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

TL;DR

This paper develops a formula for computing the motivic connected components of quotient stacks, identifies all $ ext{A}^1$-connected stacky curves, and proves the $ ext{A}^1$-connectedness of the moduli stack of elliptic curves, advancing understanding of their motivic and homotopical properties.

Contribution

It introduces a formula for motivic connected components of quotient stacks, classifies $ ext{A}^1$-connected stacky curves, and establishes $ ext{A}^1$-connectedness of the elliptic curves moduli stack.

Findings

Computed motivic connected components for orbifold quotient stacks.

Identified all $ ext{A}^1$-connected stacky curves.

Proved $ ext{A}^1$-connectedness of the moduli stack 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.