{"product_id":"9780817630881","title":"Progress in Mathematics","description":"\u003ch1\u003eProgress in Mathematics\u003c\/h1\u003e \u003ch2\u003eStevens, G.\u003c\/h2\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 \u003ch3\u003eDetails\u003c\/h3\u003e \u003cp\u003ePublished by: Birkhäuser\u003c\/p\u003e \u003cp\u003ePublication Date: 1982-01-01\u003c\/p\u003e \u003cp\u003eFormat: Paperback\u003c\/p\u003e \u003cp\u003eISBN-13: 9780817630881\u003c\/p\u003e \u003cp\u003eDOI: 10.1007\/978-1-4684-9165-4\u003c\/p\u003e \u003cp\u003eDimensions: 229cm x152cm\u003c\/p\u003e \u003cp\u003ePages: 217\u003c\/p\u003e ","brand":"Birkhäuser Boston","offers":[{"title":"Default Title","offer_id":46541443432588,"sku":"9780817630881","price":49.49,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0710\/9545\/1788\/files\/9780817630881.jpg?v=1775736364","url":"https:\/\/lateknightbooks.com\/products\/9780817630881","provider":"Late Knight Books and Services, LLC","version":"1.0","type":"link"}