Szpiro's conjecture

Modified Szpiro conjecture

Field	Number theory
Conjectured by	Lucien Szpiro
Conjectured in	1981
Implied by	abc conjecture
Consequences	abc conjecture Beal conjecture Faltings's theorem Fermat's Last Theorem Fermat–Catalan conjecture Thue–Siegel–Roth theorem Tijdeman's theorem

In number theory, Szpiro's conjecture concerns a relationship between the conductor and the discriminant of an elliptic curve. In a general form, it is equivalent to the well-known abc conjecture. It is named for Lucien Szpiro who formulated it in the 1980s.

Statement

The conjecture states that: given ε > 0, there exists a constant C(ε) such that for any elliptic curve E defined over Q with minimal discriminant Δ and conductor f, we have

$\vert \Delta \vert \leq C(\varepsilon )\cdot f^{6+\varepsilon }.$

Modified Szpiro conjecture

The modified Szpiro conjecture states that: given ε > 0, there exists a constant C(ε) such that for any elliptic curve E defined over Q with invariants c₄, c₆ and conductor f (using notation from Tate's algorithm), we have

$\max\{\vert c_{4}\vert ^{3},\vert c_{6}\vert ^{2}\}\leq C(\varepsilon )\cdot f^{6+\varepsilon }.$

References

• Lang, S. (1997), Survey of Diophantine geometry, Berlin: Springer-Verlag, p. 51, ISBN 3-540-61223-8, Zbl 0869.11051
• Szpiro, L. (1981), "Seminaire sur les pinceaux des courbes de genre au moins deux", Astérisque, 86 (3): 44–78, Zbl 0463.00009
• Szpiro, L. (1987), "Présentation de la théorie d'Arakelov", Contemp. Math., 67: 279–293, doi:10.1090/conm/067/902599, Zbl 0634.14012