Szpiro's conjecture

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.

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

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

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, we have

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

References

• S. Lang (1997). Survey of Diophantine geometry. Springer-Verlag, 51. ISBN 3-540-61223-8. 
• L. Szpiro (1981). "Seminaire sur les pinceaux des courbes de genre au moins deux". Astérisque 86 (3): 44-78.

• L. Szpiro (1987). "Présentation de la théorie d'Arakelov". Contemp. Math. 67: 279-293.

External links

• Szpiro and ABC, notes by William Stein