The Modular Invariant

M. Ram Murty and Purusottam Rath
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M. Ram Murty: Queen’s University, Department of Mathematics and Statistics
Purusottam Rath: Chennai Mathematical Institute

Chapter Chapter 14 in Transcendental Numbers, 2014, pp 65-74 from Springer

Abstract: Abstract We begin with a discussion of an important result in complex analysis called the uniformisation theorem. We have shown how to associate a ℘ $$\wp $$ -function with a given lattice L. Thus, g 2 = g 2 ( L ) , g 3 = g 3 ( L ) $$g_{2} = g_{2}(L),g_{3} = g_{3}(L)$$ can be viewed as functions on the set of lattices. For a complex number z with imaginary part ℑ(z) > 0, let L z denote the lattice spanned by z and 1. We will denote the corresponding g 2, g 3 associated with L z by g 2(z) and g 3(z). Thus, g 2 ( z ) = 60 ∑ ( m , n ) ≠ ( 0 , 0 ) ( m z + n ) − 4 , $$\displaystyle{g_{2}(z) = 60\sum _{(m,n)\neq (0,0)}(mz + n)^{-4},}$$ and g 3 ( z ) = 140 ∑ ( m , n ) ≠ ( 0 , 0 ) ( m z + n ) − 6 . $$\displaystyle{g_{3}(z) = 140\sum _{(m,n)\neq (0,0)}(mz + n)^{-6}.}$$ We set Δ ( z ) = g 2 ( z ) 3 − 27 g 3 ( z ) 2 $$\displaystyle{\varDelta (z) = g_{2}(z)^{3} - 27g_{ 3}(z)^{2}}$$ which is the discriminant of the cubic defined by the corresponding Weierstrass equation. We first prove:

Keywords: Uniformization Theorem; Weierstrass; Imaginary Quadratic Field; Standard Fundamental Domain; Usual Ideal Class Group (search for similar items in EconPapers)
Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4939-0832-5_14

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DOI: 10.1007/978-1-4939-0832-5_14

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