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Categorifying Zeta Functions for Quadratic Covers

Jon Aycock, Andrew Kobin

TL;DR

The paper develops a unified, category-theoretic framework to categorify zeta-function factorization across double covers of number fields, curves, CW complexes, and graphs, using incidence algebras and objective linear algebra. It proves a main equivalence π_*ζ_Y + ζ_X*L(π)^{-} ≅ ζ_X*L(π)^{+} that decategorifies to classical product formulas like ζ_K(s)=ζ_F(s)L(χ,s) and Z(Y,t)=Z(X,t)L(π,t), while providing local, global, and motivic variants. The work yields concrete applications: counting formulas for points on elliptic and hyperelliptic curves, prime-cycle counts on graphs, asymptotics for supersingular isogeny graphs, and a zeta-function–driven perspective on quadratic reciprocity. It sets the stage for L-functors and motivic generalizations, with potential extensions to higher-degree extensions, generalized L-functions, and ramified graph covers, offering a robust framework for intertwining arithmetic, geometry, and combinatorics through categorification.

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.

Categorifying Zeta Functions for Quadratic Covers

TL;DR

The paper develops a unified, category-theoretic framework to categorify zeta-function factorization across double covers of number fields, curves, CW complexes, and graphs, using incidence algebras and objective linear algebra. It proves a main equivalence π_*ζ_Y + ζ_X*L(π)^{-} ≅ ζ_X*L(π)^{+} that decategorifies to classical product formulas like ζ_K(s)=ζ_F(s)L(χ,s) and Z(Y,t)=Z(X,t)L(π,t), while providing local, global, and motivic variants. The work yields concrete applications: counting formulas for points on elliptic and hyperelliptic curves, prime-cycle counts on graphs, asymptotics for supersingular isogeny graphs, and a zeta-function–driven perspective on quadratic reciprocity. It sets the stage for L-functors and motivic generalizations, with potential extensions to higher-degree extensions, generalized L-functions, and ramified graph covers, offering a robust framework for intertwining arithmetic, geometry, and combinatorics through categorification.

Abstract

In various contexts, the zeta function of an object splits into a product of -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 over a finite field, we prove a collection of combinatorial formulas relating the number of ramified, split and inert points on to the overall point count of ; and for a graph , we deduce analogous combinatorial formulas for the numbers of split and inert primes in a Galois double cover . 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.
Paper Structure (41 sections, 23 theorems, 195 equations)

This paper contains 41 sections, 23 theorems, 195 equations.

Table of Contents

  1. Introduction
  2. Categorification
  3. Applications
  4. Relation to Previous Work
  5. Organization
  6. Double Covers
  7. Quadratic Number Fields
  8. Hyperelliptic Curves
  9. Topological Covers
  10. Galois Covers of Graphs
  11. Incidence Algebras and Objective Linear Algebra
  12. Incidence Algebras for Monoids
  13. The Incidence Algebra of a Number Field
  14. The Incidence Algebra of Effective $0$-Cycles for a Variety
  15. The Incidence Algebra of Effective $0$-cycles for a Graph
  16. ...and 26 more sections

Key Result

Theorem 1.1

In the incidence algebra $I(S(X))$, there is an equivalence of linear functors

Theorems & Definitions (77)

  • Theorem 1.1
  • Theorem 1.2: Theorem \ref{['thm:motivicformula']}
  • Theorem 1.3: Theorem \ref{['thm:aclsst']}, Corollary \ref{['cor:shimuraasymptotic']}
  • Proposition 2.1
  • Example 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • ...and 67 more