Sato-Tate distribution for abelian varieties with real multiplication over function fields (Q2372677)

As the abstract explains, the author calculates the monodromy groups of some universal families of abelian varieties with real multiplication by \(F={\mathbb Q}(\cos{{2\pi}\over{r}})\), over a Hilbert modular variety \({\mathcal M}_{g, 3{\mathcal L}}\) over finite fields, and then uses Deligne's equidistribution theorem to obtain a description of the Sato-Tate distribution of the associated Frobenius eigenvalues. The space \({\mathcal M}_{g, 3{\mathcal L}}\) is associated with a level-\(3\) structure sufficient to make it be a fine moduli space. The result is that as one takes larger and larger base fields the distribution of Frobenius angles converges weakly in measure to the Sato-Tate measure on \(\prod_\tau [0,\pi]\) (the product taken over the embeddings of \(F\) in \(\overline{\mathbb Q}\)). This follows from Deligne's result once one computes the geometric monodromy, which is done here by a rather general method.