4 - Function Fields with Many Rational Places

Summary

We say informally that a global function field K/F_q has many rational places if its number N(K) of rational places is reasonably close to N_q(g(K)) or to a known upper bound on N_q(g(K)). Recall from Definition 1.6.14 that N_q(g) is the maximum number of rational places that a global function field with full constant field F_q and genus g can have. We refer to Section 1.6 for upper bounds on N_q(g).

In this chapter, we employ class field theory as well as explicit function fields such as Kummer and Artin-Schreier extensions to construct global function fields with many rational places. Ray class fields (including Hilbert class fields) and narrow ray class fields are also good candidates as their Galois groups and Art in symbols are known.

Function Fields from Hilbert Class Fields

For a global function field F/F_q with a rational place ∞, the ∞’-Hilbert class field H of F with ∞‘≔ P_r\ {∞} is a finite unramified abelian extension of F in which ∞ splits completely. Moreover, [H : F] = h(F), the divisor class number of F, and the Galois group Gal(H/F) is isomorphic to the group C1(F) of divisor classes of degree zero of F. With this canonical isomorphism, the Artin symbol in H/F of a place P of F corresponds to the divisor class [P - deg(P)∞].