[過去ﾛｸﾞ] Inter-universal geometry と ABC予想 (応援スレ) 76 (1002ﾚｽ)

Inter-universal geometry と ABC予想 (応援スレ) 76 http://rio2016.5ch.net/test/read.cgi/math/1759924222/

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653: 現代数学の系譜 雑談 ◆yH25M02vWFhP [] 2025/10/22(水) 07:14:10.77 ID:Xg86O8He つづき (参考) https://en.wikipedia.org/wiki/Weierstrass_elliptic_function Weierstrass elliptic function This class of functions is also referred to as ℘-functions and they are usually denoted by the symbol ℘, a uniquely fancy script p. They play an important role in the theory of elliptic functions, i.e., meromorphic functions that are doubly periodic. A ℘-function together with its derivative can be used

to parameterize elliptic curves and they generate the field of elliptic functions with respect to a given period lattice. Motivation A cubic of the form (Cg2,g3)^C={(x,y)∈C2:y2=4x3−g2x−g3}, where g2,g3∈C are complex numbers with (g2)^3−27(g3)^2≠0, cannot be rationally parameterized.[1] Yet one still wants to find a way to parameterize it. For the quadric K={(x,y)∈R2:x2+y2=1}; the unit circle, there exists a (non-rational) parameterization using the sine function and its derivative t

he cosine function: ψ:R/2πZ→K,t↦(sin t,cos t). Because of the periodicity of the sine and cosine R/2πZ is chosen to be the domain, so the function is bijective. In a similar way one can get a parameterization of (Cg2,g3)^C by means of the doubly periodic ℘-function (see in the section "Relation to elliptic curves"). This parameterization has the domain C/Λ, which is topologically equivalent to a torus.[2] There is another analogy to the trigonometric functions. Consider the integr

al function 略 It can be simplified by substituting y=sin t and s=arcsin x: 略 That means a^−1(x)=sin x. So the sine function is an inverse function of an integral function.[3] Elliptic functions are the inverse functions of elliptic integrals. In particular, let: 略 Then the extension of u^−1 to the complex plane equals the ℘-function.[4] This invertibility is used in complex analysis to provide a solution to certain nonlinear differential equations satisfying the Painlevé property

, i.e., those equations that admit poles as their only movable singularities.[5] (引用終り) 以上 http://rio2016.5ch.net/test/read.cgi/math/1759924222/653

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