On the Distribution of the of Frobenius Elements on Elliptic Curves over Function Fields

Let C be a smooth projective irreducible curve defined over a finite field Fq of q elements and characteristic p > 3 with function field K = Fq(C). Let E/K be a non-constant elliptic curve and φ : E → C its minimal regular model. For every P ∈ C, denote by deg(P ) = [κP : Fq] its degree and qP = q deg(P . If EP = φ (P ) is an elliptic curve over κP denote by t(EP ) = qP + 1 − #EP (κP ) the trace of Frobenius of E at P . It follows from a theorem of Hasse-Weil [11, Chapter V, Theorem 2.4] t(EP ) = q 1/2 P (e iθ(EP ) + eP ) = 2q 1/2 P cos(θ(EP )) with 0 ≤ θ(EP ) ≤ π and t(EP ) ≤ 2q 1/2 P . Denote by C0 the set of points P ∈ C such that EP is an elliptic curve. In [9] we addressed the following question. Question 1.1. Let B ≥ 1 and t be integers with |t| ≤ 2q. Let π(B, t) = #{P ∈ C0 | deg(P ) ≤ B and t(EP ) = t}. How big is π(B, t)?