The orbit method and analysis of automorphic forms
Recent developments in Analytic Number Theory May 01, 2017 - May 05, 2017
Location: SLMath: Eisenbud Auditorium
Automorphic forms
Kirillov orbit model
Gross-Prasad families
Ratner's Theorem
spherical harmonics
11G07 - Elliptic curves over local fields [See also 14G20, 14H52]
11H31 - Lattice packing and covering (number-theoretic aspects) [See also 05B40, 52C15, 52C17]
11G10 - Abelian varieties of dimension $> 1$> 1 [See also 14Kxx]
11R06 - PV-numbers and generalizations; other special algebraic numbers; Mahler measure
Venkatesh
In the analytic theory of automorphic forms, especially in higher rank, one encounters complicated integrals over Lie groups which must be either evaluated or estimated. I will discuss how Kirillov's orbit method allows one to do this, at least heuristically. These ideas can often be made rigorous; I will apply it to evaluate the average value of L-functions over certain (Gross-Prasad) families, in any rank. This evaluation also uses Ratner's theorem on measure rigidity. Joint work with Paul Nelson.
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Venkatesh Notes
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Venkatesh
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12-Venkatesh.mp4
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