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 va- rieties 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.
By Jean-Louis Colliot-Thélène
By Will Sawin
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By Pierre Le Boudec
By Cyril Demarche
By Arne Smeets
By Julia Hartmann
By Olivier Wittenberg
By Antoine Chambert-Loir
