Stability and arithmetic

Abstract

Stability plays a central role in arithmetic. In this article, we explain some basic ideas and present certain constructions for our studies. There are two aspects: namely, general Class Field Theories for Riemann surfaces using semi-stable parabolic bundles and for $p$-adic number fields using what we call semi-stable filtered $(\varphi, N;\omega)$-modules; and non-abelian zeta functions for function fields over finite fields using semi-stable bundles and for number fields using semi-stable lattices.