Lower Bounds for Frobenius Traces

This paper grew out of the following question. Given an ordinary elliptic curve E//Fq over a finite field Fq, consider the sequence of integers A(n), n ≥ 1 defined by #E(Fqn) = q + 1− A(n). Is it true that as n grows we have |A(n)| → ∞? Without the hypothesis “ordinary”, the answer can be no, because for a supersingular elliptic curve, one can have A(n) = 0 on entire arithmetic progressions of n. On the other hand, all the A(n) in the supersingular case are divisible, as algebraic integers, by q, so the nonzero A(n) must have |A(n)| ≥ q. If E//Fq is ordinary, then all the A(n) are nonzero, because they are all prime to p, the characteristic of Fq, so this vanishing problem at least disappears. Now the A(n) are the traces of the iterates of a certain Frobenius endomorphism F , and this leads to the more general question of when we can assert that in the sequence |Trace(F )|, n ≥ 1, the nonzero terms tend to ∞. In the first section, we establish a quite general result of this kind, using the theorem [ESS, Thm. 1.1] of Evertse-Schlichewei-Schmidt on the generalized unit equation In the second section, we give a quantitative version of this same sort of result, based on the subspace theorem of these same authors, cf. especially [BG, Cor. 7.2.5], [E, Thm. page 228] and [ES, Thm. 3.1]. In the appendix, we give a sharper result for ordinary elliptic curves and for the classical Kloosterman sums, using the Baker-Wüstholz, theorem [BW].