分野別セミナー

In the study of totally real fields, Siegel introduced a distance from a modular point to a cusp and hence constructed corresponding fundamental domains. This distance was generalized to work for all number fields in Prof. Weng's study on non-abelian zeta functions. Motivated by this, working on product of rigid analytic upper half planes, we frist construct new distances between modular points and cusps. With the help of the correspondence between modular points and rank two bundles over curves defined over finite fields, we then obtain the following theorem.

Theorem:
A rank two bundle on a curve over finite fields is Mumford semi-stable if and only if the distances of its associated modular point to all cusps are no less than one.