Galois Representations and Modular Forms

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0070659The proposer will investigate connections between Galois representations and modular forms in the context of two fundamental problems in algebraic number theory. The first of these problems is to determine which Galois representations are associated to automorphic representations. The second problem is to relate special values of L-functions of varieties or motives to associated algebraic quantities (such as orders of Selmer groups). In the direction of the first problem the proposer intends to pursue extensions of his recent work on the modularity of various families of two-dimensional p-adic Galois representations. In particular, the proposer intends to explore possible extensions to representations of higher dimension. In the direction of the second problem, the proposer will investigate the influence of divisibility properties of constant terms of Eisenstein series on higher-rank groups (which often involve special values of L-functions) on congruences between the Hecke eigenvalues of these Eisenstein series and those of cusp forms.Both of these problems fall under the rubric "Arithmetic Geometry," a branch of mathematics which attempts to apply sophisticated mathematics to the often easy-to-state, but-hard-to-solve problems of number theory. The problems most readily tackled by arithmetic geometry include 1) counting the number of integer solutions to systems of polynomials and 2) understanding L-functions (functions attached to systems of polynomials and which encode information about their solutions). These problems are of increasing importance in applications. Recent advances in computing, cryptography, and coding theory have depended on solutions to these problems as do many proposed advances in these areas.