Algebraic geometry seminar
Wednesday September 18th, 2019
Time: 4:00 PM - 5:30 PM
Title: Motivic Euler products
Speaker: Margaret Bilu, Courant Institute of Mathematical Sciences
Location: Math Tower P-131
The Grothendieck group of varieties over a field k is the quotient of the free abelian group of isomorphism classes of varieties over k by the so-called cut-and-paste relations. It moreover has a ring structure coming from the product of varieties. Many problems in number theory have a natural, more geometric counterpart involving elements of this ring. I will focus on Manin's conjecture and on its motivic analog: the latter predicts the behavior of moduli spaces of curves of large degree on some algebraic varieties. It may be formulated in terms of the generating series of the classes of these moduli spaces in the Grothendieck ring, called the motivic height zeta function. This will lead me to explain how some power series with coefficients in the Grothendieck ring can be endowed with an Euler product decomposition and how this can be used to give a proof of the motivic version of Manin's conjecture for equivariant compactifications of vector groups. Time permitting, some further applications of motivic Euler products to problems in motivic statistics will be mentioned.