UC Irvine

For the Artin-Schreier curve yq- y = f(x) defined over a finite field Fqof q elements, the celebrated Weil bound for the number of Fqr-rational points can be sharp, especially in super-singular cases and when r is divisible. In this paper, we show how the Weil bound can be significantly improved, using ideas from moment L-functions and Katz's work on ℓ-adic monodromy calculations. Roughly speaking, we show that in favorable cases (which happens quite often), one can remove an extra √q factor in the error term. © 2010 The Author(s).