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 $\pm1.$