Adelic and l-adic Galois representations arising from abelian varieties and modular forms

The main goal of the project is to increase our theoretical and computational understanding of l-adic Galois representations arising naturally in the setting of arithmetic geometry. We focus on two types of (familes of) Galois representations: those arising from a g-dimensional abelian variety A defined over a number field K, and those arising from an eigenform f in the space of cusp forms of weight k>1 and level N. In both cases we obtain for each prime l a certain l-adic representation of (an open subgroup of) the absolute Galois group. The resulting family of representations, which we can package into a single adelic representation, is doubly important both as it encodes many arithmetic properties of the object from which it is engendered (i.e., of A/K or f), and as it sheds light on the mysterious structure of the Galois group it represents. As a step toward better understanding these families of representations, we develop techniques, in both the abelian variety and the modular form setup, for explicitly computing the image of the adelic representation, and thus also the image of each l-adic representation.

Duration of Project

Start date: 09/2010

End date: 09/2011