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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.

つづく

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

167: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/07/17(金) 17:54:41.40 ID:02nx2tCZ

>>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).
Thus, Szpiro’s conjecture in the case of Frey-Hellegouarch elliptic curves implies the abc conjecture as stated above.

3. Review of the classical modular approach

Given a triple a, b, c of coprime positive integers with a + b = c, the Frey-Hellegouarch elliptic
curve Ea,b,c is defined by the affine equation
y^2 = x(x - a)(x + b).
One directly checks that Ea,b,c is semi-stable away from 2. Furthermore (cf. p.256-257 in [89]),
ΔEa,b,c = 2^s(abc)^2 and NEa,b,c = 2^trad(abc) for integers s, t with -8 <= s <= 4 and -1 <= t <= 7.
See [28] for a detailed analysis of the local invariants at p = 2 (possibly after twisting Ea,b,c by -1).
From here, it is clear that Conjecture 1.1 implies Conjecture 1.2 and that any partial result for
Conjecture 1.1 which applies to Frey-Hellegouarch elliptic curves yields a partial result for the abc
conjecture.

18. A modular approach to Szpiro’s conjecture over number fields

References
[29] L. Dieulefait, N. Freitas, Base change for elliptic curves over real quadratic fields. Comptes Rendus Mathematique
353.1 (2015): 1-4.

[89] J. Silverman, The arithmetic of elliptic curves. Second edition. Graduate Texts in Mathematics, 106. Springer,
Dordrecht, 2009. xx+513 pp. ISBN: 978-0-387-09493-9
(引用終り)

http://rio2016.5ch.net/test/read.cgi/math/1592654877/167

185: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/07/19(日) 17:46:31.28 ID:2Y0qBKwb

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”Shimura curves”
http://www.math.columbia.edu/~chaoli/
Chao Li's homepage
http://www.math.columbia.edu/~chaoli/docs/ShimuraCurves.html
Shimura curves

In the 60s, Shimura studied certain algebraic curves as analogues of classical modular curves in order to construct class fields of totally real number fields. These curves were later coined "Shimura curves" and vastly generalized by Deligne. We will take a tour of the rich geometry and arithmetic of Shimura curves. Along the way, we may encounter tessellations of disks, quaternion algebras, abelian surfaces, elliptic curves with CM, Hurwitz curves ... and the answer to life, the universe and everything.

[-] Contents
Review of Modular Curves
Shimura curves
Moduli interpretation and class fields
Hurwitz curves

Briefly speaking, Shimura curves are simply one-dimensional Shimura varieties. I have accomplished my trivial notion task because I have told you a trivial notion. But obviously it does not help much if you do not know what the term Shimura varieties means. It only takes 5 chapters in Milne's notes in order to define them ? not too bad ? but initially Shimura invented them really because they are natural analogues of classical modular curves.

https://math.dartmouth.edu/~jvoight/articles/shimura-clay-proceedings-071707.pdf
Shimura curve computations
John Voight 1991 Mathematics Subject Classification.
Abstract. We introduce Shimura curves first as Riemann surfaces and then
as moduli spaces for certain abelian varieties. We give concrete examples of
these curves and do some explicit computations with them.
1. Introduction: modular curves

つづく

http://rio2016.5ch.net/test/read.cgi/math/1592654877/185

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