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Analytic Number Theory: Higher Order Structures (ANTHOS)
Start date: Oct 1, 2010, End date: Sep 30, 2015 PROJECT  FINISHED 

This is a proposal for research at the interface of analytic number theory, automorphic forms and algebraic geometry. Motivated by fundamental conjectures in number theory, classical problems will be investigated in higher order situations: general number fields, automorphic forms on higher rank groups, the arithmetic of algebraic varieties of higher degree. In particular, I want to focus on- computation of moments of L-function of degree 3 and higher with applications to subconvexity and/or non-vanishing, as well as subconvexity for multiple L-functions;- bounds for sup-norms of cusp forms on various spaces and equidistribution of Hecke correspondences;- automorphic forms on higher rank groups and general number fields, in particular new bounds towards the Ramanujan conjecture;- a proof of Manin's conjecture for a certain class of singular algebraic varieties.The underlying methods are closely related; for example, rational points on algebraic varietieswill be counted by a multiple L-series technique.
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