Inter-universal geometry と ABC予想 (応援スレ) 49

[過去ﾛｸﾞ] Inter-universal geometry と ABC予想 (応援スレ) 49 (1002ﾚｽ)
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555: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/10/18(日) 12:55:19.30 ID:ZLSkSSTT

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Ｐ147
[cf. the discussion preceding [Pano], Theorem 4.1].
(ii) Explicit examples of connections to classical theories: Next, we review
various explicit examples of connections between inter-universal Teichm¨uller theory, as
exposed thus far in the present paper, and various classical theories:
(1cls) Recall from the discussion of §2.10 that the notion of a “universe”, as well as
the use of multiple universes within the discussion of a single set-up in arithmetic
geometry, already occurs in the mathematics of the 1960’s, i.e., in the mathematics
of Galois categories and ´etale topoi associated to schemes [cf. [SGA1], [SGA4]].

(2cls) One important aspect of the appearance of universes in the theory of Galois
categories is the inner automorphism indeterminacies that occur when one
relates Galois categories associated to distinct schemes via a morphism between such
schemes [cf. [SGA1], Expos´e V, §5, §6, §7]. These indeterminacies may be regarded
as distant ancestors, or prototypes, of the more drastic indeterminacies ? cf.,
e.g., the indeterminacies (Ind1), (Ind2), (Ind3) discussed in §3.7, (i) ? that occur
in inter-universal Teichm¨uller theory.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1600350445/555

556: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/10/18(日) 12:55:37.72 ID:ZLSkSSTT

>>555
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Ｐ163
Phenomena that support this point of view of an “atavistic model of mathematical
development” may be seen in many of the examples discussed in §4.1, §4.2, and §4.3
such as the following:
(1atv) The very elementary construction of Belyi maps in the early 1980’s, or indeed
noncritical Belyi maps in [NCBelyi], could easily have been discovered in the
late nineteenth century [cf. §4.3, (i), (5flt); §4.3, (iii), (5PB)].
(2atv) The application of Belyi maps to Belyi cuspidalization [cf. [AbsTopII], §3]
could easily have been discovered in the mid-1990’s [cf. also (1atv)].
(3atv) The application of noncritical Belyi maps to height estimates in [GenEll], §2,
could easily have been discovered in the mid-1980’s [cf. also (1atv)].
(4atv) The Galois-theoretic interpretation of the Gaussian integral or Jacobi’s
identity furnished by inter-universal Teichm¨uller theory [cf. the discussion of §3.8;
the discussion at the end of §3.9, (iii); the discussion of the final portion of §4.1, (i)]
could easily have been discovered much earlier than in the series of papers [IUTchI],
[IUTchII], [IUTchIII], [IUTchIV].
(5atv) The interpretation of changes of universe in the context of non-ring-theoretic
“arithmetic changes of coordinates” as in the discussion of §2.10 is entirely elementary and could easily have been discovered in the 1960’s [cf. §4.1, (ii), (1cls), (2cls)].
(引用終り)
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http://rio2016.5ch.net/test/read.cgi/math/1600350445/556

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