Poincaré institute

Reinventing rational points

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Motivic Euler products

Margaret Bilu

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 over k. Many problems in number theory have a natural, more geometric counterpart involving elements of this ring. I will start by explaining how one can make sense of a notion of Euler product for some power series with coefficients in this ring. A few applications to motivic stabilisation results will be mentioned, after which I will show how this notion may be used to prove a motivic analogue of Manin's conjecture for equivariant compactifications of vector groups.