Inter-universal geometry と ABC予想 49

[過去ﾛｸﾞ] Inter-universal geometry と ABC予想 49 (1002ﾚｽ)
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928: １３２人目の素数さん [sage] 2020/04/14(火) 22:23:40.83 ID:mIlcNUlX

>>924

これか
ここは、SSの文書が公開された当時にも話題になったかも（＾＾

(C1) : Remark 5, “For fixed ... h(P) ? b.”: I can only say that it is a very challenging task to document the depth of my astonishment when I first read this Remark!
This Remark may be described as a breath-takingly  (melo ?) dramatic self-declaration, on the part of SS, of their profound ignorance of the elementary theory of heights, at the advanced undergraduate/beginning graduate level.
Indeed, the finiteness statement at the beginning of the paragraph follows immediately, by considering the j-invariant (say, multiplied by a suitable positive integer N,
which depends only on d and b) of the elliptic curve under consideration, from the finiteness of the set of complex numbers that satisfy a monic polynomial equation of degree d with coefficients ∈ Z of absolute value ? C, for some fixed real number C that depends only on d and b.
To repeat, this sort of argument lies well within the framework of advanced undergraduate/beginning graduate-level mathematics.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1586650355/928

929: １３２人目の素数さん [sage] 2020/04/14(火) 22:24:34.93 ID:mIlcNUlX

>>928

つづき

It is entirely inconceivable that any researcher with substantial experience working with heights of rational points would attempt to prove this sort of finiteness statement by invoking such a nontrivial result as Faltings’ theorem.
Anyone familiar with the proof of Faltings’ theorem will also recognize immediately that the proof of Faltings’ theorem ultimately reduces to the elementary observation reviewed above, i.e.,
that the finiteness of the set of rational points (of, say, a proper variety) of bounded height over number fields of bounded degree follows immediately from elementary considerations, namely, from the finiteness of the set of solutions of monic polynomial equations of bounded
degree with bounded coefficients ∈ Z.
(Another problem with the argument in Remark 5 is that it is never mentioned why the discriminant of k/Q is bounded.
Such a bound is necessary in order to conclude that the abelian variety A has good reduction outside a fixed finite set of primes that depends only on d and b.)
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1586650355/929

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