Two Arithmetic Applications of Additive Combinatorics

Abstract: Dirichlet's theorem on primes in arithmetic progressions is a key input into the proof of the local-global principle for quadratic forms over $\mathbb Q$. It also makes an appearance in Corollary X.6.2.1 of Silverman's "The Arithmetic of Elliptic Curves" where it is used to prove that there are infinitely many elliptic curves $E/\mathbb Q$ of rank zero. Dirichlet's theorem can be understood as a characterization of which linear polynomials take on prime values infinitely often. More generally, the multivariate Bateman-Horn conjecture makes precise predictions for how often systems of polynomials take on simultaneously prime values.

In ongoing joint work with Katy Woo, we prove some new cases of the multivariate Bateman-Horn conjecture (utilizing techniques of recent work of Green and Sawhney). We then apply our prime producing theorem towards the study of (Brauer-Manin obstructions to) local-global principles on certain conic bundles and towards a construction of rank (exactly) 2 elliptic curves in quadratic twist families. Both of these applications build on the work of too many people to list.

This project is in progress and joint with Katy Woo.

algebraic geometrynumber theory

Audience: researchers in the topic

MIT number theory seminar

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