Distribution of rational points on varieties over finite fields

Exponential sums and rational points on complete intersections

“…the corresponding result of [10]). In a recent paper [11] Fujiwara extended his method [10] to s> 1, and obtained results in the same direction as our Theorems 2 and 3. However, his bounds are weaker.…”

“…Let Tc pf!^ ' be a projective variety over a finite field F,, and V<= A^ be the affine cone over T. Let G £ F,[x,,..., x n ] be a homogeneous polynomial which does not vanish identically on T. We are interested in the following exponential sum: S q (G)= I e,(G(z)). (11) This follows from Proposition 2.1 and the Weil conjectures proved by Deligne [2]. By Proposition 2.2 T L is of dimension n-s -2, and has at most isolated singularities.…”

The Distribution of Values of Short Hybrid Exponential Sums on Curves over Finite Fields

“…This proves (8) and (10) in Theorem 2.9. If k is even, then the case for a = 2 is different from the others.…”

“…We also let H be an integer such that 1 ≤ H ≤ p, and I ⊆ [0, p − 1] an interval. Since F p -points on an affine curve is uniformly distributed (see for example Meyerson [13], Fujiwara [8], or the authors [12]), we have the following estimation of r,…”

Small zeros of quadratic congruences modulo pq

“…The result in part (ii) of Lemma 6 can also be deduced from the proof of Fujiwara's Theorem 1 [7]. In this theorem he obtains estimates for |98n V p \ for a more general class of nonsingular varieties V p .…”