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 ebook includes 22 lectures provided on the ultimate convention of the German study software "Algorithmic quantity concept and Algebra 1991-1997", subsidized by way of the Deutsche Forschungsgemeinschaft. the aim of this examine software and the assembly was once to assemble builders of laptop algebra software program and researchers utilizing computational tips on how to achieve perception into experimental difficulties and theoretical questions in algebra and quantity idea. The ebook provides an summary on algorithmic tools and effects received in this interval in general in algebraic quantity idea, commutative algebra and algebraic geometry, and workforce and illustration thought. a few of the articles illustrate the present nation of the pc algebra structures constructed with help from the learn application, for instance KANT and LiDIA for algebraic quantity concept, SINGULAR, REDLOG and INVAR for commutative algebra and invariant concept respectively, and hole, SYSYPHOS and CHEVIE for workforce and illustration theory.

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

**Sample text**

Weber. 4 39 Mordell-Weil Groups over Finite Fields We consider the following task: Let Cp be an irreducible projective regular curve of genus 9 over the finite field kp. Let Jcp (kp) be the Jacobian variety of Cpo Determine #Jcp(kp). Of course one can try to do this by counting k'-rational points of extension fields k' of kp. But since k' has to run through all extensions of degree ~ 9 it is obvious that this is only possible for very small fields kp and small genus g. The alternative is to determine the L-series Lp(Cp, t) of Cp over kp by a more structural approach.

P. Deligne and M. Rapoport. Les schemas de modules de courbes elliptiques. In W. Kuyk and P. Deligne, editors, Modular Functions in One Variable II, number 349 in Lecture Notes in Mathematics, pages 143-316, Berlin, Heidelberg, 1973. Springer-Verlag. 10. P. -P. Serre. Formes modulaires de poids 1. Ann. sci. S. 4e ser. , pages 507-530, 1974. 11. M. Eichler. The Basis Problem for Modular Forms and Traces of the Hecke Operators. In Modular Functions of one Variable I, volume 320 of Lecture Notes in Mathematics, pages 75-152, Berlin, Heidelberg, 1972.

Compo ,43:483-494, 1985. 45. J. P. Serre. Modular forms of weight one and Galois representations. In A. Frohlich, editor, Algebraic Number Fields, pages 193-268, New York, 1977. Academic Press. 46. G. Shimura. Introduction to the Arithmetic Theory of Automorphic Jilunctions. Princeton University Press, Princeton, New Jersey, 1971. 47. G. Shimura. On modular forms of half integral weight. Annals of Mathematics, 97:440-481, 1973. 48. G. Shimura. On the factors of the jacobian variety of a modular function field.

### 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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