Arithmetic on Modular Curves by G. Stevens

By G. Stevens

One of the main exciting difficulties of recent quantity idea is to narrate the mathematics of abelian forms to the distinct values of linked L-functions. a really distinct conjecture has been formulated for elliptic curves by way of Birc~ and Swinnerton-Dyer and generalized to abelian forms via Tate. The numerical facts is kind of encouraging. A weakened type of the conjectures has been proven for CM elliptic curves through Coates and Wiles, and lately bolstered through ok. Rubin. yet a normal evidence of the conjectures turns out nonetheless to be a ways off. many years in the past, B. Mazur [26] proved a susceptible analog of those c- jectures. enable N be top, and be a weight newform for r zero (N) . For a primitive Dirichlet personality X of conductor major 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 type of Af (X) modulo the "Eisenstein" excellent supplies 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 major, by means of S. Kamienny and the writer [17], and in a paper with a view to seem almost immediately, Kamienny has generalized the descent argument to this case.

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Is invariant for and view Zo E J( Define a function r g on w (E) as a differential by J( and defines a meromorphic function on with X div(g) = n • Ii (E) Since Ii (E) represents the divisor class 1 w(E) = 21Ti To prove (b) we note Let A, B <;;; homomorphism. and hence also a: Then ¢: ¢ e2 (r; A) defines a map - e 2 (r; B). I ¢ also defines a map ¢ to a 1 we have ~ ng be two subgroups of H (Y; ~ • A). We may extend and then x(E) a: and apply Proposition 1. 7 • o and ¢: A - o B be a ¢ : Div (cusps; A) - By the theorem ~-linear o.

2: HI (Y; a:) We will show that the map is defined over Define a projection

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.

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