Modular approcah to diophantine equations

Abstract

In this thesis, we study the modular approach to the Fermat’s equation xp+yp = zp, where x,y and z are co-prime integers, and p is a prime, and some generalizations. After reviewing and explaining how the modular approach can be used to deal with the Fermat-type equations, following the paper of Emmanuel Halberstadt and Alain Karus, we prove that there exists a dense subset of the set of prime numbers such that the equation axp + byp = czp has no non-trivial primitive solution. Here a,b,c are fixed pairwise co-prime odd integers and p ≥ 5 is a prime. Then we show that the equation x4 + y2 = zp has no solutions in co-prime integers when p ≥ 211 due to Jordan Ellenberg’s article. The main idea to deal with this equation is based on the modularity of Q-curves and the images of Galois representations attached such curves. This thesis was supported by TUBITAK project 117F045.