Arithmetic on Modular Curves by G. Stevens

HI (Y ;

And Hence by Poincare duality the natural map cP (f) : H ->-> H (f) is surjective. Since is a H 2 -dimensional preserves is a free 1)9 ~ Af rank 2 K (f)-vector space. 1[~ -module we have H (f) 1)9 ~ The complex conjugation involution and hence induces an involution on H (f). We have an isomorphism The spaces cP (f)± H (f)± 1)9 ~ are ) -dimensional K (f)-vector spaces. Let be the composition cP (f)± : H ->-> H (f) -0>-> H (f)± Let CPf E differential form f(z)dz. HI (X; 0:) on w(f) We view CPf as a be the cohomology class represented by the X whose pullback to the upper half plane is 1[-homomorphism CPf: H - - > 0: where 0: morphism is given the structure of h f : 1[ - 0:.

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