# Arithmetic Characteristic Curves

## TL;DR

This paper introduces arithmetic characteristic curves for arithmetic Higgs torsors over number fields, extending concepts from algebraic geometry to arithmetic settings, with applications to Hitchin fibrations and the fundamental lemma.

## Contribution

It constructs arithmetic characteristic curves using Chevalley morphisms and bases, bridging algebraic geometry and number theory in a novel way.

## Key findings

- Defines arithmetic torsors and Higgs torsors over number fields.
- Constructs arithmetic characteristic curves based on Chevalley morphisms.
- Sets the stage for future work on Hitchin fibrations and fundamental lemma applications.

## Abstract

For a split reductive group defined over a number field, we first introduce the notations of arithmetic torsors and arithmetic Higgs torsors. Then we construct arithmetic characteristic curves associated to arithmetic Higgs torsors, based on the Chevalley characteristic morphism and the existence of Chevalley basis for the associated Lie algebra. As to be expected, this work is motivated by the works of Beauville-Narasimhan on spectral curves and Donagi-Gaistgory on cameral curves in algebraic geometry. In the forthcoming papers, we will use arithmetic characteristic curves to construct arithmetic Hitchin fibrations and study the intersection homologies and perverse sheaves for the associated structures, following Ngo's approach to the fundamental lemma.