2012/03/15 by Freitas, Nuno · 2 citations
#11D41 (Primary) #11F80 #11G05 (Secondary) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1203.3371
We describe a strategy to attack infinitely many Fermat-type equations of signature (r,r,p), where r ≥ 7 is a fixed prime and p is a prime allowed to vary. We use a variant of the modular method over totally real subfields of ℚ(ζr). In particular, to a solution (a,b,c) of xr + yr =Czp we will attach several Frey curves E=E(a,b). We prove modularity of all the Frey curves and the exsitence of a constant constant Mr, depending only on r, such that for all p>Mr the representations ρE,p are absolutely irreducible. Along the way, we also prove modularity of certain elliptic curves that are semistable at all v | 3.\par Finally, we illustrate our methods by proving arithmetic statements about equations of signature (7,7,p). Among which we emphasize that, using a multi-Frey technique, we show there is some constant M such that if p > M then the equation x7 + y7 = 3zp has no non-trivial primitive solutions.