By Johannes Buchmann, Michael J. Jacobson Jr., Stefan Neis, Patrick Theobald, Damian Weber (auth.), B. Heinrich Matzat, Gert-Martin Greuel, Gerhard Hiss (eds.)
This booklet includes 22 lectures awarded on the ultimate convention of the German examine application "Algorithmic quantity thought and Algebra 1991-1997", backed through the Deutsche Forschungsgemeinschaft. the aim of this examine application and the assembly used to be to collect builders of laptop algebra software program and researchers utilizing computational ways to achieve perception into experimental difficulties and theoretical questions in algebra and quantity idea. The ebook offers an outline on algorithmic equipment and effects got in this interval generally in algebraic quantity conception, commutative algebra and algebraic geometry, and crew and illustration idea. the various articles illustrate the present nation of the pc algebra platforms constructed with help from the learn application, 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 crew 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
This was worked out and implemented by X. -J. Weber in his thesis (see ) for d::; 5. We give an example for the simplest case d = 2. For the general case we refer the reader to . 6. For N = 87 = 3·29 the genus of Xo(N) is 9 and the dimension of the space of new forms is 5. The Jacobian splits in where Al arises from an old form coming of level 29, A3 is a 3-dimensional factor and A2 is isogenous to the Jacobian of a hyperelliptic curve of genus 2 associated to the cusp form of level 87 and weight 2, with Fourier coefficients: a2 = a, al3 a3 = 4a - 3, a17 = 1, a5 = -2a = 3, .
21)). In Wang  it is shown that the pairing ImH(A, A) -+ Z corresponds to the intersection pairing on HI (Xo(N), Z). In the same paper, Wang proves that one can compute the period matrix as The next step is to decide whether A f is Q-isogenous to a principally polarized abelian variety. A sufficient condition (see ) is that holds. Let us assume now that this is the case. In general there is no criterion to decide whether a principally polarized abelian variety is isogenous to the Jacobian of a curve (Schottky problem, see ) but in the special case of hyperelliptic curves there is a result due to Mumford and Poor (): If A is simple over Q, principally polarized and certain Arithmetic of Modular Curves and Applications 37 B-series vanish, then A is isogenous to the Jacobian of an hyperelliptic curve.
Frey, editor, On Arlin's Conjecture for Odd 2-dimensional Representations, volume 1585 of Lecture Notes in Mathematics, pages 109-121, Berlin, Heidelberg, 1994. Springer-Verlag. 25. S. Lang. Survey of Diophantine Geometry. Springer-Verlag, Berlin, Heidelberg, 1997. 26. Q. Liu. Modeles minimaux des courbes de genre deux. J. Reine Angew. , 453:127-164, 1994. 27. J. Manin. Cyclotomic Fields and Modular Curves. Russian Math. Surveys, 26(6):778, 1971. 28. J. Manin. Parabolic Points and Zeta-Functions of Modular Curves.
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.)