Two applications of Weil-Serre's explicit formula

Abstract

We apply Weil-Serre's explicit formula to produce non-trivial estimates of exponential sums along a curve and of the dual distance of subfield subcodes of algebraic-geometric (AG-)codes. These bounds work while the number of rational points of a curve is large when compared to its genus thus the Weil-Bombieri bound for exponential sums being trivial.