On The Group Algebra Decomposition of a Jacobian Variety

TL;DR

This paper introduces a concrete method for decomposing Jacobian varieties of Riemann surfaces with group actions, enabling detailed geometric analysis and optimization of the decomposition kernel, especially for trigonal curves up to genus 10.

Contribution

The authors develop a new practical approach to explicitly construct and analyze group algebra decompositions of Jacobian varieties, advancing understanding of their geometric properties.

Findings

Constructed explicit decompositions for various cases

Identified decompositions with minimal kernel size

Applied method to trigonal curves up to genus 10

Abstract

Given a compact Riemann surface X with an action of a finite group G, the group algebra Q[G] provides an isogenous decomposition of its Jacobian variety JX, known as the group algebra decomposition of JX. We obtain a method to concretely build a decomposition of this kind. Our method allows us to study the geometry of the decomposition. For instance, we build several decompositions in order to determine which one has kernel of smallest order. We apply this method to families of trigonal curves up to genus 10.