[過去ﾛｸﾞ] 現代数学の系譜 工学物理雑談 古典ガロア理論も読む79 (1002ﾚｽ)
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930(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [] 2020/01/03(金) 19:55:42.47 ID:ivt0JCXh(24/37) AAS
>>929

つづき
２）
Corollary 2.3. (Diophantine Inequalities) と、Theorem A. (Diophantine Inequalities) は、ほぼ同じか

P54
Corollary 2.3. (Diophantine Inequalities)

P55
Remark 2.3.1. We take this opportunity to correct some unfortunate misprints
in [GenEll].

P63
Remark 2.3.4. Various aspects of the theory of the present series of papers
are substantially reminiscent of the theory surrounding Bogomolov’s proof of
the geometric version of the Szpiro Conjecture, as discussed in [ABKP], [Zh].
Put another way, these aspects of the theory of the present series of papers may
be thought of as arithmetic analogues of the geometric theory surrounding Bogomolov’s proof. Alternatively, Bogomolov’s proof may be thought of as a sort of
useful elementary guide, or blueprint [perhaps even a sort of Rosetta stone!],
for understanding substantial portions of the theory of the present series of papers.
The author would like to express his gratitude to Ivan Fesenko for bringing to his
attention, via numerous discussions in person, e-mails, and skype conversations
between December 2014 and January 2015, the possibility of the existence of such
fascinating connections between Bogomolov’s proof and the theory of the present
series of papers. We discuss these analogies in more detail in [BogIUT].

つづく

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931(1): 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE [] 2020/01/03(金) 19:57:31.73 ID:ivt0JCXh(25/37) AAS
>>930
つづき
３）
(PB1) の、PはParshin、BはBogomolovか

P64
In the following
discussion, we shall refer to this geometry as the Schwarz-theoretic geometry of
D. Perhaps the most fundamental difference between the proofs of Parshin and
Bogomolov lies in the fact that
(PB1) Whereas Parshin’s proof revolves around estimates of displacements arising from actions of elements of the fundamental group on a certain two dimensional complete [Kobayashi] hyperbolic complex manifold by means
of the holomorphic geometry of the Kobayashi distance, i.e., in effect,
the Schwarz-theoretic geometry of D, Bogomolov’s proof [cf. the review of Bogomolov’s proof given in [BogIUT]] revolves around estimates of
displacements arising from actions of elements of the fundamental group
on a one-dimensional real analytic manifold [i.e., a universal covering of
a copy of the unit circle S1] by means of the real analytic symplectic
geometry of the upper half-plane.
Here, it is already interesting to note that this fundamental gap, in the case of
results over complex function fields, between the holomorphic geometry applied in
Parshin’s proof of the Mordell Conjecture and the real analytic symplectic geometry
applied in Bogomolov’s proof of the Szpiro Conjecture is highly reminiscent of the
fundamental gap discussed in Remark 2.3.3, (iii), in the case of results over number
fields, between the arithmetically holomorphic nature of the proof of the Mordell
Conjecture given in [Falt] and the “arithmetically quasi-conformal” nature of
the proof of the Szpiro Conjecture [cf. Corollary 2.3] via inter-universal Teichm¨uller
theory given in the present series of papers. That is to say,

つづく

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