Szpiro's conjecture

2020 Mathematics Subject Classification: Primary: 11Gxx [MSN][ZBL]

A conjectural 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 $\epsilon > 0$, there exists a constant $C(\epsilon)$ such that for any elliptic curve $E$ defined over $\mathbb{Q}$ with minimal discriminant $\Delta$ and conductor $f$, we have $$ \vert\Delta\vert \leq C(\varepsilon ) \cdot f^{6+\varepsilon } \ . $$ The modified Szpiro conjecture states that: given $\epsilon > 0$, there exists a constant $C(\epsilon)$ such that for any elliptic curve $E$ defined over $\mathbb{Q}$ with invariants $c_4,c_6$ and conductor $f$, we have $$ \max\{\vert c_4\vert^3,\vert c_6\vert^2\} \leq C(\varepsilon )\cdot f^{6+\varepsilon } \ . $$

References

• Lang, Serge; Survey of Diophantine geometry (1997), p. 51, Springer-Verlag, Zbl 0869.11051 ISBN 3-540-61223-8
• Szpiro, L.; Seminaire sur les pinceaux des courbes de genre au moins deux, Astérisque, 86 (1981), pp. 44-78, Zbl 0463.00009
• Szpiro, L.; Présentation de la théorie d'Arakelov, Contemp. Math., 67 (1987), pp. 279-293, Zbl 0634.14012