| Arithmetic on Modular Curves 1982 Edition Contributor(s): Stevens, G. (Author) |
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ISBN: 0817630880 ISBN-13: 9780817630881 Publisher: Birkhauser OUR PRICE: $52.24 Product Type: Paperback - Other Formats Published: January 1982 |
| Additional Information |
| BISAC Categories: - Gardening - Mathematics | Algebra - General |
| Dewey: 512 |
| Series: Progress in Mathematics |
| Physical Information: 0.5" H x 6" W x 9" (0.71 lbs) 217 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: One of the most intriguing problems of modern number theory is to relate the arithmetic of abelian varieties to the special values of associated L-functions. A very precise conjecture has been formulated for elliptic curves by Birc and Swinnerton-Dyer and generalized to abelian varieties by Tate. The numerical evidence is quite encouraging. A weakened form of the conjectures has been verified for CM elliptic curves by Coates and Wiles, and recently strengthened by K. Rubin. But a general proof of the conjectures seems still to be a long way off. A few years ago, B. Mazur 26] proved a weak analog of these c- jectures. Let N be prime, and be a weight two newform for r 0 (N) . For a primitive Dirichlet character X of conductor prime to N, let i\ f (X) denote the algebraic part of L (f, X, 1) (see below). Mazur showed in 26] that the residue class of Af (X) modulo the "Eisenstein" ideal gives information about the arithmetic of Xo (N). There are two aspects to his work: congruence formulae for the values Af(X), and a descent argument. Mazur's congruence formulae were extended to r 1 (N), N prime, by S. Kamienny and the author 17], and in a paper which will appear shortly, Kamienny has generalized the descent argument to this case. |