𝐾𝑈-local zeta-functions of finite CW-complexes

Begin with the Hasse-Weil zeta-function of a smooth projective variety over Q Q . Replace the variety with a finite CW-complex, replace étale cohomology with complex K K -theory K U ∗ KU^* , and replace the p p -Frobenius operator with the p p th Adams operation on K K -theory. This simple idea yields a kind of “ K U KU -local zeta-function” of a finite CW-complex. For a wide range of finite CW-complexes X X with torsion-free K K -theory, we show that this zeta-function admits analytic continuation to a meromorphic function on the complex plane, with a nice functional equation, and whose special values in the left half-plane recover the K U KU -local stable homotopy groups of X X away from 2 2 . We then consider a more general and sophisticated version of the K U KU -local zeta-function, one which is suited to finite CW-complexes X X with nontrivial torsion in their K K -theory. This more sophisticated K U KU -local zeta-function involves a product of L L -functions of complex representations of the torsion subgroup of K U 0 ( X ) KU^0(X) , similar to how the Dedekind zeta-function of a number field factors as a product of Artin L L -functions of complex representations of the Galois group. For a wide range of such finite CW-complexes X X , we prove analytic continuation of the zeta-function, and we show that the special values in the left half-plane recover the K U KU -local stable homotopy groups of X X away from 2 2 if and only if the skeletal filtration on the torsion subgroup of K U 0 ( X ) KU^0(X) splits completely.