Szpiro's conjecture

In number theory, Szpiro's conjecture relates to 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. Szpiro's conjecture and its equivalent forms have been described as "the most important unsolved problem in Diophantine analysis" by Dorian Goldfeld, in part to its large number of consequences in number theory including Roth's theorem, the Mordell conjecture, the Fermat–Catalan conjecture, and Brocard's problem.

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,

${\displaystyle \vert \Delta \vert \leq C(\varepsilon )\cdot f^{6+\varepsilon }.}$

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),

${\displaystyle \max\{\vert c_{4}\vert ^{3},\vert c_{6}\vert ^{2}\}\leq C(\varepsilon )\cdot f^{6+\varepsilon }.}$

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.

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.

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".

• Arakelov theory

• Lang, S. (1997), Survey of Diophantine geometry, Berlin: Springer-Verlag, p. 51, ISBN 3-540-61223-8, Zbl 0869.11051
• Szpiro, L. (1981). "Propriétés numériques du faisceau dualisant rélatif". Seminaire sur les pinceaux des courbes de genre au moins deux (PDF). Astérisque. Vol. 86. pp. 44–78. Zbl 0517.14006.
• Szpiro, L. (1987), "Présentation de la théorie d'Arakelov", Contemp. Math., Contemporary Mathematics, 67: 279–293, doi:10.1090/conm/067/902599, ISBN 9780821850749, Zbl 0634.14012