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Mathematics of Computation

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Real and imaginary quadratic representations
of hyperelliptic function fields

Authors: Sachar Paulus and Hans-Georg Rück
Journal: Math. Comp. 68 (1999), 1233-1241
MSC (1991): Primary 11R58, 14Q05; Secondary 11R65, 14H05, 14H40
Published electronically: February 15, 1999
MathSciNet review: 1627817
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Abstract: A hyperelliptic function field can be always be represented as a real quadratic extension of the rational function field. If at least one of the rational prime divisors is rational over the field of constants, then it also can be represented as an imaginary quadratic extension of the rational function field. The arithmetic in the divisor class group can be realized in the second case by Cantor's algorithm. We show that in the first case one can compute in the divisor class group of the function field using reduced ideals and distances of ideals in the orders involved. Furthermore, we show how the two representations are connected and compare the computational complexity.

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Additional Information

Sachar Paulus
Affiliation: Institut für Theoretische Informatik, TU Darmstadt, Alexanderstraße 10, 64283 Darmstadt (Germany)

Hans-Georg Rück
Affiliation: Institut für Experimentelle Mathematik, Universität GH Essen, Ellernstr.29, 45326 Essen (Germany)

Keywords: Hyperelliptic curves, divisor class groups, real quadratic model
Received by editor(s): July 24, 1997
Received by editor(s) in revised form: November 3, 1997, and January 20, 1998
Published electronically: February 15, 1999
Article copyright: © Copyright 1999 American Mathematical Society

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