On the bounded generation of arithmetic SL2

Abstract

Let K be a number field and S be a finite set of primes of K containing the archimedean valuations. Let 𝒪 be the ring of S-integers in K Morgan, Rapinchuck, and Sury [A. V. Morgan et al, Algebra Number Theory 12, 1949-1974 (2018)] have proved that if the group of units [Formula: see text] is infinite, then every matrix in SL2(𝒪) is a product of at most 9 elementary matrices. We prove that under the additional hypothesis that K has at least 1 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 SL2(𝒪) is the product of at most 5 elementary matrices if K has at least 1 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.