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166(4): 現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/07/17(金)17:54 ID:02nx2tCZ(1/4) AAS
>>164 追加

外部ﾘﾝｸ[pdf]:arxiv.org
SHIMURA CURVES AND THE ABC CONJECTURE
HECTOR PASTEN Date: July 6, 2018.
(抜粋)
Abstract. We develop a general framework to study Szpiro’s conjecture and the abc conjecture by
means of Shimura curves and their maps to elliptic curves, introducing new techniques that allow us
省19

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167(4): 現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/07/17(金)17:54 ID:02nx2tCZ(2/4) AAS
>>166

つづき

A classical construction of Frey [36] shows that Szpiro’s conjecture implies the abc conjecture:
To a triple of coprime positive integers a, b, c with a + b = c one associates the Frey-Hellegouarch
elliptic curve Ea,b,c given by the affine equation y^2 = x(x ? a)(x + b).
Then ΔE and NE are equal to (abc) ^2 and rad(abc) respectively,
up to a bounded power of 2 (cf. Section 3 for details and references).
省18

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168(1): 現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/07/17(金)17:58 ID:02nx2tCZ(3/4) AAS
>>167

訂正
elliptic curve Ea,b,c given by the affine equation y^2 = x(x ? a)(x + b).
　↓
elliptic curve Ea,b,c given by the affine equation y^2 = x(x - a)(x + b).

補足
Given a triple a, b, c of coprime positive integers with a + b = c, the Frey-Hellegouarch elliptic
省8

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169(1): 現代数学の系譜 雑談 ◆yH25M02vWFhP 2020/07/17(金)18:00 ID:02nx2tCZ(4/4) AAS
>>166
補足
SHIMURA CURVES AND THE ABC CONJECTURE
HECTOR PASTEN Date: July 6, 2018.
(抜粋)
Abstract. We develop a general framework to study Szpiro’s conjecture and the abc conjecture by
means of Shimura curves and their maps to elliptic curves, introducing new techniques that allow us
省6

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