An algorithm for computing Hilbert modular varieties

We describe an algorithm for computing an analogue of the modular polynomial for elliptic curves, for principally polarised (p.p.) abelian varieties with real multiplication (RM). Given a p.p. abelian variety with RM by $\mathcal{O}_{F}$, where $\mathcal{O}_{F}$ is the maximal order in a totally real number field $F$, for any totally positive prime $\mu$ in $\mathcal{O}_{F}$, one can find a "$\mu$-isogenous" p.p. abelian variety with RM by $\mathcal{O}_{F}$. We describe an algorithm to compute a birational model for an algebraic variety which parametrises $\mu$-isogeny classes of p.p. abelian varieties with RM by $\mathcal{O}_{F}$.