Characteristic classes for curves of genus one

TL;DR

This paper computes the cohomology of the moduli stack of genus one curves, revealing characteristic classes that serve as invariants for families of such curves, with results on their vanishing and explicit examples.

Contribution

It provides explicit calculations of cohomology groups and characteristic classes for genus one curves, including vanishing results and concrete examples.

Findings

Cohomology groups of M_1 computed with coefficients in Z[1/2] and Z.

Identification of characteristic classes as invariants of genus one curve families.

Examples demonstrating non-vanishing characteristic classes.

Abstract

We compute the cohomology of the stack M_1 with coefficients in Z[1/2], and in low degrees with coefficients in Z. Cohomology classes on M_1 give rise to characteristic classes, cohomological invariants of families of curves of genus one. We prove a number of vanishing results for those characteristic classes, and give explicit examples of families with non-vanishing characteristic classes.