Refinements on higher order Weil–Oesterlé bounds via a Serre type argument

Abstract

Abstract Weil's theorem gives the most standard bound on the maximum number of points of a curve of genus over a finite field . This bound was improved by Ihara and Oesterlé for larger genus. A recent point of view allows one to recover these bounds by solving a sequence of semi‐definite programs, and the first two steps lead to Weil's and Ihara's bounds. Another refinement of Weil's bound was obtained by Serre, involving arithmetic constraints. In this article, we combine these two approaches and propose a strengthening of Ihara's bound, based on an argument similar to Serre's refinement. We obtain a closed formula that generically improves upon Ihara's bound, even in the range where it was the best explicit bound so far. This provides new upper bounds on for infinitely many pairs .