${\mathbb{A}}^1$-connected stacky curves and the Brauer group of moduli of elliptic curves

Abstract

Given a smooth scheme X with an action by an affine algebraic group G, we give a formula to compute the Nisnevich sheaf of the motivic connected components of the quotient stack [X/G] in the case of an orbifold. We apply it to identify all the ${\mathbb{A}}^1$-connected stacky curves and prove the ${\mathbb{A}}^1$ -connectedness of the moduli stack of elliptic curves. We also prove homotopy purity for the smooth algebraic stacks and recover a computation of the Picard group of stacky orbifold curves by computing their mixed motive.

Corrected Theorem 1.1, streamlined section 3 and some other minor changes. Comments still welcome