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Suppose we have an elliptic surface $\pi: \mathscr E \to C$ over a curve over a finite field $\mathbb F_q$. We consider only the places on $C$ over which we have good reduction.

Over any point $x\in C$, we get an elliptic curve $\mathscr E_x$ and we consider the multiset of the eigenvalues for the Frobenius action on the first etale cohomology and after normalizing by $q^{\deg(x)/2}$, these eigenvalues will lie on the unit circle.

What does the distribution of eigenvalues look like as we let $\deg(x) \to \infty$? We can ask the same question for Abelian varieties (or even for other varieties) over a curve.

This is an analogue of the well known Sato-Tate distribution but to my knowledge, results on the Sato-Tate are proven by essentially analytic methods so I am not sure if we have any results for this question.

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    $\begingroup$ It's $SU_2$ in the non-isotrivial case or a finite group in the isotrivial case. This was proven by Deligne, in his Weil II paper, long before Sato-Tate was proven. $\endgroup$ – Will Sawin Oct 10 at 0:18

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