>>527 追加 >宇宙際Teichmuller理論 >[7] The Mathematics of Mutually Alien Copies: from Gaussian Integrals to Inter-universal Teichmuller Theory. PDF NEW !! (2020-12-23) >https://www.kurims.kyoto-u.ac.jp/~motizuki/Alien%20Copies,%20Gaussians,%20and%20Inter-universal%20Teichmuller%20Theory.pdf
Mutually Alien Copies に関連しそうなところを、下記に引用すると 1）N ・ h “=～” h N be a fixed natural number > 1 2）qN “=～” q 3）“alien” is that of its original Latin root, i.e., a sense of abstract, tautological “otherness”. とか、そのまま読むと、望月ワールド全開で、NHKスペシャル見ているから「同じものを別と見て、かつ同一視する」でしたか、ああこのことかと思いました 普通に読むと、読めないでしょうね ” Gaussian integral に繋げないんだろう”と好意的に読むと、気持ちは分かりますがね（これ数学として成り立つ？ｗ） ここ、説明の一つの山でしょね
(引用開始) P3 Introduction Let N be a fixed natural number > 1. Then the issue of bounding a given nonnegative real number h ∈ R?0 may be understood as the issue of showing that N ・ h is roughly equal to h, i.e., N ・ h “=～” h [cf. §2.3, §2.4]. When h is the height of an elliptic curve over a number field, this issue may be understood as the issue of showing that the height of the [in fact, in most cases, fictional!] “elliptic curve” whose q-parameters are the N-th powers “qN ” of the q-parameters “q” of the given elliptic curve is roughly equal to the height of the given elliptic curve, i.e., that, at least from the point of view of [global] heights, qN “=～” q [cf. §2.3, §2.4].
In order to verify the approximate relation qN “=～” q, one begins by introducing two distinct - i.e., two “mutually alien” - copies of the conventional scheme theory surrounding the given initial Θ-data. Here, the intended sense of the descriptive “alien” is that of its original Latin root, i.e., a sense of abstract, tautological “otherness”.
These two mutually alien copies of conventional scheme theory are glued together - by considering relatively weak underlying structures of the respective conventional scheme theories such as multiplicative monoids and profinite groups - in such a way that the “qN ” in one copy of scheme theory is identified with the “q” in the other copy of scheme theory. This gluing is referred to as the Θ-link. Thus, the “qN ” on the left-hand side of the Θ-link is glued to the “q” on the right-hand side of the Θ-link, i.e., qNLHS “=” qRHS [cf. §3.3, (vii), for more details]. Here, “N” is in fact taken not to be a fixed natural number, but rather a sort of symmetrized average over the values j2, where j = 1,...,l*, and we write l* def = (l ? 1)/2. Thus, the left-hand side of the above display {qj2LHS}j bears a striking formal resemblance to the Gaussian distribution. One then verifies the desired approximate relation qN “=～” q by computing {qj2LHS}j - not in terms of qLHS [which is immediate from the definitions!], but rather - in terms of [the scheme theory surrounding] qRHS [which is a highly nontrivial matter!]. (引用終り) 以上
