Reflection principles for class groups

TL;DR

This paper establishes new reflection principles relating class groups of number fields and Picard groups of curves, confirming a conjecture for specific Galois groups and employing sheaf cohomology techniques.

Contribution

It introduces novel reflection principles applicable to various Galois groups, proving a conjecture for 2-rank equality in certain subfields and extending the framework to both number fields and function fields.

Findings

Proves a conjecture of Lemmermeyer for A4 subfields.

Establishes reflection principles for multiple Galois groups.

Uses sheaf cohomology to connect Galois modules and class groups.

Abstract

We present several new examples of reflection principles which apply to both class groups of number fields and picard groups of of curves over $\mathbb{P}^{1}/\mathbb{F}_{p}$. This proves a conjecture of Lemmermeyer about equality of 2-rank in subfields of $A_{4}$, up to a constant not depending on the discriminant in the number field case, and exactly in the function field case. More generally we prove similar relations for subfields of a Galois extension with group $G$ for the cases when $G$ is $S_{3}$, $S_{4}$, $A_{4}$, $D_{2l}$ and $\mathbb{Z}/l\mathbb{Z}\rtimes\mathbb{Z}/r\mathbb{Z}$. The method of proof uses sheaf cohomology on 1-dimensional schemes, which reduces to Galois module computations.