Arithmetic field theory via pro-p duality groups

TL;DR

This paper constructs a general framework for arithmetic TQFTs by using pro-$p$ Poincaré duality groups and relative PD structures to define pro-$p$ cobordisms, culminating in a complete classification of $(1+1)$ pro-$p$ TQFTs via $ ext{Aut}(\mathbb{Z}_p)$-extended Frobenius algebras. It develops a pro-$p$ Dijkgraaf–Witten theory and provides explicit, representation-theoretic formulas to count Galois extensions of local $p$-adic fields with prescribed finite $p$-group Galois groups, thereby connecting geometric cobordism notions with arithmetic invariants. The approach hinges on a pants-decomposition style analysis of pro-$p$ PD$^2$ pairs, gluing theorems for pro-$p$ PD structures, and a categorical bridge to cospans of pro-$p$ groupoids, enabling both structural classification and concrete calculation of arithmetic invariants. The results extend arithmetic topology by offering a unified, purely algebraic framework to study cobordisms and TQFTs alongside classical arithmetic objects, with potential for further generalizations and deformations of DW theories.

Abstract

Using the theory of pro-p groups and relative Poincaré duality, we define a type of cobordism category well suited to arithmetic topology. We completely classify topological quantum field theories on these two-dimensional versions of our cobordism categories. This classification uses Frobenius algebras with extra operations corresponding to automorphisms of the p-adic integers. We look in more detail at the example of arithmetic Dijkgraff--Witten theory for a finite gauge p-group in this setting. This allows us to deduce formulae counting Galois extensions of local p-adic fields whose Galois groups are the given gauge group.

Theorems & Definitions (71)

• Theorem 1.1
• Corollary 1.2
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Theorem 2.7
• Definition 2.8