Zeta function of F-gauges and special values
Shubhodip Mondal
TL;DR
This work delivers an unconditional framework for the special values of zeta functions attached to smooth proper varieties over finite fields by working with dualizable $F$-gauges. Central to the approach are the stable Bockstein characteristic and the syntomic-invariant $\mu_{\text{syn}}(M,r)$, which together determine the $p$-adic size of the leading term of $\zeta(M,s)$ at an integer $r$ via $|\lim_{s\to r} \zeta(M,s)/(1-q^{r-s})^{\rho}|_p = 1/(\mu_{\text{syn}}(M,r) q^{\chi(M,r)})$. The paper develops a robust toolkit—Bockstein complexes, a stable Bockstein formalism, and a descent spectral sequence for syntomic cohomology—that unifies and extends Milne–Ramachandran-type results, applies to $p$-divisible groups, and yields concrete consequences for surfaces over finite fields. In particular, it recovers and strengthens classical special-value formulas in an unconditional setting and provides new links between $p$-adic invariants, the Nygaard filtration, and Hodge data, with applications to Artin–Tate-type conjectures for surfaces.
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 Dieudonné theory using $F$-gauges.