2025/01/09 by Mondal, Shubhodip
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT) #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2501.05027
openalex publication_date 2025/01/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.