An Arithmetic Day

Program:

09:00 - 10:00. Coffee

10:00 - 11:00  Quentin Gazda (École Polytechnique, Palaiseau) 

Motivic cohomology for function fields

Abstract : The category of Anderson t-motives is widely considered as the function field counterpart of the unknown category of mixed motives. After discussing anachronic reasons why one should agree on this, we shall explain how to make profit of this analogy to define motivic cohomology for function fields — aka t-motivic cohomology — as certain submodules of extensions in the category of t-motives. I will then present recent computations for Carlitz twists — the counterpart of Tate twists — obtained jointly with Andreas Maurischat, and discuss conjectural relations to special zeta values.

11:00 - 12:00  Tuan Ngo Dac (Université de Caen, Normandie)

On Hopf algebras and multiple zeta values

Abstract: Multiple zeta values (MZV’s for short) in positive characteristic were introduced by Thakur as analogues of classical multiple zeta values of Euler. In this talk we first review Euler’s MZV’s and some Hopf algebras related to these values, and state important conjectures such as those of Ihara-Kaneko-Zagier, Hoffman and Zagier.  Then we explain our recent work on algebraic structures of MZV’s in positive characteristic. 

14:00 - 15:00  Dinesh Thakur (University of Rochester)  

Collaborating versus exceptional primes in function fields

Abstract: We will describe some interesting arithmetic phenomena which are possible due to collaboration of primes and some other usual phenomena which have exceptional primes. 

15:00 - 16:00  Daniel Disegni (Ben-Gurion University of the Negev) 

Algebraic cycles and p-adic L-functions for Rankin-Selberg motives

Abstract: The conjectures of Beilinson-Bloch-Kato predict that for a (smooth, proper) variety of dimension 2N-1 over a number field, the existence of nontrivial algebraic cycles of “arithmetic middle dimension” N-1 should be detected by L-functions. Moreover, the relevant Selmer group should be generated by algebraic cycles.

I will talk about a “case study” that confirms a variant in p-adic coefficients, for a certain product of unitary Shimura varieties uniformised by complex unit balls. The result comes from a formula for the p-adic height of an explicit cycle (for curves, the cycle is a Heegner point). The proof is based on a comparison of relative-trace formulas. (Joint work with Wei Zhang.)