By G. Stevens

One of the main exciting difficulties of recent quantity concept is to narrate the mathematics of abelian types to the designated values of linked L-functions. a truly unique conjecture has been formulated for elliptic curves by way of Birc~ and Swinnerton-Dyer and generalized to abelian kinds by way of Tate. The numerical proof is kind of encouraging. A weakened kind of the conjectures has been demonstrated for CM elliptic curves through Coates and Wiles, and lately bolstered by means of okay. Rubin. yet a basic evidence of the conjectures turns out nonetheless to be a ways off. many years in the past, B. Mazur [26] proved a vulnerable analog of those c- jectures. enable N be leading, and be a weight newform for r zero (N) . For a primitive Dirichlet personality X of conductor major to N, enable i\ f (X) denote the algebraic a part of L (f , X, 1) (see below). Mazur confirmed in [ 26] that the residue classification of Af (X) modulo the "Eisenstein" perfect provides information regarding the mathematics of Xo (N). There are points to his paintings: congruence formulae for the values Af(X) , and a descent argument. Mazur's congruence formulae have been prolonged to r 1 (N), N leading, through S. Kamienny and the writer [17], and in a paper in an effort to look almost immediately, Kamienny has generalized the descent argument to this case.

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**Extra info for Arithmetic on Modular Curves**

**Example text**

II' M. write and Mp is an integral domain . for cp: H± - A a be a maximal ideal for which 'II'P -module. ;;; P, there is a which makes the diagram j commutative. Proof: H± Let f be an eigenform for which 'jl(f) I: P. : ~ A P In case (ii), 'll(f) P P - is the kernel of localization at P P. Hence the map H: is an isomorphism. " p P to be an Eisenstein prime. is an integral domain is equivalent to 26 saying that is reduced at at ,:I P contains precisely one minimal prime ideal ,:I (1. e. rad ("][",:1) = 0).

3). 0 = O. We have for all a E ~ prime to m 29 Definition 1. 6. 1 is motivated by the following proposition due to Birch [2]. PROPOSITION 1. 6. 3: m prime to N For a primitive Dirichlet character and a cusp form T6<:) L(f, X, 1) f for - 217i r, 1: 1 a =0 x(aN 2 ) 1 ioo X of conductor f(z) dz a N2 m where T6<) L: m -1 x(a) e217ia/m a=O is the usual Gauss sum. 1'ian/me n m > 1, 30 We then have 'T(X) • L(f, X, 1) - 217i 1 iao f (z) dz o X o m Let A be an arbitrary abelian group and let qJ Let E X be a primitive Dirichlet character of conductor and let onto a A ~11: 11: [X] ->-> A [X] 11: [ X ] -module Definition 1.

F) is the ring generated in coefficients of f. }(f). 14). :: End (Af/q» Af/q> and the homo- is the restriction map. By passing to the dual abelian varieties we obtain an epimorphism Jac (X) / ~ ->-> Jf/~ where Jf/q> is the variety dual to H HI(X;'Zl) H (f) H I Of ; 'Zl) Af/q>. Let where the notation on the right denotes the singular homology groups of the manifold of II-valued points. 21 By the definition of H1(PiCO(X);Z) ° H I (Pic (X); Z) Af is injective. ~ H I (X; Z) the natural map HI (A f ; I'L) C-> HI(Af;Z)~HJ(Jf;Z) But canonically.