Mark Shusterman, July 23, 2020

Title: The quadratic Bateman-Horn conjecture over function fields

Abstract: Are there infinitely many natural numbers n with n2+1 a prime? In a joint work in progress with Will Sawin we show that for some finite fields F, there are infinitely many monic polynomials f in F[u] for which f2+u is prime (i.e. monic irreducible). After surveying some earlier works, I'll explain how to reduce the problem to a question of cancellation in an incomplete exponential sum. Via the Grothendieck-Lefschetz trace formula, this will lead us to bounding the cohomology of certain sheaves on the complement of a hyperplane arrangement in affine space. Video of talk