Hostname: page-component-cb9f654ff-d5ftd Total loading time: 0 Render date: 2025-08-25T05:16:31.215Z Has data issue: false hasContentIssue false
  • English
  • Français

Picard Groups and Refined Discrete Logarithms

Published online by Cambridge University Press:  01 February 2010

W. Bley  and
M. Endres
W. Bley
Affiliation:
Institut für Mathematik, Universität Augsburg, Universitätsstrasse 8, D-86135 Augsburg, Germany, bley@math.uni-augsburg.de
M. Endres
Affiliation:
Institut für Mathematik, Universität Augsburg, Universitätsstrasse 8, D-86135 Augsburg, Germany, me@mendres.org

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let K denote a number field, and G a finite abelian group. The ring of algebraic integers in K is denoted in this paper by $/cal{O}_K$, and $/cal{A}$ denotes any $/cal{O}_K$-order in K[G]. The paper describes an algorithm that explicitly computes the Picard group Pic($/cal{A}$), and solves the corresponding (refined) discrete logarithm problem. A tamely ramified extension L/K of prime degree l of an imaginary quadratic number field K is used as an example; the class of $/cal{O}_L$ in Pic($/cal{O}_K[G]$) can be numerically determined.

Information

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 8 , 2005 , pp. 1 - 16
Copyright
Copyright © London Mathematical Society 2005

References

1.Ayala, E. J. and Schertz, R., ‘Eine Bemerkung Zur Galoismodulstrutur in Strahlklassenkörpern über imaginär-quadratischen Zahlkörpern’, J. Number Theory 44 (1994) 41–46.CrossRefGoogle Scholar
2.Batut, C., Belabas, K., Bernardi, D. and Cohen, H., ‘User's guide to PARI/GP’, M. Olivier, 2000, http://pari.math.u-bordeaux.fr.Google Scholar
3.Bley, W., ‘Computing associated orders and Galois generating elements of unit lattices’, J. Number Theory 62 (1997) 242–256.CrossRefGoogle Scholar
4.Bley, W., ‘An algorithmic approach to determining local and global module structures’, Appendix to [7].Google Scholar
5.Bley, W. and Boltje, R., ‘Lubin–Tate formal groups and module structure over Hopf orders’, J. Théor. Nombres Bordx. 11 (1999) 269–305.CrossRefGoogle Scholar
6.Bley, W. and Burns, D., ‘Über arithmetische assoziierte Ordnungen’, J. Number Theory 58 (1996) 361–387.CrossRefGoogle Scholar
7.Burns, D., ‘On the equivariant structure of ideals in abelian extensions of local fields’, Comment. Math. Helv. 75 (2000) 1–44.CrossRefGoogle Scholar
8.Cassou-Nogués, P. and Taylor, M., Elliptic functions and rings of integers, Progr. Math. 66 (Birkhäuser, 1986).Google Scholar
9.Cohen, H., A course in computational algebraic number theory, Grad. Texts in Math. 138 (Springer, New York, 1995).Google Scholar
10.Cohen, H., Advanced topics in computational number theory, Grad. Texts in Math. 193 (Springer, New York, 2000).Google Scholar
11.Curtis, C. and Reiner, I., Methods of representation theory, vol. I, Wiley Classics Lib. (Wiley-Interscience, New York, 1994).Google Scholar
12.Eisenbud, D., Commutative algebra with a view towards algebraic geometry, Grad. Texts in Math. 150 (Springer, New York, 1995).Google Scholar
13.Greither, C., ‘On normal integral bases in ray class fields over imaginary quadratic fields’, Acta Arith. 83 (1997) 315–329.CrossRefGoogle Scholar
14.Halter-Koch, F., ‘Die Klassengruppe einer kommutativen Ordnung’, Math. Nachr. 168 (1994) 97–108.CrossRefGoogle Scholar
15.Huckaba, J. A., Commutative rings with zero divisors (Marcel Dekker, 1988).Google Scholar
16.Klüners, J. and Pauli, S., ‘Computing residue class rings and Picard groups of arbitrary orders’, preprint, 2003, www.math.tu-berlin.de/~pauli.Google Scholar
17.Koch, H., Algebraische Zahlentheorie (Vieweg, 1997).CrossRefGoogle Scholar
18.Neukirch, J., Algebraische Zahlentheorie (Springer, Heidelberg, 1992).CrossRefGoogle Scholar
19.Schertz, R., ‘Galoismodulstruktur und Elliptische Funktionen’, J. Number Theory 39 (1991)285–326.CrossRefGoogle Scholar
20.Taylor, M. J., ‘Mordell-Weil groups and the Galois module structure of rings of integers’, Ill. J. Math. 32 (1988) 428–452.Google Scholar