Integral points on varieties defined by matrix factorization into elementary matrices

Abstract

Let K be a number field and S be a finite set of valuations of K containing the archimedean valuations. Let O be the ring of S-integers. For A∈SL2(O) and k≥1, we define matrix-factorization varieties Vk(A) over O which parametrize factoring A into a product of k elementary matrices beginning with lower triangular; the equations defining Vk(A) are written in terms of Euler's continuant polynomials. We show that the Vk(A) are rational (k−3)-folds with an inductive fibration structure. We combine this geometric structure with arithmetic results to study the Zariski closure of the O-points of Vk(A). We prove that for k≥4 the O-points on Vk(A) are Zariski dense if Vk(A)(O)≠∅ assuming the group of units O× is infinite. This can then be combined with results on factoring into elementary matrices for SL2(O). One result is that for k≥9 the O-points on Vk(A) are Zariski dense if O× is infinite.