Title:

Diophantine applications of Serre's modularity conjecture over general number fields

The thesis starts with two expository chapters. In the first one we discuss abelian varieties with potential good reduction and their Galois representations. We proceed by presenting some of the theory concerning eigenforms for GL2 over number fields in the second chapter. The last two chapters correspond to two papers I wrote during the course of my PhD studies. Both of these are concerned with applications of Serre's modularity conjecture (see Conjecture 2.2.1) to Diophantine equations over number fields. As a corollary to Theorem 3.1.1, we show that assuming Conjecture 2.2.1, Fermat's Last Theorem holds over Q(i). Theorems 3.1.1 and 3.5.1 concern Fermat's equation with prime exponent and cover all the nine quadratic imaginary fields of class number 1. They are under the assumption of Serre's modularity conjecture. For a large class (heuristically most) of irreducible binary cubic forms F(x; y) ε Z[x; y], Bennett and Dahmen proved that the generalized superelliptic equation F(x; y) = zl has at most finitely many solutions in x; y ε Z coprime, z ε Z and exponent l _ 5 prime. Their proof uses, among other ingredients, modularity of certain mod l Galois representations and Ribet's level lowering theorem. In Chapter 3, we considered the same problem over general number fields K. Theorem 4.1.1 shows that, a similar (but more restrictive) criterion on the binary form F ε OK[x; y] ensures that the generalised superelliptic equation has finitely many proper solutions in OK. This result is conjectural on Serre's modularity conjecture (see Conjecture 2.2.1) and on a version of EichlerShimura (see Conjecture 2.4.1). When K is totally real and Galois, we can make use of modularity theorems to obtain Theorem 4.1.2, independent of Conjecture 2.2.1.
