### Kloosterman sums and Siegel zeros

Heilbronn Number Theory Seminar

24th January 2018, 4:00 pm – 5:00 pm

Howard House, 4th Floor Seminar Room

Kloosterman sums arise naturally in the study of the

distribution of various arithmetic objects in analytic number theory.

The 'vertical' Sato-Tate law of Katz describes their distribution over a

fixed field F_p, but the equivalent 'horizontal' distribution as the

base field varies over primes remains open. We describe work showing

cancellation in the sum over primes if there are exceptional

Siegel-Landau zeros. This is joint work with Sary Drappeau, relying on a

fun blend of ideas from algebraic geometry, the spectral theory of

automorphic forms and sieve theory.

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