Workshop on Algebraic Curves Over Finite Fields

Let L(t) = 1+a1t+ · · ·+a2gt be the numerator of the zeta function of an algebraic curve C defined over the finite field Fq of genus g. We show that the coefficients ar of L(t) satisfy certain inequalities. Conversely, for any integers a1, . . . , am satisfying these inequalities and all sufficiently large integers g there exist curves of genus g, whose L-polynomial satisfies the following congruence. L(t) ≡ 1 + a1t+ · · ·+ amt (mod t). In fact, this result is equivalent to the following statement: for any non-negative integers b1, . . . , bm and all sufficiently large integers g there exist curves of genus g having exactly bj points of degree j, for 1 ≤ j ≤ m. This is a joint work with Henning Stichtenoth. Reference. N. Anbar, H. Stichtenoth, Curves of every genus with a prescribed number of rational points, Bulletin of the Brazilian Mathematical Society, 44 (2013), 173–193. “Obtaining towers using Drinfeld modules” Peter Beelen Technical University of Denmark