Quadratic recursive towers of function fields over F2

Abstract

Let F = (Fn)n≥0 be a quadratic recursive tower of algebraic function fields over the finite field F2 , i.e. F is a recursive tower such that [Fn : Fn−1] = 2 for all n ≥ 1. For any integer r ≥ 1, let βr(F) := limn→∞ Br(Fn)/g(Fn), where Br(Fn) is the number of places of degree r and g(Fn) is the genus, respectively, of Fn/F2 . In this paper we give a classification of all rational functions f(X, Y ) ∈ F2(X, Y ) that may define a quadratic recursive tower F over F2 with at least one positive invariant βr(F). Moreover, we estimate β1(F) for each such tower.