Zeta function of F-gauges and special values

TL;DR

This paper unconditionally formulates and proves conjectures relating the zeta functions of algebraic structures over finite fields using novel techniques involving $F$-gauges and stack theory.

Contribution

It provides the first unconditional proof of conjectures connecting zeta functions and $F$-gauges over finite fields, introducing the concept of stable Bockstein characteristic.

Findings

Unconditional proof of conjectures for $F$-gauges over finite fields

Application of stacky approach to $F$-gauges by Drinfeld and Bhatt--Lurie

Introduction of stable Bockstein characteristic

Abstract

In 1966, Tate proposed the Artin--Tate conjectures, which expresses special values of zeta function associated to surfaces over finite fields. Conditional on the Tate conjecture, Milne--Ramachandran formulated and proved similar conjectures for smooth proper schemes over finite fields. The formulation of these conjectures already relies on other unproven conjectures. In this paper, we give an unconditional formulation of these conjectures for dualizable $F$-gauges over finite fields and prove them. In particular, our results also apply unconditionally to smooth proper varieties over finite fields. A key new ingredient is the notion of ``stable Bockstein characteristic" that we introduce. Our proof uses techniques from the stacky approach to $F$-gauges recently introduced by Drinfeld and Bhatt--Lurie and the author's recent work on Dieudonn\'e theory using $F$-gauges.