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538: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/10/17(土) 16:31:00.47 ID:02Kfs2KS

下記、Goldfeld, Modular forms, elliptic curves, and the ABC-conjecture が、なかなか良いね

https://ja.wikipedia.org/wiki/%E3%82%B9%E3%83%94%E3%83%AD%E4%BA%88%E6%83%B3
スピロ予想

脚注
3^ D. Goldfeld, Modular forms, elliptic curves, and the ABC-conjecture.
http://www.math.columbia.edu/~goldfeld/
DORIAN GOLDFELD
http://www.math.columbia.edu/~goldfeld/Papers.html
Selected Publications of Dorian Goldfeld
http://www.math.columbia.edu/~goldfeld/ABC-Conjecture.pdf
Modular Forms, Elliptic Curves, and the ABC Conjecture,  (2003) pdf

§1. The ABC-Conjecture.
The ABC-conjecture was first formulated by David Masser and Joseph Osterl´e (see
[Ost]) in 1985. Curiously, although this conjecture could have been formulated in the
last century, its discovery was based on modern research in the theory of function fields
and elliptic curves, which suggests that it is a statement about ramification in arithmetic
algebraic geometry. The ABC-conjecture seems connected with many diverse and well
known problems in number theory and always seems to lie on the boundary of what is
known and what is unknown. We hope to elucidate the beautiful connections between
elliptic curves, modular forms and the ABC-conjecture.
Conjecture (ABC). Let A, B, C be non-zero, pairwise relatively prime, rational integers
satisfying A + B + C = 0. Define
N = Πp|ABC p
to be the squarefree part of ABC. Then for every ε > 0, there exists κ(ε) > 0 such that
max(|A|, |B|, |C|) < κ(ε)N1+ε.
A weaker version of the ABC-conjecture (with the same notation as above) may be given
as follows.
Conjecture (ABC) (weak). For every ε > 0, there exists κ(ε) > 0 such that
|ABC| 1/3 < κ(ε)N1+ε.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1592654877/538

539: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/10/17(土) 16:31:24.91 ID:02Kfs2KS

>>538
つづき

P7
§4. Conjectures which are equivalent to ABC.

Conjecture. (Szpiro, 1981) Let E be an elliptic curve over Q which is a global minimal
model with discriminant Δ and conductor N. Then for every ε > 0, there exists κ(ε) > 0
such that
Δ < κ(ε)N6+ε. We show that Szpiro’s conjecture above is equivalent to the weak ABC-conjecture.
Let
A, B, C be coprime integers satisfying A + B + C = 0 and ABC 6= 0. Set N = Πp|ABCp.
Consider the Frey-Hellegouarch curve
EA,B : y2 = x(x - A)(x + B). A minimal model for EA,B has discriminant (ABC)2・ 2-s and conductor N ・ 2-t for
certain absolutely bounded integers s, t, (see Frey [F1]). Plugging this data into Szpiro’s
conjecture immediately shows the equivalence.

[F1] FREY, G., Links between stable elliptic curves and certain diophantine equations,
Annales Universiatis Saraviensis, Vol 1, No. 1 (1986), 1-39.
[F2] FREY, G., Links between elliptic curves and solutions of A-B=C, Journal of the
Indian Math. Soc. 51 (1987), 117-145.
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1592654877/539

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