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

This e-book comprises 22 lectures awarded on the ultimate convention of the German study software "Algorithmic quantity idea and Algebra 1991-1997", subsidized by means of the Deutsche Forschungsgemeinschaft. the aim of this learn software and the assembly was once to compile builders of machine algebra software program and researchers utilizing computational how to achieve perception into experimental difficulties and theoretical questions in algebra and quantity conception. The ebook provides an summary on algorithmic equipment and effects received in this interval typically in algebraic quantity conception, commutative algebra and algebraic geometry, and crew and illustration concept. the various articles illustrate the present kingdom of the pc algebra structures built with help from the study software, for instance KANT and LiDIA for algebraic quantity conception, SINGULAR, REDLOG and INVAR for commutative algebra and invariant thought respectively, and hole, SYSYPHOS and CHEVIE for crew and illustration theory.

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Norm. Super. , IV. Ser. 19, 409-468 (1986). 6. J. Cremona. Algorithms for Modular Elliptic Curves. Cambridge University Press, Cambridge, 1992. 7. J. E. Cremona. Computing the degree of modular parametrization of a modular elliptic curve. Math. Compo ,64:1235-1250, 1995. 8. H. Darmon, F. Diamond, and R. Taylor. Fermat's Last Theorem. Current Development in Mathematics, pages 1-170, Boston, 1995. International Press. 46 G. Frey, M. Miiller 9. P. Deligne and M. Rapoport. Les schemas de modules de courbes elliptiques.

Universal Fourier expansions of modular forms. In G. Frey, editor, On Arlin's Conjecture for Odd 2-dimensional Representations, volume 1585 of Lecture Notes in Mathematics, pages 59-94, Berlin, Heidelberg, 1994. 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.

D(E) i= and j(E) = (3-1. If v((3) ~ 0, E has good reduction (v((3) = 0) or split multiplicative reduction (v((3) > 0, cf. 8)) with f(EIK) = or 1, respectively. ° ° Let therefore v((3) = -k with some natural number k, and put k = 6l - m with m E {O, 1, ... , 5}. 2) = m. Write jj:= 7r61 (3 = h m 7rm + h m +l 7r m +1 +... where v(7r 61 (3) with hi E k(v) <-+ K. 3 Theorem. 2) is minimal in each of the following cases: (a) m = 1 (b) m = 3 (c) m = 5 (d) m = 2, h3 i= (e) m = 4, 1 ~ 2, h5 i= (f) m = 0, hI i= 0.