$KU$-local zeta-functions of finite CW-complexes

Abstract:

Begin with the Hasse-Weil zeta-function of a smooth projective variety over $\mathbb {Q}$. Replace the variety with a finite CW-complex, replace étale cohomology with complex $K$-theory $KU^*$, and replace the $p$-Frobenius operator with the $p$th Adams operation on $K$-theory. This simple idea yields a kind of “$KU$-local zeta-function” of a finite CW-complex. For a wide range of finite CW-complexes $X$ with torsion-free $K$-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 $KU$-local stable homotopy groups of $X$ away from $2$.

We then consider a more general and sophisticated version of the $KU$-local zeta-function, one which is suited to finite CW-complexes $X$ with nontrivial torsion in their $K$-theory. This more sophisticated $KU$-local zeta-function involves a product of $L$-functions of complex representations of the torsion subgroup of $KU^0(X)$, similar to how the Dedekind zeta-function of a number field factors as a product of Artin $L$-functions of complex representations of the Galois group. For a wide range of such finite CW-complexes $X$, we prove analytic continuation of the zeta-function, and we show that the special values in the left half-plane recover the $KU$-local stable homotopy groups of $X$ away from $2$ if and only if the skeletal filtration on the torsion subgroup of $KU^0(X)$ splits completely.