Szpiro's conjecture

Modified Szpiro conjecture

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 c₄, c₆ and conductor f (using notation from Tate's algorithm),

${\displaystyle max\{|c_4{|}^3,|c_6{|}^2\}\le C(\epsilon )\cdot f^{6+\epsilon }.}$

abc conjecture

The abc conjecture originated as the outcome of attempts by Joseph Oesterlé and David Masser to understand Szpiro's conjecture, and was then shown to be equivalent to the modified Szpiro's conjecture.

Consequences

Szpiro's conjecture and its modified form are known to imply several important mathematical results and conjectures, including Roth's theorem, Faltings's theorem, Fermat–Catalan conjecture, and a negative solution to the Erdős–Ulam problem.

Claimed proofs

In August 2012, Shinichi Mochizuki claimed a proof of Szpiro's conjecture by developing a new theory called inter-universal Teichmüller theory (IUTT). 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".