# Sato-Tate over function fields

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.

• 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. – Will Sawin Oct 10 at 0:18