Riemann-Roch and motives for arithmetic problems

Our project “Riemann-Roch and Motives for Arithmetic Problems” aims to develop techniques in the area of Motives and the Riemann-Roch to attack arithmetic problems.

To be more concrete we aim to attack:- The integral Riemann-Roch: At SGA VI Grothendieck developed his landmark Riemann-Roch result stating an integral version of it as an open question.

Later on, research of Fulton, MacPherson and Pappas raised Grothendieck original conjecture to a more complete statement related to traces, which is known today only in the complex geometric setting.

We aim to prove this conjecture in its full generality. -The discrete Riemann-Roch: At SGA5 Grothendieck proved his wellknown Ogg-Shafarevich formula computing the Euler characteristic of a constructible sheaf over curve in terms of the genus, the Swan conductor and therank. This formula plays a central role in the original strategy to prove the Weyl conjectures.

Grothendieck also conjectured that this formula would fit into a Riemann-Roch type theorem for the K-group of étale constructible sheaves and general schemes, which he called the “discrete Riemann-Roch”.

We aim to attack this theorem from the motivic point of view.-Intersection theory in the arithmetic setting: A major objective of Algebraic Geometry is to define a product algebraic cycles forin the arithmetic setting. So far, this product has being defined with rational coefficients.

The first definition, due to Gillet-Soulé, was achieved throughout the Adam’s operations, the Adams Riemann-Roch and theGrothendieck-Riemann-Roch.

We aim to explore some of Gillet-Soulé’s ideas and the arithmetic bivariant integral version of the Riemann-Roch to explore a definition of the intersection product of cycles after killing certain torsion on the Chow groups related to the codimension of the cycle