Joint distribution of spins

TL;DR

This paper investigates the joint distribution of spin symbols, providing insights into their behavior and addressing longstanding conjectures in number theory.

Contribution

It offers a new analysis of spin symbol distributions and disproves a conjecture on governing fields for class group ranks under certain assumptions.

Findings

Disproved the conjecture of Cohn and Lagarias on governing fields.

Provided a negative answer to the existence of governing fields for the 16-rank of class groups.

Analyzed the joint distribution of spin symbols in number fields.

Abstract

We answer a question of Iwaniec, Friedlander, Mazur and Rubin on the joint distribution of spin symbols. As an application we give a negative answer to a conjecture of Cohn and Lagarias on the existence of governing fields for the $16$-rank of class groups under the assumption of a short character sum conjecture.