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.

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

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

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

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

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

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).[1] However, the papers have not been accepted by the mathematical community as providing a proof of the conjecture,[2][3][4] 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".[5][6][7]

References

1. Ball, Peter (10 September 2012). "Proof claimed for deep connection between primes". Nature. doi:10.1038/nature.2012.11378. Retrieved 19 April 2020.
2. Revell, Timothy (September 7, 2017). "Baffling ABC maths proof now has impenetrable 300-page 'summary'". New Scientist.
3. Conrad, Brian (December 15, 2015). "Notes on the Oxford IUT workshop by Brian Conrad". Retrieved March 18, 2018.
4. Castelvecchi, Davide (8 October 2015). "The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof". Nature. 526 (7572): 178–181. Bibcode:2015Natur.526..178C. doi:10.1038/526178a. PMID 26450038.
5. Scholze, Peter; Stix, Jakob. "Why abc is still a conjecture" (PDF). Retrieved September 23, 2018. (updated version of their May report)
6. Klarreich, Erica (September 20, 2018). "Titans of Mathematics Clash Over Epic Proof of ABC Conjecture". Quanta Magazine.
7. "March 2018 Discussions on IUTeich". Retrieved October 2, 2018. Web-page by Mochizuki describing discussions and linking consequent publications and supplementary material

• 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