Adelic and 1-adic Galois representations arising from abelian varieties and modular forms
Final Report Abstract
The main goal of the project was to increase our theoretical and computational understanding of l-adic Galois representations arising naturally in the setting of arithmetic geometry. We were mainly concerned with 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 can define for each prime l a certain l-adic representation ρl of (an open subgroup of) the absolute Galois group Gal(Q/Q). The resulting family (ρl ), which we can package into a single adelic representation ρ = ρl , 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 sought to develop techniques, in both the abelian variety and the modular form setup, for explicitly computing the image of the adelic representation ρ, and thus also the image of each l-adic representation ρl. I consider my inability to develop the necessary theoretical tools for computing explicit examples of abelian surfaces (or abelian varieties of higher dimension for that matter) with surjective adelic Galois representations to be a major failure of this project. That said, I do not consider this project to be a total failure.
Publications
- Elliptic curves with surjective adelic Galois representations, Experiment. Math. 19 (2010), no. 4, 495-507
Aaron Greicius
