Equations defining Jacobians with Real Multiplication

TL;DR

This paper characterizes conditions on genus 2 curves for their Jacobians to have real multiplication by quadratic orders, generalizing classical results and providing explicit cases.

Contribution

It generalizes Humbert's classical results to all discriminants for Jacobians of genus 2 curves with real multiplication.

Findings

Derived explicit equations for Jacobians with real multiplication.

Extended classical results to all discriminants.

Provided explicit examples for specific cases.

Abstract

If $C:y^2=x(x-1)(x-a_1)(x-a_2)(x-a_3)$ is genus $2$ curve a natural question to ask is: Under what conditions on $a_1,a_2,a_3$ does the Jacobian $J(C)$ have real multiplication by $\mathbb{Z}[\sqrt{\Delta}]$ for some $\Delta>0$. Over a hundred years ago Humbert gave an answer to this question for $\Delta=5$ and $\Delta=8$. In this paper we use work of Birkenhake and Wilhelm along with some classical results in enumerative geometry to generalize this to all discriminants, in principle. We also work it out explicitly in a few more cases.