Zeta function of F-gauges and special values

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.