The structure of the tautological ring in genus one

TL;DR

This paper proves that in genus one, the tautological ring matches the even cohomology ring and all relations derive from WDVV and Getzler's relations, clarifying its algebraic structure.

Contribution

It establishes the equivalence of the tautological ring and even cohomology in genus one and identifies the fundamental relations generating all others.

Findings

Tautological ring equals even cohomology ring in genus one

All relations follow from WDVV and Getzler's relations

Provides a complete description of the ring's structure

Abstract

We prove Getzler's claims about the tautological ring of stable curves of genus one, i.e. that the tautological ring coincides with the even cohomology ring, and that all relations between generators follow from the WDVV relation and Getzler's relation.