>>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)