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Washington State
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Combinatorics, Linear Algebra and Number Theory Seminar

Department of Mathematics and Statistics

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October 11, Monday, 4:10 - 5:00 PM

Jack Buttcane

Department of Mathematics and Statistics

University of Maine

Combinatorics, Linear Algebra and Number Theory Seminar

Department of Mathematics and Statistics

Zoom

October 11, Monday, 4:10 - 5:00 PM

Jack Buttcane

Department of Mathematics and Statistics

University of Maine

Title:
Automorphic forms and exponential sums on GL(3) (and
beyond)

Abstract: The modular forms of weight k for GL(2) may be thought of as type of Maass form with weight, with some renormalization, and the concept of "Maass forms with weight" extends to automorphic forms with "ramification at infinity" on reductive groups. Similarly, we can extend the idea of Poincare series, Kloosterman sums and Bessel functions. I will discuss the GL(3) case of Kuznetsov formulas, aka relative trace formulas, that connect these things. In particular, I will show how the arithmetic Kuznetsov formula demonstrates cancellation in smooth sums of the long-element Kloosterman sums and what needs to be done to extend the result, if possible, to the hyper-Kloosterman sums. Time permitting, I'll discuss the situation on GL(n) and more general reductive groups.

Abstract: The modular forms of weight k for GL(2) may be thought of as type of Maass form with weight, with some renormalization, and the concept of "Maass forms with weight" extends to automorphic forms with "ramification at infinity" on reductive groups. Similarly, we can extend the idea of Poincare series, Kloosterman sums and Bessel functions. I will discuss the GL(3) case of Kuznetsov formulas, aka relative trace formulas, that connect these things. In particular, I will show how the arithmetic Kuznetsov formula demonstrates cancellation in smooth sums of the long-element Kloosterman sums and what needs to be done to extend the result, if possible, to the hyper-Kloosterman sums. Time permitting, I'll discuss the situation on GL(n) and more general reductive groups.