Cyclotomic Function Fields with Many Rational Places

Abstract

Let $A = \mathbb{F}{_{q}}[T]$ be the polynomial ring in the variable T and $K = \mathbb{F}{_{q}}[T]$ the rational function field over $\mathbb{F}{_{q}}$ (the finite field with q elements), and let K _∞ be the completion of K at the place $\infty : = \frac{1}{T}$. Furthermore let C be the completion of a fixed algebraic closure of K _∞.

We aim to construct extensions K ⊂ K′ ⊂ C with many rational places relative to the genus g(K′) of K′.

As a first step we consider the cyclotomic fields K(n)/K with n ∈ A, which are generated analogously to the classical cyclotomic fields over ℚ. Then we consider certain decomposition fields and their intersections. Here we know a lower bound for the number of rational places. We get explicit formulas to calculate the genus, but they depend on the relative position of some subgroups of the multiplicative group (A/(n))* of the ring A/(n). So the concrete calculation of examples must be done by computer. With a special program we made a systematic search for q = 2 and found three improved lower bounds.