Computational aspects of modular forms and p-adic Galois representations

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The theory of automorphic forms is one of the core topics of algebraic number theory. It has a long history and is at the same time, with such spectacular successes as the proof of Fermat s last theorem, the Sato-Tate conjecture or the Serre conjecture, one of the most flourishing topics. Presently many new directions are explored. Basic computer algorithms axe an invaluable tool to yield experimental data which further advance the theory. The present proposal provides one approach to obtain and implement such algorithms. It also explains a number of interesting experiments which we hope to perform. The approach applies to number as well as function fields. Its relation to classical automorphic forms is explained by the Langlands program - and in often conjectural. The intention is to compute Galois representations associated to automorphic via their Hecke eigensystems. The key tool is the algorithmic computation of quotients of Bruhat-Tits buildings by cocompact arithmetic subgroups. The result is a finite simplicial set and thus a finite combinatorial object. Automorphic forms are sections of local systems on these finite quotients. Their systems of Hecke eigenvalues can be computed combinatorially.