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Some Diophantine analogies between Dirichlet series and polynomials (Vesselin Dimitrov, Caltech)
March 25, 2025 @ 12:15 pm - 1:10 pm
I will present an integral — requiring no character twists — converse theorem for recognizing when is a Dirichlet series with algebraic integer coefficients equal to the L-function of a modular form. This refines the unbounded denominators conjecture of Atkin and Swinnerton-Dyer. Analogies with basic function field arithmetic then suggest a quantitative refinement which precludes a pair of GL(2) automorphic L-functions with closely matched up zeros. I will explain how to get at such a theorem.
