{"product_id":"arithmetic-on-modular-curves-9780817630881","title":"Arithmetic on Modular Curves","description":"\u003cp\u003e • Author(s): G. Stevens\u003cbr\u003e • Publisher: Springer\u003cbr\u003e • Publisher Imprint: Birkhauser\u003cbr\u003e • BISAC: General\u003c\/p\u003e\u003cp\u003e\u003c\/p\u003e\u003cp\u003eOne 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.\u003c\/p\u003e","brand":"Springer","offers":[{"title":"Paperback","offer_id":45281282785431,"sku":"9780817630881","price":3672.0,"currency_code":"INR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0666\/3471\/1191\/files\/9780817630881.webp?v=1769298775","url":"https:\/\/atlanticbooks.com\/products\/arithmetic-on-modular-curves-9780817630881","provider":"Atlantic Books","version":"1.0","type":"link"}