The quadratic Bateman-Horn conjecture over function fields

Abstract: Are there infinitely many natural numbers $n$ with $n^2+1$ 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 $f^2 + u$ 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.

algebraic geometrynumber theory

Audience: researchers in the topic

MAGIC (Michigan - Arithmetic Geometry Initiative - Columbia)

Series comments: Description: Research seminar in arithmetic geometry

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