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 types to the detailed values of linked L-functions. a truly targeted conjecture has been formulated for elliptic curves by means of Birc~ and Swinnerton-Dyer and generalized to abelian kinds by means of Tate. The numerical proof is sort of encouraging. A weakened type of the conjectures has been demonstrated for CM elliptic curves through Coates and Wiles, and lately bolstered through ok. Rubin. yet a basic facts of the conjectures turns out nonetheless to be some distance off. many years in the past, B. Mazur [26] proved a susceptible analog of those c- jectures. allow N be major, and be a weight newform for r zero (N) . For a primitive Dirichlet personality X of conductor top to N, allow 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" excellent provides information regarding the mathematics of Xo (N). There are facets 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, via S. Kamienny and the writer [17], and in a paper to be able to seem presently, Kamienny has generalized the descent argument to this case.

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Extra resources for Arithmetic on Modular Curves

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We will do this in the periods of next chapter. 6) and Drinfeld (6]. Proof of Theorem 1. 8. 2: HI (Y; a:) We will show that the map is defined over Define a projection

For a prime ideal the completion of M PROPOSITION 1. 5. ::; 'II' at P. Let A 'II'-homomorphism. ::; '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.

Of E € K(f) [X] K(f) [X] • n;gn X as the algebraic part of Of course this depends on the choice of the periods n~. In subsequent sections we will construct explicit cohomology classes cP with values in finite 1[-modules. By calculating the associated special values we obtain congruences satisfied by the universal special value A(X). In case the conditions of Proposition 1. 5. 1 are satisfied we also obtain congruences satisfied by the algebraic part of cusp forms f. L(f, X, 1) for appropriate 32 ~ o Points of finite order in Pic (X(r»: 1.

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