Get-Together Math Seminar

Speaker: Satoshi Kondo (Kavli IPMU)
Title: On the rational K_2 of a curve of GL_2 type over the function field of a curve over a finite field
Date (JST): Thu, Jul 05, 2012, 15:30 - 17:00
Place: Seminar Room A
Abstract: We consider the K_2 group (higher K-theory of Quillen) tensored with the rational numbers Q of a proper smooth curve X over the function field k of a curve C over a finite field. Take a regular model Y that is proper and flat over C (at least one such exists). We assume that the 1st 'etale cohomology (or the l-adic Tate module of the Jacobian of X tensored by the algebraic closure of Q_l) is the direct sum of two dimensional irreducible ones as a representation of the absolute Galois group of k. We then show that the boundary map from the rational K_2 of X to the rational G_1 of the union of the special fibers (i.e., the union of the fibers of Y at various closed points of C) in the localization sequence of G-theory (similar to that of K-theory) is surjective. Examples satisfying the condition above are elliptic curves, Drinfeld modular curves, and the moduli of D-elliptic sheaves of rank 2.