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

TL;DR

The work builds a bridge between fixed points of operator representations from Drinfeld modules and number fields with class number one via a non-abelian class field theory framework involving noncommutative tori. It derives explicit Lambert $W$-function expressions for the defining parameters $\alpha_j$ and shows that $K\cong k$ exactly when these expressions hold, offering a Gauss-class-number-one type classification. The results yield concrete counts for fields with class number one (finite for imaginary $r=1$, infinite otherwise) and connect the Gauss conjecture for real quadratic fields to this non-abelian, operator-theoretic perspective. This provides new conceptual and computational tools for identifying real-quadratic fields of class number one and illustrates deep links between noncommutative geometry, Drinfeld modules, and algebraic 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.

Theorems & Definitions (16)

• Theorem 1.1
• Corollary 1.2
• Remark 1.3
• Lemma 2.1
• Theorem 2.2
• Corollary 2.3
• Lemma 3.1
• proof
• Lemma 3.2
• proof