Szpiro's conjecture

For every \(\varepsilon > 0\) there are finitely many elliptic curves over \(\mathbb{Q}\) with conductor \(N\) and minimal discriminant \(\Delta\) violating \(|\Delta| \le C(\varepsilon) N^{6+\varepsilon}\). Equivalently, the abc conjecture up to constants.

Szpiro posed it in the early 1980s in the language of elliptic curves and reduction; Masser–Oesterlé's abc (1985) turned out to be equivalent up to constants. Mochizuki's inter-universal Teichmüller theory (2012–2021) claims a proof, but the claim remains disputed — Scholze–Stix identified what they see as an unbridgeable gap, and the community has not accepted the argument. The conjecture therefore sits in the rare state of simultaneously open and claimed-proved, with the dispute itself part of the record.