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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. We may view qJE By Theorem 1. 8. 2 Let Periods (E) R(E)* = {¢ E E ~ is the unique map which qJE as a cohomology class qJE I H (X; A (E» CIl • R (E).

Since the involution 011 H' ,H (X; a:) T-homomorphisms L* is a free (1 ± L *) on H\X;a:) rank 2 map 1 H' HO(X;Ol) ° isafree maps 1ra: -module. 1 H' Ta:-module ° isomorphically In fact, the isomorphically onto We have then, PROPOSITION 1. 2: rank 1 The modules H\X; Z/:)± and H1(X; :Z)± are o T-modules. The statement for the homology groups follows from Poincare duality. }(f) = image (h f ). }(f) is the ring generated in coefficients of f. }(f). 14). :: End (Af/q» Af/q> and the homo- is the restriction map.

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Arithmetic on Modular Curves by G. Stevens


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