Equivariant Chow-Witt groups and moduli stacks of elliptic curves

TL;DR

This paper develops equivariant Chow-Witt groups to study the Chow-Witt rings of moduli stacks of elliptic curves, providing new geometric insights and computations for quotient stacks and classifying stacks.

Contribution

It introduces equivariant Chow-Witt groups and computes the Chow-Witt rings of moduli stacks of elliptic curves and classifying stacks of cyclic groups, offering new geometric interpretations.

Findings

Computed the Chow-Witt ring of the moduli stack of elliptic curves.

Provided a geometric interpretation of new generators in the Chow-Witt ring.

Determined the Chow-Witt ring of the classifying stack of μ_{2n}.

Abstract

We introduce equivariant Chow-Witt groups in order to define Chow-Witt groups of quotient stacks. We compute the Chow-Witt ring of the moduli stack of stable (resp. smooth) elliptic curves, providing a geometric interpretation of the new generators. Along the way, we also determine the Chow-Witt ring of the classifying stack of $\mu_{2n}$.