[過去ﾛｸﾞ] IUTを読むための用語集資料集スレ (1002ﾚｽ)

IUTを読むための用語集資料集スレ http://rio2016.5ch.net/test/read.cgi/math/1592654877/

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164: 現代数学の系譜 雑談 ◆yH25M02vWFhP [] 2020/07/17(金) 07:44:20.30 ID:kAj+yiGd >>93 『ABC予想入門』（黒川、小山 PHPサイエンス・ワールド新書 2013） P202には 楕円曲線 y^2 = x(x-a)(x+b) の導手が N = rad(abc) . (radは、根基) となると、 書いてあるね その証明が http://www.numdam.org/article/AST_1990__183__19_0.pdf D. W. MASSER Note on a conjecture of Szpiro Asterisque, tome 183 (1990), p. 19-23 にあるね（たぶんｗ） なるほどね http://rio2016.5ch.net/test/read.cgi/math/1592654877/164

166: 現代数学の系譜 雑談 ◆yH25M02vWFhP [] 2020/07/17(金) 17:54:06.93 ID:02nx2tCZ >>164 追加 https://arxiv.org/pdf/1705.09251.pdf 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 to obtain several unconditional results for these conjectures. A main difficulty in the theory is the lack of q-expansions, which we overcome by making essential use of suitable integral models and CM points. Our proofs require a number of tools from Arakelov geometry, analytic number theory, Galois representations, complex-analytic estimates on Shimura curves, automorphic forms, known cases of the Colmez conjecture, and results on generalized Fermat equations. 1.1. The problems. Let us briefly state the motivating problems; we take this opportunity to introduce some basic notation. Precise details will be recalled in Section 3. For an elliptic curve E over Q we write ΔE for the absolute value of its minimal discriminant and NE for its conductor. In the early eighties, Szpiro formulated the following conjecture: Conjecture 1.1 (Szpiro’s conjecture; cf. [91]). There is a constant κ > 0 such that for all elliptic curves E over Q we have ΔE < NκE. The radical rad(n) of a positive integer n is defined as the product of the primes dividing n without repetition. Let’s recall here a simple version of the abc conjecture of Masser and Oesterl´e. Conjecture 1.2 (abc conjecture). There is a constant κ > 0 such that for all coprime positive integers a, b, c with a + b = c we have abc < rad(abc)κ. Both conjectures are open. There are stronger versions in the literature (cf. [76]), but we keep these simpler formulations for the sake of exposition. つづく http://rio2016.5ch.net/test/read.cgi/math/1592654877/166

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