Resumen de Arithmetic properties of Abelian varieties under Galois conjugation

Xavier Guitart

• This thesis is concerned with several arithmetic properties of abelian varieties that are isogenous to their Galois conjugates, To be more precise, the central object of study are abelian k-varieties, especially in the case where k is a number field. That is, abelian varieties over the algebraic closure of k that are equivariantly isogenous to all of their Galois conjugates.

  The interest in Q-varieties arose in connection with the Shimura-Taniyama conjecture about modularity of elliptic curves over Q, and its generalizations to higher dimensional varieties over Q and to varieties over number fields. Indeed, the absolutely simple factors of the modular abelian varieties attached to classical modular forms are Q-varieties. More generally, if k is a totally real number field, then the absolutely simple factors of the varieties attached to Hilbert modular forms over k are k-varieties.

  In Chapter 2 we consider abelian varieties with field of moduli k up to isogeny. A theorem of Ribet characterizes under what conditions such a variety is isogenous to a variety defined over k. Using this result, we identify two obstructions to descend the field of definition in terms of a set of Galois cohomology classes attached to the variety. In this way we obtain a characterization of the descent property which is suitable for practical computations.

  In Chapter 3 we apply the results of the previous chapter to abelian k-varieties. We characterize their fields of definition up to isogeny, in terms of a Galois cohomology class canonically attached to them. We also describe their fields of complete definition (that is, where the endomorphisms and the isogenies are defined).

  In Chapter 4 we study fields of definition of abelian k-varieties of the first kind. First, we perform the technical computations that are needed to determine in practice their minimal fields of definition. Then we illustrate the techniques developed in the previous chapters with some concrete examples of building blocks with quaternionic multiplication, for which we explicitly compute their minimal fields of definition.

  In Chapter 5 we study abelian k-varieties A defined over k whose endomorphisms defined over k are a maximal subfield of the full endomorphism algebra of A. We call them Ribet-Pyle varieties. The main result is that a Ribet-Pyle variety is isogenous to a power of some k-variety and, conversely, that every k-variety is the absolutely simple factor of some Ribet-Pyle variety. Applying this to varieties over k of GL_2-type, we obtain a description of their absolutely simple factors that generalizes the results of K. Ribet and E. Pyle in the case k=Q. We also study restrictions of scalars of k-varieties. To be more precise, we compute their algebra of endomorphisms defined over k, and we characterize when they are isogenous to products of Ribet-Pyle varieties over k.

  In Chapter 6 we study some properties of abelian Q-varieties related to their modularity. The main result is a characterization of the abelian varieties over a number field K such that L(B/K;s) is a product of L-series of classical modular forms over Q. The varieties satisfying this property, which we have called strongly modular, are the ones that are useful in most of the applications of modularity.

  Finally, in Chapter 7 we present some explicit examples of strongly modular abelian varieties. They are Jacobians of genus 2 curves given by explicit equations over number fields, and we deduce their strong modularity just as a consequence of their geometric and arithmetic properties. In some cases we are able to identify the corresponding modular forms giving the L-series of the varieties.