現代数学の系譜工学物理雑談古典ガロア理論も読む79

[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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751: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/12/29(日) 10:01:47.32 ID:uR3g5aDb

>>750

（参考）
https://en.wikipedia.org/wiki/Frey_curve
Frey curve
(抜粋)
Frey curve or Frey?Hellegouarch curve is the elliptic curve
y^2=x(x-a^l)(x+b^l)
associated with a (hypothetical) solution of Fermat's equation
a^l+b^l=c^l.
The curve is named after Gerhard Frey.

History
Yves Hellegouarch (1975) came up with the idea of associating solutions  (a,b,c)}(a,b,c) of Fermat's equation with a completely different mathematical object: an elliptic curve. If ? is an odd prime and a, b, and c are positive integers such that
a^l+b^l=c^l,
then a corresponding Frey curve is an algebraic curve given by the equation
y^2=x(x-a^l)(x+b^l)
or, equivalently
y^2=x(x-a^l)(x-c^l).
This is a nonsingular algebraic curve of genus one defined over Q, and its projective completion is an elliptic curve over Q.

(Gerhard Frey 1982) called attention to the unusual properties of the same curve as Hellegouarch, which became called a Frey curve.
This provided a bridge between Fermat and Taniyama by showing that a counterexample to Fermat's Last Theorem would create such a curve that would not be modular.
The conjecture attracted considerable interest when Frey (1986) suggested that the Taniyama?Shimura?Weil conjecture implies Fermat's Last Theorem. However, his argument was not complete.
In 1985, Jean-Pierre Serre proposed that a Frey curve could not be modular and provided a partial proof of this. This showed that a proof of the semistable case of the Taniyama-Shimura conjecture would imply Fermat's Last Theorem. Serre did not provide a complete proof and what was missing became known as the epsilon conjecture or ε-conjecture.
In the summer of 1986, Ribet (1990) proved the epsilon conjecture, thereby proving that the Taniyama?Shimura?Weil conjecture implies Fermat's Last Theorem.

http://rio2016.5ch.net/test/read.cgi/math/1573769803/751

752: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/12/29(日) 10:28:53.79 ID:uR3g5aDb

>>751

（参考：これ分り易いかも）
https://inference-review.com/article/fukugen
Fukugen
Ivan Fesenko
Published on September 28, 2016 in Volume 2, Issue 3.
On Shinichi Mochizuki’s Inter-universal Teichmuller Theory
(抜粋)
THE SUNLIGHT is strong in Kyoto, even in winter. In December of 2014, I visited Shinichi Mochizuki at the Research Institute for Mathematical Sciences to discuss his inter-universal Teichmuller theory (IUT).1 A distinguished mathematician and a leading figure in anabelian geometry,
Mochizuki first made his papers about IUT available at the end of August, 2012. Their study has proved challenging.

A term that is frequently used in mathematical discussions about anabelian geometry and IUT is fukugen, which may be translated as restoration or as reconstruction, and which, like so many words in a foreign language, cannot be truly translated.

It must be used without translation. But isn’t this true of mathematics itself?

IUT contributes to a new view of the numbers. This may sound as if Mochizuki had announced, rather than executed, a program in pure mathematics. But IUT yields proofs of several outstanding problems in number theory: the strong Szpiro conjecture for elliptic curves,
Vojta’s conjecture for hyperbolic curves, and the Frey conjecture for elliptic curves.

And it settles the famous Oesterle?Masser or abc conjecture.2

The abc conjecture is easy to state and difficult to prove. Prime numbers are defined in terms of multiplication in the ring of integers.

http://rio2016.5ch.net/test/read.cgi/math/1573769803/752

760: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/12/29(日) 18:15:21.09 ID:uR3g5aDb

>>751
＜Frey curve＞
https://en.wikipedia.org/wiki/Abc_conjecture#cite_ref-1
abc conjecture
(抜粋)
The precise statement is given below. The abc conjecture originated as the outcome of attempts by Oesterle and Masser to understand the Szpiro conjecture about elliptic curves.[1]

Citations
[1]
https://www.maths.nottingham.ac.uk/plp/pmzibf/notesoniut.pdf
Fesenko, Ivan (2015), "Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki" (PDF), European Journal of Mathematics, 1 (3): 405?440, doi:10.1007/s40879-015-0066-0.
(抜粋)
P4
1.3. Conjectural inequalities for the same property.
(a) the effective Mordell conjecture ? a conjectural extension of the Faltings?Mordell theorem which involves an effective bound on the height of rational points of the curve C over the number field K in the Faltings theorem in terms of data associated to C and K,
(b) the Szpiro conjecture, see below,
(c) the Masser?Oesterle conjecture, a.k.a. the abc conjecture (whose statement over Q is well known^6 , and which has an extension to arbitrary algebraic number fields, see Conj. 14.4.12 of [6]),
(d) the Frey conjecture, see Conj. F.3.2(b) of [15],
(f) arithmetic Bogomolov?Miyaoka?Yau conjectures (there are several versions).

The Szpiro conjecture was stated several years before^7 the work of Faltings, who learned much about the subject related to his proof from Szpiro.
Using the Frey curve^8, it is not difficult to show that (c) and (d) are equivalent and that they imply (b), see e.g. see sect. F3 of [15] and references therein.
Using Belyi maps as in 1.1, one can show the equivalence of (c) and (a).
For the equivalence of (c) and (e) see e.g. Th. 14.4.16 of [6]
and [47]. For implications (e) ⇒ (f) see [48].

Footnote
^8 y^2 = x(x+a)(x?b) where a,b,a+b are non-zero coprime integers

http://rio2016.5ch.net/test/read.cgi/math/1573769803/760

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