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MA342J: Introduction to Modular Forms Tutorial 5, April 5 Families of elliptic curves and modular forms Consider the one-parametric family of cubic curves given by Et : y 2 = x3 − 2x2 + (1 − t)x (t ∈ C) . a) Determine singular values of t for this family, i.e. those t0 for which Et0 is not an elliptic curve b) Write j-invariant for this family j(Et ) as a rational function of t. Check that the map t 7→ j(t) is generically 3 : 1. c) Show that in this family one can find a representative of every isomorphism class of elliptic curves. d) Consider the modular function t(z) = 64 ∆(2z) ∆(z) + 64∆(2z) j(t(z)) = 1 + 744 + O(q) q on Γ0 (2). Check that e) Show that the point (0, 0) ∈ Et is 2-torsion, which explains appearance of the group Γ0 (2). f) Explain that the integrals Z √ 1− t Ω1 (t) = 0 Z dx p 0 Ω2 (t) = −∞ x3 − 2x2 + (1 − t)x , dx p 3 x − 2x2 + (1 − t)x are periods for this family. In fact, Ω2 (t(z)) is a square root of an Eisenstein series of weight 2 on Γ0 (2) and Ω1 (t(z)) is its multiple by a simple rational function in z. 1