p-adic Motives and Special Values of Zeta Functions

Explore an advanced mathematical lecture examining the unconditional formulation and proof of Artin-Tate conjectures for special values of zeta functions associated to surfaces over finite fields. Delve into the historical context beginning with Tate's 1966 proposal and the subsequent work by Milne and Ramachandran on smooth proper schemes over finite fields, understanding how previous formulations relied on unproven conjectures. Learn about the innovative approach using F-Gauges theory, originally introduced by Fontaine-Jannsen and further developed by Bhatt-Lurie and Drinfeld, which serves as a candidate for p-adic motives theory. Discover the central role of stable Bockstein characteristics in this mathematical framework and how these concepts combine to provide an unconditional proof of these fundamental conjectures in algebraic geometry and number theory.