Categorifying Zeta Functions for Quadratic Covers

Abstract

In various contexts, the zeta function of an object splits into a product of $L$-functions. We categorify this product formula for quadratic covers of objects in the following contexts: quadratic extensions of number fields, ramified double covers of algebraic curves, ramified double covers of topological spaces and Galois double covers of graphs. Our unified approach utilizes objective linear algebra in the abstract incidence algebra of each object, interpreted appropriately. We also provide several applications: for a hyperelliptic curve $C$ over a finite field, we prove a collection of combinatorial formulas relating the number of ramified, split and inert points on $C$ to the overall point count of $C$; and for a graph $G$, we deduce analogous combinatorial formulas for the numbers of split and inert primes in a Galois double cover $\widetilde{G}\rightarrow G$. We then use the formulas for graphs to deduce asymptotic counts of cycles in supersingular isogeny graphs and certain associated dual graphs of special fibers of Shimura curves. Finally, we analyze quadratic reciprocity from the perspective of zeta functions.