# Self-contained reference for projective embedding of moduli of polarized abelian varieties via modular forms

I've been working on reading and understanding Arakelov's '71 paper and he uses the fact that the moduli space of complex abelian varieties of dimension $$g$$ with polarization of degree $$d$$ admits an embedding into some projective space $$\mathbb{P}_{\mathbb{C}}^N$$ by modular forms of weight $$m >\!\!> 0$$. My understanding is this is a consequence of Satake/Baily-Borel compactification but I'd like to see the details of this, as it's pretty far outside of my wheelhouse. Are there any self-contained explanations of this?

• A self-contained explanation of an even stronger result is Moret-Bailly's Asterisque "Pinceaux de varietes abeliennes" ; see numdam.org/issue/AST_1985__129__1_0.pdf – Ariyan Javanpeykar Nov 2 '19 at 22:26
• @AriyanJavanpeykar: Do you have a more precise reference? The book has 269 pages, so some pointer is needed... – Bombyx mori Nov 17 '19 at 5:22