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593: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/10/25(日) 09:23:08.65 ID:eIdDsFH8

>>592
つづき
In particular, when one computes the height of a rational point of the projective line
minus three points as a suitable weighted sum of the valuations of the q-parameters of
the corresponding elliptic curve, one may ignore, up to bounded discrepancies, contributions to the height that arise, say, from the archimedean valuations or from the
nonarchimedean valuations that lie over some “exceptional” prime number such as 2.

§ 2.2. Arithmetic degrees as global integrals

§ 2.7. The apparatus and terminology of mono-anabelian transport
Example 2.6.1 is exceptionally rich in structural similarities to inter-universal
Teichm¨uller theory, which we proceed to explain in detail as follows. One way to understand these structural similarities is by considering the quite substantial portion of
terminology of inter-universal Teichm¨uller theory that was, in essence, inspired by
Example 2.6.1:
(i) Links between “mutually alien” copies of scheme theory: One central
aspect of inter-universal Teichm¨uller theory is the study of certain “walls”, or “filters”
− which are often referred to as “links” − that separate two “mutually alien”
copies of conventional scheme theory [cf. the discussions of [IUTchII], Remark
3.6.2; [IUTchIV], Remark 3.6.1]. The main example of such a link in inter-universal
Teichm¨uller theory is constituted by [various versions of] the Θ-link. The log-link also
plays an important role in inter-universal Teichm¨uller theory. The main motivating
example for these links which play a central role in inter-universal Teichm¨uller theory
is the Frobenius morphism ΦηX of Example 2.6.1. From the point of view of the
discussion of §1.4, §1.5, §2.2, §2.3, §2.4, and §2.5, such a link corresponds to a change of coordinates.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1592654877/593

594: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/10/25(日) 09:23:32.03 ID:eIdDsFH8

>>593
つづき

§ 2.10. Inter-universality: changes of universe as changes of coordinates
One fundamental aspect of the links [cf. the discussion of §2.7, (i)] − namely, the
Θ-link and log-link − that occur in inter-universal Teichm¨uller theory is their incompatibility with the ring structures of the rings and schemes that appear in their
domains and codomains. In particular, when one considers the result of transporting
an ´etale-like structure such as a Galois group [or ´etale fundamental group] across such
a link [cf. the discussion of §2.7, (iii)], one must abandon the interpretation of such
a Galois group as a group of automorphisms of some ring [or field] structure [cf.
[AbsTopIII], Remark 3.7.7, (i); [IUTchIV], Remarks 3.6.2, 3.6.3], i.e., one must regard
such a Galois group as an abstract topological group that is not equipped with any
of the “labelling structures” that arise from the relationship between the Galois group
and various scheme-theoretic objects. It is precisely this state of affairs that results in
the quite central role played in inter-universal Teichm¨uller theory by results in
[mono-]anabelian geometry, i.e., by results concerned with reconstructing
various scheme-theoretic structures from an abstract topological group that “just
happens” to arise from scheme theory as a Galois group/´etale fundamental group.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1592654877/594

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