Arithmetic Study Group: Automorphicity, Mean-Periodicity and Higher Adelic Duality
4 February 2014 13:00 in CM107
Associated to an algebraic variety over a number field, one has a family of Hasse-Weil L-functions. Such L-functions are "motivic", and, according to the Langlands program, should be identified with their "automorphic" counterparts. One of the consequences of such an identification would be the conjectural analytic continuation and functional equation of the L-functions. On the other hand, by so-called "converse theorems", such analytic properties are a stepping stone to general automorphic properties. In practice it is very difficult to prove that a general Hasse-Weil L-function comes from an automorphic representation. The most up-to-date results have a heavy dependence on the base number field, the Euler characteristic and the dimension - A recent example of such a statement being "an elliptic curve over a totally real field is potentially modular". By viewing an algebraic curve as the generic fibre of an arithmetic surface, I will show how to understand certain analytic properties of Hasse-Weil L-functions in terms of "mean-periodicity", regardless of base field or genus, and provide comparisons to the conjectural automorphicity of the generic fibre. Time permitting, I will show how to interpret mean-periodicity as a statement of analytic two-dimensional adelic duality.
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