Download Arithmetic on Modular Curves by G. Stevens PDF

By G. Stevens

One of the main exciting difficulties of recent quantity concept is to narrate the mathematics of abelian types to the designated values of linked L-functions. a truly unique conjecture has been formulated for elliptic curves by way of Birc~ and Swinnerton-Dyer and generalized to abelian kinds by way of Tate. The numerical proof is kind of encouraging. A weakened kind of the conjectures has been demonstrated for CM elliptic curves through Coates and Wiles, and lately bolstered by means of okay. Rubin. yet a basic 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. enable N be leading, and be a weight newform for r zero (N) . For a primitive Dirichlet personality X of conductor major 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 classification of Af (X) modulo the "Eisenstein" perfect 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, through S. Kamienny and the writer [17], and in a paper in an effort to look almost immediately, Kamienny has generalized the descent argument to this case.

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II' M. write and Mp is an integral domain . for cp: H± - A a be a maximal ideal for which 'II'P -module. ;;; P, there is a which makes the diagram j commutative. Proof: H± Let f be an eigenform for which 'jl(f) I: P. : ~ A P In case (ii), 'll(f) P P - is the kernel of localization at P P. Hence the map H: is an isomorphism. " p P to be an Eisenstein prime. is an integral domain is equivalent to 26 saying that is reduced at at ,:I P contains precisely one minimal prime ideal ,:I (1. e. rad ("][",:1) = 0).

3). 0 = O. We have for all a E ~ prime to m 29 Definition 1. 6. 1 is motivated by the following proposition due to Birch [2]. PROPOSITION 1. 6. 3: m prime to N For a primitive Dirichlet character and a cusp form T6<:) L(f, X, 1) f for - 217i r, 1: 1 a =0 x(aN 2 ) 1 ioo X of conductor f(z) dz a N2 m where T6<) L: m -1 x(a) e217ia/m a=O is the usual Gauss sum. 1'ian/me n m > 1, 30 We then have 'T(X) • L(f, X, 1) - 217i 1 iao f (z) dz o X o m Let A be an arbitrary abelian group and let qJ Let E X be a primitive Dirichlet character of conductor and let onto a A ~11: 11: [X] ->-> A [X] 11: [ X ] -module Definition 1.

F) is the ring generated in coefficients of f. }(f). 14). :: End (Af/q» Af/q> and the homo- is the restriction map. By passing to the dual abelian varieties we obtain an epimorphism Jac (X) / ~ ->-> Jf/~ where Jf/q> is the variety dual to H HI(X;'Zl) H (f) H I Of ; 'Zl) Af/q>. Let where the notation on the right denotes the singular homology groups of the manifold of II-valued points. 21 By the definition of H1(PiCO(X);Z) ° H I (Pic (X); Z) Af is injective. ~ H I (X; Z) the natural map HI (A f ; I'L) C-> HI(Af;Z)~HJ(Jf;Z) But canonically.

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