Lambert $W$-function and Gauss class number one conjecture

TL;DR

This paper links fixed points of a specific function in representation theory to real quadratic fields with class number one, providing a novel approach to the Gauss conjecture.

Contribution

It establishes a correspondence between fixed points of a function related to Drinfeld modules and class number one real quadratic fields, offering a new proof of the Gauss conjecture.

Findings

Fixed points correspond to class number one real quadratic fields.

Provides a solution to the Gauss conjecture for these fields.

Connects representation theory with number theory.

Abstract

We study fixed points of a function arising in a representation theory of the Drinfeld modules by the bounded linear operators on a Hilbert space. We prove that such points correspond to number fields of the class number one. As an application, one gets a solution to the Gauss conjecture for the real quadratic fields of class number one.