Zeta functions of algebraic varieties and special values

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 relied on other unproven conjectures. In this talk, I will discuss an unconditional formulation and proof of these conjectures. A key new ingredient is the notion of “stable Bockstein characteristic” that we will introduce.