Asymptotically Optimal Towers of Curves over Finite Fields

Abstract

Let (X k ) k ∈ ℕ be a series of algebraic curves over the finite field \( \mathbb{F}_q \), with N(X k) rational points, and whose genera g(X k) tend to infinity. It is called asymptotically optimal if the ratio N(X k )/g(X k ) tends to its largest possible value q 1/2 - 1. We show that “almost every” such series constructed from (classical elliptic or Drinfeld) modular curves is asymptotically optimal, provided that q is a square.