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An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves

IIT Madras, , Prof. T.E. Venkata Balaji

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Lecture 36: The Weight Two Modular Form assumes Real Values on the Imaginary Axis

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An Introduction to Riemann Surfaces and Algebraic Curves Complex 1-Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit httpwww.nptel.iitm.ac.insyllabus111106044Goals In the last few lectures, we constructed an analytic function on the upper half-plane which is a modular form of weight two, i.e., an analytic function that is invariant under the action of the congruence-mod-2 subgroup of the unimodular group. We saw that the effect of a general element of the unimodular group on this weight two modular form can be understood by just computing the effect under each of five unimodular elements representing pre-images of the five non-trivial elements in the quotient by the congruence-mod-2 subgroup. We saw that these computations resulted in five simple and beautiful functional equations satisfied by the weight two modular form. Using these functional equations diligently, in the present and the forthcoming lectures, we obtain a suitable region in the upper half-plane on which the mapping properties of the weight two modular form can be studied. In this lecture we show that the weight two modular form assumes only real values on the imaginary axis, which will turn out to be a boundary for such a regionKeywords Upper half-plane, invariants for complex tori, complex torus associated to a lattice (or) grid in the plane, doubly-periodic meromorphic function (or) elliptic function associated to a lattice, Weierstrass phe-function associated to a lattice, ordinary differential equation satisfied by the Weierstrass phe-function, automorphic function (or) automorphic form, weight two modular function (or) modular form, congruence-mod-2 subgroup of the unimodular group, special linear group, finite group, kernel of a group homomorphism, normal subgroup, zeros of the derivative of the Weierstrass phe-function

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Sam

Excellent course helped me understand topic that i couldn't while attendinfg my college.

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Dembe

Great course. Thank you very much.

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