Explicit methods for algebraic automorphic forms

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The Langlands Programme is one of the most ambitious research programmes in modern mathematics. The programme is a vast web of conjectures which connect analysis, algebraic geometry, representation theory and number theory. At the heart of this programme are automorphic forms (analytic objects) and Galois representations (geometric objects). The two sets are related by a series of conjectures whose level of precision varies according to our understanding. The programme warrant our attention as it has far reaching consequences. This project will develop explicit methods for algebraic automorphic forms. By providing concrete examples of automorphic to work with, it will help improve our understanding of many of conjectures in this field. The PI is particularly interested in the mod p Langlands, as he has had some success in this case with the group GL(2).