Download Arithmetic on Modular Curves by G. Stevens PDF

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 2 , O(f, s) '" -21Tny/N y s ~ a 10 e L n 0 y n =1 CD t· i' ~ (A)S . 2. A Calculus of Special Values: Let f be an arbitrary weight 2 modular form of some level.

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

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