Equations defining Jacobians with Real Multiplication

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.