Vanishing Theorem on Arithmetic Surfaces

摘要

In the paper the author proves an analog for arithmetic surfaces of the Kodaira-Ramanujam vanishing theorem. It says that if L(bar) is a hermitian line bundle on such a surface X which is positive on the associated Riemann surface and semi-positive on any divisor, the logarithm of the L(sup 2) -norm of any non torsion element e in H(sup 1)(X,L(sup 1)) is bounded below by a multiple the arithmetic self-intersection of L(bar). The analogy with the classical geometric statement is explained.