Even sets of nodes and Gauss genus theory

TL;DR

This paper reveals a surprising connection between a lemma about even sets of nodes on surfaces and Gauss's formula on the 2-torsion of quadratic field class groups, showing a deep interplay between algebraic geometry and number theory.

Contribution

It demonstrates that a lemma from algebraic geometry can be directly applied to prove a classical number theory result, bridging two mathematical areas.

Findings

The lemma applies almost verbatim to Gauss's formula.

A new perspective on the relationship between surface nodes and class groups.

Potential for further cross-disciplinary applications.

Abstract

We observe that a lemma used in the study of even sets of nodes on surfaces applies almost verbatim to prove a celebrated formula of Gauss on the 2-torsion of the class group of a quadratic field.