Arithmetic properties of Abelian varieties under Galois conjugation

摘要: 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 study k-varieties, especially in case where k a number field. That is, over algebraic closure equivariantly all conjugates. The interest Q-varieties arose connection Shimura-Taniyama conjecture about modularity elliptic curves Q, and its generalizations higher dimensional Q fields. Indeed, absolutely simple factors modular attached classical forms Q-varieties. More generally, if totally real field, then Hilbert k-varieties. In Chapter 2 we consider field moduli up isogeny. A theorem Ribet characterizes under what conditions such variety defined k. Using this result, identify two obstructions descend definition terms set cohomology classes variety. In way obtain characterization descent property which suitable for practical computations. In 3 apply results previous chapter k-varieties. We characterize fields isogeny, class canonically them. also describe complete (that endomorphisms isogenies defined). In 4 k-varieties first kind. First, perform technical computations needed determine practice minimal definition. Then illustrate techniques developed chapters some concrete examples building blocks quaternionic multiplication, explicitly compute definition. In 5 whose maximal subfield full endomorphism algebra A. call them Ribet-Pyle varieties. The main result power k-variety and, conversely, every factor Applying GL_2-type, description generalizes K. E. Pyle k=Q. restrictions scalars k, when they products k. In 6 related modularity. K L(B/K;s) product L-series Q. satisfying property, have called strongly modular, ones useful most applications Finally, 7 present explicit They Jacobians genus given by equations fields, deduce strong just as consequence geometric properties. cases able corresponding giving