating a solution (a, b, c) to the previous equation to a solution (a, b, c1) of another Diophantine equation with coefficients in Q( √ 13). We then construct Frey-curves associated with (a, b, c1) and we prove modularity of them in order to apply the modular approach via Hilbert cusp forms over Q( √ 13). We also prove a modularity result for elliptic curves over totally real cyclic number fields of interest by itself.

Criteria for Irreducibility of mod p Representations of Frey Curves

“…In such a setting, one usually knows that the elliptic curve in question has semistable reduction outside a given set of primes, and one often knows some primes of potentially good reduction. We illustrate this, by giving an improvement to a recent theorem of Dieulefait and Freitas [6] on the equation x 13 + y 13 = Cz p . In a forthcoming paper [2], the authors apply our Theorem 1 together with modularity and level-lowering theorems to completely solve the equation x 2n ± 6x n + 1 = y 2 in integers x, y, n with n ≥ 2, after associating this to a Frey elliptic curve over Q( √ 2).…”

“…In [6], Dieulefait and Freitas, used the modular method to attack certain Fermat-type equations of the form x 13 + y 13 = Cz p , for infinitely many values of C. They attach Frey curves (independent of C) over Q( √ 13) to primitive solutions of these equations, and prove irreducibility of the mod p representations attached to these Frey curves, for p > 7 and p = 13, 37 under the assumption that the isogeny signatures are (0, 0) or (12,12). Here we improve on the argument by dealing with the isogeny signature (0, 12), (12,0) and also by dealing with p = 37.…”

“…13); this has class number 1. Let E = E (a,b) /K be the Frey curve defined in [6]. Then, the Galois representation ρ E,p is irreducible.…”

“…be the Frey curves attached to it in [6]. As explained in [6], but now using Theorem 3 above instead of Theorem 4.1 in loc. cit., we obtain an isomorphism (6.2)ρ E,p ∼ρ f,p , where p | p and f ∈ S 2 (2 i w 2 ) for i = 3, 4.…”

“…Attention is now shifting towards Diophantine equations where the Frey elliptic curves are defined over totally real fields (for example [2], [6], [7], [8]). One now finds in the literature some of the necessary modularity (e.g.…”