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

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

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Lecture 38: The Weight Two Modular Form Decays Exponentially in a Neighbourhood of Infinity III

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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 of Lecture 37 Part A 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, we want to find a suitable region in the upper half-plane on which the mapping properties of this weight two modular form may be easily studied. In the last couple of lectures we saw 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 region, and that the weight two modular form vanishes at infinity. In part A of this lecture we estimate that this vanishing at infinity is in fact an exponential decay. This estimation is actually a computation of the Fourier coefficient that matters most in the Fourier development of the weight two modular form which has period two. This estimation is critical for the study of the mapping properties which will be completed in part B of this lectureKeywords for Lecture 37 Part A 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, singular part of the Laurent expansion, pole of order two, uniform convergence, Weierstrass M-test, removable singularity, entire function, periodic function, period of a function, singly periodic function, Liouvilles theorem, Fourier coefficient, Fourier development

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