Szpiro's conjecture

Modified Szpiro conjecture

Field	Number theory
Conjectured by	Lucien Szpiro
Conjectured in	1981
Equivalent to	abc conjecture
Consequences	Beal conjecture Faltings's theorem Fermat's Last Theorem Fermat–Catalan conjecture Roth's theorem Tijdeman's theorem

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(ε) such that for any elliptic curve E defined over Q with minimal discriminant Δ and conductor f, we have

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

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), we have

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

Claimed proofs

Main article: Abc conjecture § 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".

References

• Lang, S. (1997), Survey of Diophantine geometry, Berlin: Springer-Verlag, p. 51, ISBN 3-540-61223-8, Zbl 0869.11051
• Szpiro, L. (1981), "Seminaire sur les pinceaux des courbes de genre au moins deux", Astérisque, 86 (3): 44–78, Zbl 0463.00009
• Szpiro, L. (1987), "Présentation de la théorie d'Arakelov", Contemp. Math., 67: 279–293, doi:10.1090/conm/067/902599, Zbl 0634.14012