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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325: １３２人目の素数さん [] 2021/02/28(日) 09:31:59.41 ID:c9K39yvS

>>314
コメントありがとう
検索キーワードとして、重要ですね
例えば
Legendre form elliptic curve x(x-1)(x-λ) 検索ヒット 約 97 件 （0.59 秒）
つまり、x(x-1)(x-λ)について、いろんな先行する研究がある。つまり、良い性質があるってことだね

（参考）
https://en.wikipedia.org/wiki/Modular_lambda_function
Modular lambda function
The relation to the j-invariant is[6][7]
j(τ )= 256(1-λ (1-λ ))^3/{(λ (1-λ ))^2}= 256(1-λ +λ ^2)^3/{λ ^2(1-λ )^2} .
which is the j-invariant of the elliptic curve of Legendre form  y^2=x(x-1)(x-λ ) y^2=x(x-1)(x-λ )

https://www2.math.kyushu-u.ac.jp/~mkaneko/
https://www2.math.kyushu-u.ac.jp/~mkaneko/papers.html
金子昌信 論文
17. Supersingular j-invariants, hypergeometric series, and Atkin's orthogonal polynomials (with D. Zagier), AMS/IP Studies in Advanced Mathematics, vol. 7, 97--126, (1998). pdf
https://www2.math.kyushu-u.ac.jp/~mkaneko/papers/atkin.pdf
SUPERSINGULAR j-INVARIANTS, HYPERGEOMETRIC
SERIES, AND ATKIN’S ORTHOGONAL POLYNOMIALS
M. Kaneko and D. Zagier
§1. Introduction.
The polynomial describing supersingularity in terms of the λ-invariant of E (defined
by writing E over K¯ in Legendre form y
2 = x(x - 1)(x - λ)) has a well-known and
simple explicit expression, but a convenient expression for the polynomial expressing
the condition of supersingularity directly in terms of the j-invariant (i.e., in terms of
a Weierstrass model over K, without numbering the 2-torsion points over K¯ ) is less
easy to find. In this (partially expository) paper, we will describe several different
ways of constructing canonical polynomials in Q[j] whose reductions modulo p give ssp(j).

http://rio2016.5ch.net/test/read.cgi/math/1613784152/325

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