# Szpiro's conjecture

From Citizendium, the Citizens' Compendium

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

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 *c*_{4}, *c*_{6} and conductor *f*, we have