Riemann-Roch and Riemann-Hurwitz theorems for global fields

TL;DR

This paper extends classical theorems from algebraic geometry, specifically Riemann-Roch and Riemann-Hurwitz, to global fields of any characteristic using geometry of numbers techniques.

Contribution

It introduces a novel approach to generalize fundamental theorems to broader classes of global fields, unifying characteristic zero and positive characteristic cases.

Findings

Extended Riemann-Roch theorem to all global fields

Generalized Riemann-Hurwitz formula for arbitrary characteristic

Applied geometry of numbers to algebraic geometry theorems

Abstract

In this paper, we use counting theorems from the geometry of numbers to extend the Riemann-Roch theorem and the Riemann-Hurwitz formula to global fields of arbitrary characteristic.