By G. Stevens

One of the main fascinating difficulties of recent quantity conception is to narrate the mathematics of abelian kinds to the distinct values of linked L-functions. a truly special conjecture has been formulated for elliptic curves through Birc~ and Swinnerton-Dyer and generalized to abelian types by way of Tate. The numerical facts is sort of encouraging. A weakened kind of the conjectures has been validated for CM elliptic curves by way of Coates and Wiles, and lately bolstered through ok. Rubin. yet a common evidence of the conjectures turns out nonetheless to be far off. many years in the past, B. Mazur [26] proved a vulnerable analog of those c- jectures. enable N be best, and be a weight newform for r zero (N) . For a primitive Dirichlet personality X of conductor top to N, permit i\ f (X) denote the algebraic a part of L (f , X, 1) (see below). Mazur confirmed in [ 26] that the residue category of Af (X) modulo the "Eisenstein" excellent 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, by way of S. Kamienny and the writer [17], and in a paper so one can look presently, Kamienny has generalized the descent argument to this case.

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2: (a) D(f, s) = i • r(s) • (211) (b) D(f, s) e 00 Ji ~ f(z)y -s s -1 L(f, s) (i

T . J, - U We may assume for exactly one choice of value of k. t The double coset decom- IS (~ ~) , r x ] E cusps. t. t) y. ex+ y lr~ since Hence, (L, N) = 1. 4. 'JI'-modules and Periods of Cusp Forms: Let r r algebra of be of type (N l' N 2). as in § 1. 2. For a field Definition 1. 4. 4). have rank r 2) There is a field free 'JI' K-module ;J-module For a A 'JI'-module acts invertibly on K of characteristic of M rank r . M 1[K M 0 be the Hecke 1[ for 'JI' ®~ K. of finite type is said to such that M ®z K is a 0 on which an involution L acts, let M+ {m € MIL(m) m} M {m € M/L(m) - m} M+ M/(l - L)M M M/(l + L)M M, then M+ "" M+ and K write and ®z

T) y. ex+ y lr~ since Hence, (L, N) = 1. 4. 'JI'-modules and Periods of Cusp Forms: Let r r algebra of be of type (N l' N 2). as in § 1. 2. For a field Definition 1. 4. 4). have rank r 2) There is a field free 'JI' K-module ;J-module For a A 'JI'-module acts invertibly on K of characteristic of M rank r . M 1[K M 0 be the Hecke 1[ for 'JI' ®~ K. of finite type is said to such that M ®z K is a 0 on which an involution L acts, let M+ {m € MIL(m) m} M {m € M/L(m) - m} M+ M/(l - L)M M M/(l + L)M M, then M+ "" M+ and K write and ®z