# The class of the d-elliptic locus in genus 2

## TL;DR

This paper computes the Chow class of genus 2 curves with a degree d map to genus 1 curves, revealing quasi-modularity and classifying admissible covers, with applications to counting d-elliptic curves.

## Contribution

It provides the first explicit computation of the Chow class for the d-elliptic locus in genus 2 and classifies admissible covers, extending previous results and revealing new modular properties.

## Key findings

- Chow class of the d-elliptic locus in genus 2 computed
- Classification of Harris-Mumford admissible covers provided
- Number of d-elliptic curves in a general family determined

## Abstract

We compute the rational Chow class of the locus of genus 2 curves admitting a d-to-1 map to a genus 1 curve, recovering a result of Faber-Pagani when d=2. The answer exhibits quasi-modularity properties similar to those in the Gromov-Witten theory of a fixed genus 1 curve. Along the way, we give a classification of Harris-Mumford admissible covers of a genus 1 curve by a genus 2 curve, which may be of independent interest. As an application of the main calculation, we compute the number of d-elliptic curves in a very general family of genus 2 curves obtained by fixing five branch points of the hyperelliptic map and varying the sixth.