By Dixon M., Kurdachenko L., Subbotin I.

ISBN-10: 0470496363

ISBN-13: 9780470496367

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Extra info for Algebra and number theory. An integrated approach

Example text

This was worked out and implemented by X. -J. Weber in his thesis (see [52]) for d::; 5. We give an example for the simplest case d = 2. For the general case we refer the reader to [52]. 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 [51] 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 [51]) 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 [38]) but in the special case of hyperelliptic curves there is a result due to Mumford and Poor ([41]): 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.