Arithmetic and motivic statistics via zeta functions
Margaret Bilu (NYU)
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.
Audience: researchers in the topic
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