Stability and Arithmetic

TL;DR

This paper explores the concept of stability in arithmetic, discussing constructions in class field theories, p-adic fields, and zeta functions through semi-stable structures.

Contribution

It introduces new frameworks for studying stability in arithmetic via semi-stable bundles, filtered modules, and lattices across various mathematical contexts.

Findings

Development of class field theories using semi-stable parabolic bundles and filtered modules.

Construction of non-abelian zeta functions via semi-stable bundles and lattices.

Unification of stability concepts across different arithmetic settings.

Abstract

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