## Download PDF by G. Stevens: Arithmetic on Modular Curves

By G. Stevens

ISBN-10: 0817630880

ISBN-13: 9780817630881

ISBN-10: 1468491652

ISBN-13: 9781468491654

One of the main exciting difficulties of recent quantity conception is to narrate the mathematics of abelian types to the unique values of linked L-functions. a truly distinct conjecture has been formulated for elliptic curves by means of Birc~ and Swinnerton-Dyer and generalized to abelian types via Tate. The numerical proof is kind of encouraging. A weakened type of the conjectures has been confirmed for CM elliptic curves via Coates and Wiles, and lately bolstered through ok. Rubin. yet a common 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. allow N be top, and be a weight newform for r zero (N) . For a primitive Dirichlet personality X of conductor top 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 category of Af (X) modulo the "Eisenstein" excellent offers information regarding the mathematics of Xo (N). There are elements 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 best, via S. Kamienny and the writer [17], and in a paper on the way to seem almost immediately, Kamienny has generalized the descent argument to this case.

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**Extra resources for Arithmetic on Modular Curves**

**Example text**

The analytically continues to a meromorphic s-plane(Ogg [33], Theorem 1 pg. 1-5). a f i OO 0 is the function of s-1 f (z) Y dz ~ z s defined by x + iy. The lemma (b), (c) shows that the integral converges absolutely for Re(s) > 2 and hence D(f, s) defines an analytic function of s in this region. Let then PROPOSITION 2. 1. 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

Following Mazur ([ 26], ~6) we define the universal special value associated to as follows. X to be a relative homology class 28 Definition 1. 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.

Let be the composition cP (f)± : H ->-> H (f) -0>-> H (f)± Let CPf E differential form f(z)dz. HI (X; 0:) on w(f) We view CPf as a be the cohomology class represented by the X whose pullback to the upper half plane is 1[-homomorphism CPf: H - - > 0: where 0: morphism is given the structure of h f : 1[ - 0:. 1[-module We will write inherited from the homo- Periods (f) for the image of CPf. 22 From the definition of Af it follows that CPf factors through cP (f). Hence we have a commutative diagram Let Then each y± be the generators of the free y E H 181 ~ + Ct, /1 E 1l"~.

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