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4 - Function Fields with Many Rational Places

Published online by Cambridge University Press:  05 April 2013

Harald Niederreiter
Affiliation:
National University of Singapore
Chaoping Xing
Affiliation:
National University of Singapore
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Summary

We say informally that a global function field K/Fq has many rational places if its number N(K) of rational places is reasonably close to Nq(g(K)) or to a known upper bound on Nq(g(K)). Recall from Definition 1.6.14 that Nq(g) is the maximum number of rational places that a global function field with full constant field Fq and genus g can have. We refer to Section 1.6 for upper bounds on Nq(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/Fq with a rational place ∞, the ∞’-Hilbert class field H of F with ∞‘≔ Pr\ {∞} 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)∞].

Type
Chapter
Information
Rational Points on Curves over Finite Fields
Theory and Applications
, pp. 76 - 121
Publisher: Cambridge University Press
Print publication year: 2001

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