Derived ℓ-adic zeta functions

Abstract

Let K0(Vk) be the Grothendieck group of k-varieties. Campbell and Zakharevich have constructed a higher algebraic K-theory spectrum K(Vk) such that π0K(Vk)=K0(Vk). In this paper we construct non-trivial classes in the higher homotopy groups of K(Vk) when k is finite or a subfield of C. To do this we give a recipe for lifting motivic measures K0(Vk)→K0(E) to maps of spectra K(Vk)→K(E). We consider two special cases: the classical local zeta function, thought of as a homomorphism K0(VFq)→K0(End(Qℓ)), and the compactly-supported Euler characteristic, thought of as a homomorphism K0(VC)→K0(Q). We use lifts of these motivic measures to prove that the Grothendieck spectrum of varieties contains nontrivial geometric information in its higher homotopy groups by showing that the map S→K(Vk) is nontrivial in higher dimensions when k is finite or a subfield of C, and, moreover, that when k is finite this map is not surjective on higher homotopy groups.