Symmetry, splitting rational places in extensions of function fields and generalization of the Hermitian function field

Let F/K be an algebraic function field in one variable over a finite field of constants K, i.e., F is a finite algebraic extension of K(x) where x ∈ F is transcendental over K. Let E be a finite separable extension of F . Let N(E) and g(E) denote the number of places of degree one (or rational places), and the genus, respectively, of E. Let [E : F ] denote the degree of this extension. In recent years, there has been a spurt of interest in algebraic function fields with many rational places, or, equivalently, curves over finite fields with many rational points. The initial impetus for this interest came from applications to coding theory, wherein, in 1981, Goppa [8] discovered that function fields with many rational places could be used to construct long codes, whose parameters could then be ascertained using the Riemann-Roch theorem. Since then, more applications of such function fields have been discovered [25]. Equally importantly, function fields with many rational places are an interesting mathematical problem in their own right, with connections to several well studied problems in arithmetical algebraic geometry, and have been recognised as such. Consequently, various aspects of such function fields have been studied. Many authors have also written on this subject in the language of curves over finite fields. One technique to produce function fields with many rational places is to somehow split many rational places of the projective line in an extension, while keeping the rise in genus low. The technique existing in literature that can be used to achieve this uses class field theory and was introduced by Serre [17, 18, 19]. For practical applications of such function fields, however, it is necessary that the constructions be explicit, in that generators and