On the constant D(q) defined by Homma

Abstract

Let X be a projective, irreducible, nonsingular algebraic curve over the finite field Fq with q elements and let |X(Fq)| and g( X) be its number of rational points and genus respectively. The Ihara constant A(q) has been intensively studied during the last decades, and it is defined as the limit superior of |X(Fq)|/g( X) as the genus of X goes to infinity. In 2012 Homma defined an analogue D(q) of A(q), where the nonsingularity of X is dropped and g( X) is replaced with the degree of X. We will call D(q) Homma's constant. In this paper, upper and lower bounds for the value of D(q) are found.