Classical and modular methods applied to Diophantine equations

Abstract

Deep methods from the theory of elliptic curves and modular forms have been used to prove Fermat's last theorem and solve other Diophantine equations. These so-called modular methods can often benefit from information obtained by other, classical, methods from number theory; and vice versa. In our work we are interested in explicitly solving Diophantine equations, especially generalized Fermat equations. We construct certain families of Frey curves and prove irreducibility results for the Galois representation associated to the p-torsion points of some of these curves for small primes p. This allows us to use modular methods to solve, amongst other equations, the generalized Fermat equations with coefficients 1 and signatures (3,3,5) and (2,10,3). By combining classical arguments from algebraic number theory with modular methods, we solve the generalized Fermat equation with coefficients 1 and signature (2,62,3). Using classical methods, we obtain an algorithm to solve the generalized Fermat equations with signature (2,3,5) and arbitrary nonzero integer coefficients. An algorithm for solving these equations was obtained earlier. However, using our algorithm we are able to prove that the local-to-global principle does not hold for signature (2,3,5). Furthermore, our algorithm allows input obtained by modular methods.