Images of Galois representations in mod p Hecke algebras

E-print/Working paper (2017)

Images of Galois representations and p-adic models of Shimura curves

Doctoral thesis (2016)

The thesis treats two questions situated in the Langlands program, which is one of the most active and important areas in current number theory and arithmetic geometry. The first question concerns the ... [more ▼]

The thesis treats two questions situated in the Langlands program, which is one of the most active and important areas in current number theory and arithmetic geometry. The first question concerns the study of images of Galois representations into Hecke algebras coming from modular forms over finite fields, and the second one deals with p-adic models of Shimura curves and its bad reduction. Consequently, the thesis is divided in two parts. The first part is concerned with the study of images of Galois representations that take values in Hecke algebras of modular forms over finite fields. The main result of this part is a complete classification of the possible images of 2-dimensional Galois representations with coefficients in local algebras over finite fields under the hypotheses that: (i) the square of the maximal ideal is zero, (ii) that the residual image is big (in a precise sense), and (iii) that the coefficient ring is generated by the traces. In odd characteristic, the image is completely determined by these conditions; in even characteristic the classification is much richer. In this case, the image is uniquely determined by the number of different traces of the representation, a number which is given by an easy formula. As an application of these results, the existence of certain p-elementary abelian extensions of big non-solvable number fields can be deduced. Whereas some aspects of class field theory are accessible through this approach, it can be applied to huge fields for which standard techniques totally fail. The second part of the thesis consists of an approach to p-adic uniformisations of Shimura curves X(Dp,N) through a combination of different techniques concerning rigid analytic geometry and arithmetic of quaternion orders. The results in this direction lean on two methods: one is based on the information provided by certain Mumford curves covering Shimura curves and the second one on the study of Eichler orders of level N in the definite quaternion algebra of discriminant D. Combining these methods, an explicit description of fundamental domains associated to p-adic uniformisation of families of Shimura curves of discriminant Dp and level N ≥ 1, for which the one-sided ideal class number h(D,N) is 1, is given. The method presented in this thesis enables one to find Mumford curves covering Shimura curves, together with a free system of generators for the associated Schottky groups, p-adic good fundamental domains and their stable reduction-graphs. As an application, general formulas for the reduction-graphs with lengths at p of the considered families of Shimura curves can be computed. [less ▲]

Mumford curves covering p-adic Shimura curves and their fundamental domains

E-print/Working paper (2016)

We give an explicit description of fundamental domains associated to the p-adic uniformisa- tion of families of Shimura curves of discriminant Dp and level N ≥ 1, for which the one-sided ideal class ... [more ▼]

We give an explicit description of fundamental domains associated to the p-adic uniformisa- tion of families of Shimura curves of discriminant Dp and level N ≥ 1, for which the one-sided ideal class number h(D,N) is 1. The obtained results generalise those in [19, Ch. IX] for Shimura curves of discriminant 2p and level N = 1. The method we present here enables us to find Mumford curves covering Shimura curves, together with a free system of generators for the associated Schottky groups, p-adic good fundamental domains and their stable reduction- graphs. This is based on a detailed study of the modular arithmetic of an Eichler order of level N inside the definite quaternion algebra of discriminant D, for which we generalise classical results of Hurwitz [20]. As an application, we prove general formulas for the reduction-graphs with lengths at p of the considered families of Shimura curves. [less ▲]

Deligne's theorem

in Rotger, Víctor (Ed.) Galois representations associated to modular forms (n.d.)