Polynomial parametrization of $\text{SL}_2$ over global function field rings

Source abstract

We combine the universal polynomial matrix identities of Larsen--Nguyen with Trost's arithmetic lemmas to obtain polynomial parametrizations of $\text{SL}_2(\mathcal{O}_{K,S})$ for every global function field $K$ and every nonempty finite set of places $S$. The additional ingredient is a function-field cancellation lemma, proved using ray class fields and Chebotarev in odd characteristic and the perfectness of finite residue fields in characteristic two. The proof also gives a polynomial matrix over $\mathbb{Z}$ that works simultaneously for all these rings.