No abstract

Let n 3. This paper is concerned with the equation a 3 + b 3 = c n , which we attack using a combination of the modular approach (via Frey curves and Galois representations) with obstructions to the solutions that are of Brauer-Manin type. We shall show that there are no solutions in coprime, non-zero integers a, b, c, for a set of prime exponents n having Dirichlet density 28 219 44 928 ≈ 0.628, and for a set of exponents n having natural density 1.

This paper is devoted to the generalized Fermat equation xp + yq = zr, where p, q and r are integers, and x, y and z are nonzero coprime integers. We begin by surveying the exponent triples (p, q, r), including a number of infinite families, for which the equation has been solved to date, detailing the techniques involved. In the remainder of the paper, we attempt to solve the remaining infinite families of generalized Fermat equations that appear amenable to current techniques. While the main tools we employ are based upon the modularity of Galois representations (as is indeed true with all previously solved infinite families), in a number of cases we are led via descent to appeal to a rather intricate combination of multi-Frey techniques.

We study modular Galois representations mod p m . We show that there are three progressively weaker notions of modularity for a Galois representation mod p m : We have named these "strongly", "weakly", and "dc-weakly" modular. Here, "dc" stands for "divided congruence" in the sense of Katz and Hida. These notions of modularity are relative to a fixed level M . Using results of Hida we display a level-lowering result ("strippingof-powers of p away from the level"): A mod p m strongly modular representation of some level Np r is always dc-weakly modular of level N (here, N is a natural number not divisible by p). We also study eigenforms mod p m corresponding to the above three notions. Assuming residual irreducibility, we utilize a theorem of Carayol to show that one can attach a Galois representation mod p m to any "dc-weak" eigenform, and hence to any eigenform mod p m in any of the three senses. We show that the three notions of modularity coincide when m = 1 (as well as in other particular cases), but not in general.

We describe a criterion for showing that the equation s 2 +y 2p = α 3 has no non-trivial proper integer solutions for specific primes p > 7. This equation is a special case of the generalized Fermat equation x p + y q + z r = 0. The criterion is based on the method of Galois representations and modular forms together with an idea of Kraus for eliminating modular forms for specific p in the final stage of the method (1998). The criterion can be computationally verified for primes 7 < p < 10 7 and p = 31.