A quadratically enriched zeta function

Zeta funtion
A1-homotopy theory

Appears in collection : 2023 - T2 - WS1 - GAP XVIII: Homotopy algebras and Higher Structures

For a smooth variety over a finite field, we enrich the logarithmic derivative of the zeta function to a power series with coefficients in the Grothendieck--Witt group of stable isomorphism classes of unimodular modular forms, using traces of powers of Frobenius in A1-homotopy theory. (It is not a motivic measure applied to Professor Kapranov's.) In analogy with the celebrated connection between the Betti numbers of associated complex manifolds, we show the quadratically enriched logarithmic zeta function to be connected to the Betti numbers of the associated real manifolds. We show a Lefshetz fixed point theorem for cellular varieties with a recent cohomology theory of F. Morel and A. Sawant, proving a rationality result for cellular varieties. This is joint work with Margaret Bilu, Wei Ho, Padma Srinivasan, and Isabel Vogt.