On the bounded generation of arithmetic ${\rm SL}_2$

TL;DR

This paper improves bounds on the number of elementary matrices needed to generate any element of ${ m SL}_2$ over rings of S-integers in number fields, under various hypotheses including GRH.

Contribution

It refines previous results by reducing the maximum number of elementary matrices required, depending on the properties of the number field and hypotheses like GRH.

Findings

At most 8 elementary matrices without GRH under certain conditions.

At most 5 elementary matrices assuming GRH with a real embedding.

At most 6 elementary matrices assuming GRH if S contains a finite place.

Abstract

Let $K$ be a number field and ${\mathcal O}$ be the ring of $S$-integers in $K$. Morgan, Rapinchuck, and Sury have proved that if the group of units ${\mathcal O}^{\times}$ is infinite, then every matrix in ${\rm SL}_2({\mathcal O})$ is a product of at most $9$ elementary matrices. We prove that under the additional hypothesis that $K$ has at least one real embedding or $S$ contains a finite place we can get a product of at most $8$ elementary matrices. If we assume a suitable Generalized Riemann Hypothesis, then every matrix in ${\rm SL}_2({\mathcal O})$ is the product of at most $5$ elementary matrices if $K$ has at least one real embedding, the product of at most $6$ elementary matrices if $S$ contains a finite place, and the product of at most $7$ elementary matrices in general.