Szpiro's conjecture


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

Original statement

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

Modified Szpiro conjecture

The modified Szpiro conjecture states that: given ε > 0, there exists a constant C such that for any elliptic curve E defined over Q with invariants c4, c6 and conductor f, we have

Claimed proofs

In August 2012, Shinichi Mochizuki claimed a proof of Szpiro's conjecture by developing a new theory called inter-universal Teichmüller theory. However, the papers have not been accepted by the mathematical community as providing a proof of the conjecture, with Peter Scholze and Jakob Stix concluding in March 2018 that the gap was "so severe that … small modifications will not rescue the proof strategy".