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MODULAR FORMS-page95

# MODULAR FORMS-page95 - 91 8.4 Find all normal subgroups...

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91 8.4 Find all normal subgroups Γ of Γ(1) for which the genus of the modular curve X (Γ) is equal to 0. [Hint: Use Theorem 10.4 and prove that r 2 = μ Γ / 2 , r 3 = μ Γ / 3 , r | μ Γ ]. 8.5 Generalize the Hurwitz formula to any non-constant holomorphic map f : X Y of compact Riemann surfaces. 8.6 Show that the Moebius transformation τ → - 1 /Nτ defines a holomorphic auto- morphism of finite order 2 of the modular curve X 0 ( N ). Give an interpretation of this automorphism if one identifies the points of X 0 ( N ) with isomorphism clases of pairs ( E, H ) as in Theorem 8.7. 8.7 Let Γ 1 ( N ) = { α β γ δ « Γ 0 ( N ) : α 1 mod N } . Give an analogue of Theorems 8.6 and 8.7 for the curve H / Γ 1 ( N ). 8.8 Using Riemann-Roch theorem prove that any compact Riemann surface of genus 0 is isomorphic to P 1 ( C ). 8.9 Using Riemann-Roch theorem prove that any compact Riemann surface of genus 1 is isomorphic to a complex torus C / Λ. 8.10 Compute the dimension of the space M 1 ( X 0 (11)).
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• Fall '09
• ONTONKONG
• Elliptic Curve, Riemann surface, Algebraic curve, Riemann–Roch theorem, compact Riemann surface, modular curve

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