Spiegelungssatz: a combinatorial proof for the 4-rank

TL;DR

This paper presents a combinatorial proof of the Spiegelungssatz, an inequality relating the 4-ranks of narrow ideal class groups of quadratic fields, and characterizes cases of equality through an affine system.

Contribution

It offers the first combinatorial proof of the Spiegelungssatz and introduces an affine system to describe equality cases.

Findings

Proof of the Spiegelungssatz using combinatorial methods

Characterization of equality cases via affine systems

Enhanced understanding of 4-rank relationships in quadratic fields

Abstract

The Spiegelungssatz is an inequality between the (4)-ranks of the narrow ideal class groups of the quadratic fields (\mathbb{Q}(\sqrt{D})) and (\mathbb{Q}(\sqrt{-D})). We provide a combinatorial proof of this inequality. Our interpretation gives an affine system of equations that allows to describe precisely some equality cases.