By Johannes Buchmann, Michael J. Jacobson Jr., Stefan Neis, Patrick Theobald, Damian Weber (auth.), B. Heinrich Matzat, Gert-Martin Greuel, Gerhard Hiss (eds.)

ISBN-10: 3540646701

ISBN-13: 9783540646709

ISBN-10: 364259932X

ISBN-13: 9783642599323

This publication includes 22 lectures offered on the ultimate convention of the German study application "Algorithmic quantity idea and Algebra 1991-1997", subsidized via the Deutsche Forschungsgemeinschaft. the aim of this examine application and the assembly used to be to assemble builders of computing device algebra software program and researchers utilizing computational easy methods to achieve perception into experimental difficulties and theoretical questions in algebra and quantity idea. The publication offers an summary on algorithmic tools and effects got in this interval generally in algebraic quantity idea, commutative algebra and algebraic geometry, and workforce and illustration idea. a number of the articles illustrate the present country of the pc algebra platforms constructed with aid from the learn software, for instance KANT and LiDIA for algebraic quantity conception, SINGULAR, REDLOG and INVAR for commutative algebra and invariant idea respectively, and hole, SYSYPHOS and CHEVIE for team and illustration theory.

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**Extra info for Algorithmic Algebra and Number Theory: Selected Papers From a Conference Held at the University of Heidelberg in October 1997**

**Sample text**

8 2 (z) J. Basmaji implemented this algorithm in his thesis [3] and computed the form FE for many elliptic curves E. He uses this to get a lot of empirical material concerning the distribution of values of L-series at s = 1 (and so, according to the conjecture of Birch-Swinnerton-Dyer, of orders of Selmer groups) of families of twists of modular elliptic curves. It would be very interesting to have a better theoretical understanding of these data. Cusp forms of weight 1: Another weight which is not accessible directly is k = 1.

Springer-Verlag. 33. J. F. Mestre. La methode des graphes. Exemples et applications. In Proceedings of the international conference on class numbers and fundamental units of algebraic number fields, pages 217-242, Katata/ Japan, 1986. 34. J. F. Mestre. Construction de courbes de genre 2 a partir de leurs modules. In T. Mora and C. Traverso, editors, Effective Methods in Algebraic Geometry, volume 94 of Progress in Mathematics, pages 313-334. Birkhauser Verlag, 1991. 35. L. Mai and M. R. Murty. The Phragmen-Lindelof theorem and modular elliptic curves.

Prop. ) Using the function field analogue of the Shimura-Taniyama-Weil correspondence and some properties of Drinfeld modular curves, we obtain a complete classification of these curves (Thm. 4, Cor. 6). In some cases, dealt with in sect. 6, we are able to determine the torsion of the Mordell-Weil group and the possible isogenies between such curves. The results of this paper had their origin in calculations of Hecke operators in spaces of automorphic forms over function fields. These were performed in the framework of the "Schwerpunktprogramm Algorithmische Zahlentheorie und Algebra" of Deutsche Forschungsgemeinschaft (DFG), whose support is gratefully acknowledged.

### Algorithmic Algebra and Number Theory: Selected Papers From a Conference Held at the University of Heidelberg in October 1997 by Johannes Buchmann, Michael J. Jacobson Jr., Stefan Neis, Patrick Theobald, Damian Weber (auth.), B. Heinrich Matzat, Gert-Martin Greuel, Gerhard Hiss (eds.)

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