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w(E) HI (Y; Zl) V w(E) - - > HI (X; Zl) - - > 0 l~E 0 - - > R(E) - - > Periods (E) - - - > A (E) - - - > 0 41 where A (E) dfn - Periods(E); and R (E) , makes the diagram commute.

A Calculus of Special Values: Let f be an arbitrary weight 2 modular form of some level. particular interest to us are the special values of s L(f, s) at s =0 Of and = 1. We have the following simple proposition. 2. 1. 2 (a) D(f, s) = i ' r(s) • (211') We obtain (a) by comparing the residues at s =1 proves (b). s -s =0 modular forms to an action of the group ring a: [G L;(

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Arithmetic on Modular Curves by G. Stevens


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