Conference "Global Fields"

Adelic construction of the Chern Class

Roman Budylin (Moscow)

Wednesday 26 October, 10:30 - 11:30

Abstract

For the surface over finite field there is the functional equation for the $L$-function of an abelian unramified character $\chi$, $$L_X(s,\chi)=\epsilon(\chi)q^{c_2(X)}q^{-sc_2(X))}L_X(2-s,\chi^{-1}),$$ As $L$-function is the product over points it is interesting to decompose the factor in the functional equation into the product. In my talk I give the formula for the second Chern class $c_2(E)$ of vector bundle on the surface in the form of the sum over all flags $x\in C$, where point $x$ lies on the irreducible curve $C$. It is the generalisation of the formula for the $c_1(X)$ of the curve $X$: $$c_1(X)=\sum_x \nu_x(w)x,$$ where w is a differential form and $\nu_X$ is the valuation in the point $x$.