Inter-universal geometry と ABC予想 (応援スレ) 52

[過去ﾛｸﾞ] Inter-universal geometry と ABC予想 (応援スレ) 52 (1002ﾚｽ)
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301: １３２人目の素数さん [] 2021/02/27(土) 12:38:32.69 ID:f+hU2HEr

>>277
南出論文
楕円曲線 “y2 = x(x-1)(x-λ)”
the Legendre form
“λ-line”
ここらが重要キーワードですね

http://www.kurims.kyoto-u.ac.jp/~motizuki/Explicit%20estimates%20in%20IUTeich.pdf
Explicit Estimates in Inter-universal Teichmuller Theory. PDF NEW!! (2020-11-30) いわゆる南出論文
P2
Introduction

Theorem.
7We shall regard X as the “λ-line” - i.e.,
we shall regard the standard coordinate on X as the “λ” in the Legendre
form “y2 = x(x-1)(x-λ)” of the Weierstrass equation defining an elliptic
curve -

P34
Corollary 5.2. (Construction of suitable µ6-initial Θ-data) Write X
for the projective line over Q; D ⊆ X for the divisor consisting of the
three points “0”, “1”, and “∞”; (Mell)Q for the moduli stack of elliptic
curves over Q. We shall regard X as the “λ-line” - i.e., we shall regard
the standard coordinate on X as the “λ” in the Legendre form “y2 =x(x - 1)(x - λ)” of the Weierstrass equation defining an elliptic curve -

http://rio2016.5ch.net/test/read.cgi/math/1613784152/301

302: １３２人目の素数さん [] 2021/02/27(土) 12:51:48.66 ID:f+hU2HEr

>>301
>the Legendre form
>ここらが重要キーワードですね

参考
https://en.wikipedia.org/wiki/Legendre_form
Legendre form

Contents
1 Definition
2 Numerical evaluation
3 References

The respective complete elliptic integrals are obtained by setting the amplitude, Φ, the upper limit of the integrals, to π/2.

The Legendre form of an elliptic curve is given by
y2 = x(x-1)(x-λ)

Numerical evaluation
The classic method of evaluation is by means of Landen's transformations. Descending Landen transformation decreases the modulus   k k towards zero, while increasing the amplitude Φ. Conversely, ascending transformation increases the modulus towards unity, while decreasing the amplitude. In either limit of k, zero or one, the integral is readily evaluated.

Most modern authors recommend evaluation in terms of the Carlson symmetric forms, for which there exist efficient, robust and relatively simple algorithms. This approach has been adopted by Boost C++ Libraries, GNU Scientific Library and Numerical Recipes.[3]

http://rio2016.5ch.net/test/read.cgi/math/1613784152/302

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