Download PDF by G. Stevens: Arithmetic on Modular Curves
By G. Stevens
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  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 , 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
The analytically continues to a meromorphic s-plane(Ogg , 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"~.
Arithmetic on Modular Curves by G. Stevens