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This is a question from a colleague.

Let $A_{g,d,n}$ be the coarse moduli space over $\mathbb Z$ (of the moduli stack, or assume $n\ge3$ if you like) of abelian schemes of relative dimension $g,$ with a polarization of degree $d^2$ and level-$n$-structure. Then what is $\Gamma(A_{g,d,n},\mathscr O^*)?$ The case when $n=1$ is what we are mainly interested. (I guess otherwise $\mathbb Z[1/n]$ is the correct base.)

I was told that the case $d=1$ had been solved: the only global units are $\pm1.$

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For $g=d=1$ and $n>4$, there are lots of (modular) units. Manin proved that the difference of any two cusps is torsion in the Jacobian. I don't know anything about $g>1$ but it's a nice question. In your remark about $d=1$ are you assuming $n=1$ also? I suspect you might have to, even for $g>1$. – Felipe Voloch Jan 28 2011 at 20:09
If you allow yourself to invert all the primes whose squares divide $d$, $A_{g,d,1}$ (as a stack) admits a compactification (the so-called minimal or Baily-Borel-Satake compactification) with normal, geometric irreducible fibers, and whose boundary has codimension at least $2$. In particular, one does not expect to find any interesting units in this case. – Keerthi Madapusi Pera Feb 3 at 12:33

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