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See detailImages of Galois representations in mod p Hecke algebras
Amoros Carafi, Laia UL

E-print/Working paper (2017)

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See detailImages of Galois representations with values in mod p Hecke algebras
Amoros Carafi, Laia UL

Presentation (2017, January 24)

I will explain how one can construct Galois representations that take values in local mod $p$ Hecke $\mathbb{F}_q$-algebras. I will then show how one can compute explicitly the image of such a Galois ... [more ▼]

I will explain how one can construct Galois representations that take values in local mod $p$ Hecke $\mathbb{F}_q$-algebras. I will then show how one can compute explicitly the image of such a Galois representation assuming that the corresponding residual representation has big image. Finally I will show some computations of these images in concrete examples. [less ▲]

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See detailImages of Galois representations and p-adic models of Shimura curves
Amoros Carafi, Laia UL

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 ▲]

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See detailFrom modular curves to Shimura curves: a p-adic approach
Amoros Carafi, Laia UL

Presentation (2016, November 24)

Shimura curves arise as a natural generalisation of elliptic curves. As modular curves, they are constructed as Riemann surfaces, and they turn out to have structure of algebraic curve, i.e. they can be ... [more ▼]

Shimura curves arise as a natural generalisation of elliptic curves. As modular curves, they are constructed as Riemann surfaces, and they turn out to have structure of algebraic curve, i.e. they can be described by some algebraic equations with coefficients in some finite extension of Q. Number theorists are interested in the reductions modulo p of these equations. The problem is that these equations are very difficult to compute. I will describe a method to find these reductions without actually knowing the equation. [less ▲]

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See detailMumford curves covering p-adic Shimura curves and their fundamental domains
Amoros Carafi, Laia UL; Milione, Piermarco

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 ▲]

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See detailDeligne's theorem
Amoros Carafi, Laia UL

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

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