18/09/24 13:12:30.15 b5cCvILu.net
>>60
つづき
(Smm) Suppose that A and B are positive real numbers, which are defined so as
to satisfy the relation
?2B = ?A
(which corresponds to the Θ-link). One then proves a theorem
?2B <= ?2A + 1
(which corresponds to the multiradial representation of [IUTchIII],
Theorem 3.11). This theorem, together with the above defining relation,
implies a bound on A
?A <= ?2A + 1, i.e., A <= 1
(which corresponds to [IUTchIII], Corollary 3.12). From the point of view
of this (very rough!) summary of IUTch, the misunderstandings of SS
amount to the assertion that
the theory remains essentially unaffected even if one takes A = B,
which implies (in light of the above defining relation) that A = B = 0,
in contradiction to the initial assumption that A and B are positive real
numbers. In fact, however, the essential content (i.e., main results) of
IUTch fail(s) to hold under the assumption “A = B”; moreover, the
“contradiction” A = B = 0 is nothing more than a superficial consequence
of the extraneous assumption “A = B” and, in particular, does not imply
the existence of any flaws whatsoever in IUTch. (We refer to (SSIdEx),
(ModEll), (HstMod) below for a “slightly less rough” explanation of the
essential logical structure of an issue that is closely related to the extraneous
assumption “A = B” in terms of
・ complex structures on real vector spaces
or, alternatively (and essentially equivalently), in terms of the well-known
classical theory of
・ moduli of complex elliptic curves.
Additional comparisons with well-known classical topics such as
・ the invariance of heights of elliptic curves over number fields
with respect to isogeny,
・ Grothendieck’s definition of the notion of a connection, and
・ the differential geometry surrounding SL2(R)
may be found in §16.)
つづく