Arithmetic and motivic statistics via zeta functions

Abstract: The Grothendieck group of varieties over a field $k$ is the quotient of the free abelian group on isomorphism classes of algebraic varieties over k by the so-called cut-and-paste relations. Many results in number theory have a natural motivic analogue which can be formulated in the Grothendieck ring of varieties. For example, Poonen's finite field Bertini theorem has a motivic counterpart due to Vakil and Wood, though none of the two statements can be deduced from the other. We describe a conjectural way to unify the number-theoretic and motivic statements (when the base field is finite) in this and other examples, and will provide some evidence towards it. A key step is to reformulate everything in terms of convergence of zeta functions of varieties in several different topologies. This is joint work with Ronno Das and Sean Howe.

algebraic geometry

Audience: researchers in the topic

( video )

Stanford algebraic geometry seminar

Series comments: The seminar was online for a significant period of time, but for now is solely in person. More seminar information (including slides and videos, when available): agstanford.com