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On Subfields of the Hermitian Function Field

Published online by Cambridge University Press:  04 December 2007

Arnaldo Garcia ,
Henning Stichtenoth  and
Chao-Ping Xing
Arnaldo Garcia
Affiliation:
Instituto de Matématica Pura e Aplicada IMPA, 22460-320 Rio de Janeiro RJ, Brazil. e-mail: [email protected]
Henning Stichtenoth
Affiliation:
Universitäat GH Essen, FB 6, Mathematik u. Informatik, 45117 Essen, Germany. e-mail: [email protected]
Chao-Ping Xing
Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China; and Department of Information Systems and Computer Science, The National University of Singapore, 10 Lower Kent Ridge Crescent, Singapore 119260. e-mail: [email protected]
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Abstract

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The Hermitian function field H= K(x,y) is defined by the equation yq+y=xq +1 (q being a power of the characteristic of K). Over K=${\mirrored F}$q2 it is a maximal function field; i.e. the number N(H) of ${\mirrored F}$q2-rational places attains the Hasse–Weil upper boundN(H)=q2+1+2g(H)·q. All subfields K[subnE ] E⊆H are also maximal. In this paper we construct a large number of nonrational subfields E⊆H, by considering the fixed fields H$^{\mathcal G}$ under certain groups ${\mathcal G}$ of automorphisms of H/K. Thus we obtain many integers g[ges ]0 that occur as the genus of some maximal function field over ${\mirrored F}$q2.

Keywords

function fieldsrational placesfinite fields
Type
Research Article
Information
Compositio Mathematica , Volume 120 , Issue 2 , January 2000 , pp. 137 - 170
Copyright
© 2000 Kluwer Academic Publishers