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Resumen de L-functions and artin representations attached to twisted abelian varieties

Francesc Fité Árbol académico

  • Given an abelian variety A defined over a number field k, it is a general principle that its Hasse-Weil L-function L(A,s) encodes arithmetic properties of A. Modularity, functional equations, the Birch and Swinnerton-Dyer conjecture, and Sato-Tate distributions are examples of this deep relation. In the present thesis, we explore relations between Hasse-Weil L-functions of abelian varieties and Artin representations of finite groups. This can be thought as a contribution towards the understanding of the behaviour of these fundamental properties under the effect of twist.

    Chapter 1 is devoted to preliminary definitions and results. In Chapter 2, for abelian varieties A and A' defined over a number field k and isogenous over a finite Galois extension L/k, we define a rational Artin representation q(A,A';L/k) of the group Gal(L/k). This Artin representation shows a global relation between the corresponding L-functions. This global relation is induced by the inclusion of QG_k]-modules VA') q(A,A';L/k) ¿ VA). We establish the fundamental properties of q(A,A';L/k).

    In Chapter 3, we explore further properties of q(C,C')=q(A,A';L/k) when A and A' are the Jacobians of two twisted curves C and C' defined over the number field k. More precisely, if K denotes the field of definition of the group of automorphisms Aut(C), we introduce:

    (i) a representation q_C, which only depends on C, and (ii) an Artin representation q_¿, for every isomorphism ¿: C' ---> C defined over a finite Galois extension L/k, of the group Gal(L/k) isomorphic to q(C,C'), such that q_¿ factors through q_C whenever L satisfies that End_L^0(J(C))=End_K^0(J(C)). Furthermore, we show how this property can be used to compute the decomposition of q(C,C') ¿VC) into simple QG_k]-modules, when J(C) decomposes as the power of an elliptic curve and certain hypothesis are satisfied.

    In Chapter 4, we apply the previous results to curves of genus 2 to obtain the decomposition of q(C,C') ¿ VC). Thanks to this decomposition, for every prime P unramified in L, from the local factor L_P(C/k,T) and the representation q(C,C'), we are able to determine either the twisted local factor L_P(C'/k,T) or the product L_P(C'/k,T) · L_P(C'/k,-T).

    In Chapter 5, we analyze Sato-Tate distributions for abelian surfaces A defined over Q. Let K denote the minimal field of definition of End(A). Our guess is that the Sato-Tate distribution of the traces of the local factors of A is determined by the Artin representation q(A,A;K/Q), introduced in the previous chapters. Then, we describe a ceratin field M_A K and we consider the representation q(A)=q(A,A;K/M_A). We prove that, under the above natural guess, if q(A) is isomorphic to q(A'), for an abelian surface A' defined over Q and [M_A:Q]=[M_ A':Q], then A and A' have the same Sato-Tate distribution of traces. We obtain 26 possible isomorphism classes for the pair (q(A),[M_A:Q]). It follows that there exist at most 26 Sato-Tate distributions of the traces of an abelian surface defined over Q. We find 26 curves defined over Q, whose Jacobians follow each of these 26 distributions. This refutes a conjecture of Kedlaya-Sutherland, which asserted that the number of Sato-Tate distributions of abelian surfaces defined over Q is 23. This chapter is a partial exposition of a work which is in progress and we thank J-P. Serre for driving our attention to this direction.

    In Chapter 6, as an example that the techniques introduced do apply to higher dimension than two, for each curve C' in a family of twists of a certain genus 3 curve C, we explicitly compute the Artin representation attached to the Jacobians of C and C' and show how this Artin representation can be used to determine the L-function of the curve C' in terms of the L-function of C. Moreover, from this Artin representation, we are able to compute the moments of the Sato-Tate distributions of the traces of the local factors of the curve C'.


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