By G. Stevens

One of the main exciting difficulties of recent quantity idea is to narrate the mathematics of abelian forms to the distinct values of linked L-functions. a really distinct conjecture has been formulated for elliptic curves by way of Birc~ and Swinnerton-Dyer and generalized to abelian forms via Tate. The numerical facts is kind of encouraging. A weakened type of the conjectures has been proven for CM elliptic curves through Coates and Wiles, and lately bolstered through ok. Rubin. yet a normal evidence of the conjectures turns out nonetheless to be a ways off. many years in the past, B. Mazur [26] proved a susceptible analog of those c- jectures. enable N be top, and be a weight newform for r zero (N) . For a primitive Dirichlet personality X of conductor major to N, allow i\ f (X) denote the algebraic a part of L (f , X, 1) (see below). Mazur confirmed in [ 26] that the residue type of Af (X) modulo the "Eisenstein" excellent supplies information regarding the mathematics of Xo (N). There are facets 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 major, by means of S. Kamienny and the writer [17], and in a paper with a view to seem almost immediately, Kamienny has generalized the descent argument to this case.

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**Example text**

Is invariant for and view Zo E J( Define a function r g on w (E) as a differential by J( and defines a meromorphic function on with X div(g) = n • Ii (E) Since Ii (E) represents the divisor class 1 w(E) = 21Ti To prove (b) we note Let A, B <;;; homomorphism. and hence also a: Then ¢: ¢ e2 (r; A) defines a map - e 2 (r; B). I ¢ also defines a map ¢ to a 1 we have ~ ng be two subgroups of H (Y; ~ • A). We may extend and then x(E) a: and apply Proposition 1. 7 • o and ¢: A - o B be a ¢ : Div (cusps; A) - By the theorem ~-linear o.

2: HI (Y; a:) We will show that the map is defined over Define a projection

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.