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166: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/07/17(金) 17:54:06.93 ID:02nx2tCZ

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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.

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http://rio2016.5ch.net/test/read.cgi/math/1592654877/166

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