Explicit Riemann-Roch spaces in the Hilbert class field

TL;DR

This paper investigates the structure and algorithmic properties of Riemann-Roch spaces associated with divisors on algebraic curves over finite fields, particularly in the context of unramified abelian covers and their Galois modules.

Contribution

It provides a detailed analysis of the module structure of Riemann-Roch spaces in the Hilbert class field, including conditions for freeness and algorithmic applications.

Findings

Riemann-Roch spaces form free modules over group rings under mild conditions.

Algorithmic methods for computing these modules are developed.

Applications include explicit descriptions of class field towers and related algebraic structures.

Abstract

Let $\mathbf K$ be a finite field, $X$ and $Y$ two curves over $\mathbf K$, and $Y\rightarrow X$ an unramified abelian cover with Galois group $G$. Let $D$ be a divisor on $X$ and $E$ its pullback on $Y$. Under mild conditions the linear space associated with $E$ is a free ${\mathbf K}[G]$-module. We study the algorithmic aspects and applications of these modules.