By G. Stevens

ISBN-10: 0817630880

ISBN-13: 9780817630881

ISBN-10: 1468491652

ISBN-13: 9781468491654

One of the main interesting difficulties of recent quantity conception is to narrate the mathematics of abelian types to the distinct values of linked L-functions. a really specified conjecture has been formulated for elliptic curves through Birc~ and Swinnerton-Dyer and generalized to abelian forms through Tate. The numerical facts is sort of encouraging. A weakened type of the conjectures has been demonstrated for CM elliptic curves by means of Coates and Wiles, and lately reinforced by way of ok. Rubin. yet a basic facts 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 leading 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 type of Af (X) modulo the "Eisenstein" excellent offers 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 top, by means of S. Kamienny and the writer [17], and in a paper to be able to seem almost immediately, Kamienny has generalized the descent argument to this case.

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**Extra resources for Arithmetic on Modular Curves**

**Sample text**

6. 1: (a) For two cusps X,y E lP\~) <;;;:K~ let be the relative homology class represented by the projection to oriented geodesic path joining x to of the y. (b) For a primitive Dirichlet character X f. 1 to X of conductor m prime N, let Remark 1. 6. 2: There is a natural inclusion of the absolute homology of X in the relative homology: The special value A(X) actually represents an absolute homology class. To see this, consider the long exact relative homology sequence We need only show But since ~(A(X» (m, N) = 1, (see §t.

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

These results yield another (more explicit) proof of Theorem 1. 8. 2. 1. L-functions: Let f be a weight 2 modular form of level definition and functional equation of the Suppose f associated to '" L: n=O an q~ e With f as above, and 0(n 1 +e:) ... € a (b) f(iy) O(y -2-e:) as y ... 0 (c) f(iy) O(e -yiN) as y ... '" as n for the function > 0 , '" (a) n f. 211' i z/N aO(f) = a O be the constant term and write f(z) LEMMA 2. 1. 1: Proof: L-function has the Fourier expansion f(z) Let N. We review the This is well known: (a) Heeke ([ 15], Satz 6) (b) Ogg ([ 33 ] , Proposition 1 Page 1-3) .

### Arithmetic on Modular Curves by G. Stevens

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