Distribution of the Traces of Frobenius on Elliptic Curves over Function Fields

Let C be a smooth irreducible projective curve defined over a finite field Fq of q elements of characteristic p > 3 andK = Fq(C) its function field and φE : E → C the minimal regular model of E/K. For each P ∈ C denote EP = φ −1 E (P ). The elliptic curve E/K has good reduction at P ∈ C if and only if EP is an elliptic curve defined over the residue field κP of P . This field is a finite extension of Fq of degree deg(P ). Let t(EP ) = q deg(P ) + 1−#EP (κP ) be the trace of Frobenius at P . By Hasse-Weil’s theorem (cf. [10, Chapter V, Theorem 2.4]), t(EP ) is the sum of the inverses of the zeros of the zeta function of EP . In particular, |t(EP )| ≤ 2q deg(P . Let C0 ⊂ C be the set of points of C at which E/K has good reduction and C0(Fqk) the subset of Fqk-rational points of C0. Question 1. Let k ≥ 1 and t be integers and suppose |t| ≤ 2q. Let π(k, t) = #{P ∈ C0(Fqk) | t(EP ) = t}. How big is π(k, t)?