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 &epsilon; &gt; 0, there exists a constant C(&epsilon;) such that for any elliptic curve E defined over 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; &gt; 0, there exists a constant C(&epsilon;) such that for any elliptic curve E defined over Q with invariants c4, c6 and conductor f, we have


 * $$ \max\{\vert c_4\vert^3,\vert c_6\vert^2\} \leq C(\varepsilon )\cdot f^{6+\varepsilon }. \, $$