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

We calculate the monodromy groups of some universal families of abelian varieties with real multiplication by $\mathbf{Q}(\cos\frac{2\pi}{r})$, over certain Hilbert type modular varieties over finite fields. Using Deligne’s equidistribution theorem, we obtain the Sato-Tate law for the distribution of Frobenius eigenvalues of these universal families of abelian varieties.

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Published 1 January 2007