Szpiro's conjecture explained

Modified Szpiro conjecture
Field:	Number theory
Conjectured By:	Lucien Szpiro
Conjecture Date:	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 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,^[1] 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.^{[2] [3] [4] [5]}

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,

\vert\Delta\vert \leq C(\varepsilon ) ⋅ 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),

max \{\vert c ^{3 ,\vert} _{4 \vert} c ^{2 \}} _{6 \vert} \leq C(\varepsilon ) ⋅ f ^{6+\varepsilon} .

abc conjecture

The abc conjecture originated as the outcome of attempts by Joseph Oesterlé and David Masser to understand Szpiro's conjecture,^[6] 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,^[7] Faltings's theorem,^[8] Fermat–Catalan conjecture,^[9] 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).^[10] However, the papers have not been accepted by the mathematical community as providing a proof of the conjecture,^{[11] [12] [13]} 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".^{[14] [15] [16]}

See also

• Arakelov theory

Bibliography

• • Book: Szpiro, L. . Seminaire sur les pinceaux des courbes de genre au moins deux. Propriétés numériques du faisceau dualisant rélatif. Astérisque . 86 . 3 . 1981 . 0517.14006 . 44–78.

Notes and References

1. Goldfeld . Dorian . Dorian M. Goldfeld . 1996 . Beyond the last theorem . . 4 . September . 26–34 . 25678079 . 10.1080/10724117.1996.11974985 .
2. Enrico . Bombieri. Enrico Bombieri . Roth's theorem and the abc-conjecture . Preprint . 1994 . ETH Zürich .
3. Elkies . N. D. . Noam Elkies . ABC implies Mordell . International Mathematics Research Notices. 1991 . 1991 . 99–109 . 10.1155/S1073792891000144 . 7 . free .
4. Book: Pomerance, Carl . Carl Pomerance . Computational Number Theory . The Princeton Companion to Mathematics . . 2008 . 361–362 .
5. Andrzej . Dąbrowski . On the diophantine equation x! + A = y² . Nieuw Archief voor Wiskunde, IV. . 14 . 321–324 . 1996 . 0876.11015 .
6. .
7. Book: 10.1007/978-3-0348-0859-0_13. Lecture on the abc Conjecture and Some of Its Consequences. Mathematics in the 21st Century. Springer Proceedings in Mathematics & Statistics. 2015. Waldschmidt. Michel. 98. 211–230. 978-3-0348-0858-3. https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/abcLahoreProceedings.pdf.
8. Elkies . N. D. . Noam Elkies . ABC implies Mordell . International Mathematics Research Notices. 1991 . 1991 . 99–109 . 10.1155/S1073792891000144 . 7 . free.
9. Book: Pomerance, Carl . Carl Pomerance . Computational Number Theory . The Princeton Companion to Mathematics . Princeton University Press . 2008 . 361–362 .
10. Ball . Peter . 10 September 2012. Proof claimed for deep connection between primes . Nature . 10.1038/nature.2012.11378 . 19 April 2020. free .
11. New Scientist. Baffling ABC maths proof now has impenetrable 300-page 'summary'. Timothy. Revell. September 7, 2017.
12. Web site: Notes on the Oxford IUT workshop by Brian Conrad . Brian . Conrad . Brian Conrad. December 15, 2015 . March 18, 2018.
13. Castelvecchi . Davide . 8 October 2015 . The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof . Nature . 526 . 7572. 178–181 . 10.1038/526178a . 2015Natur.526..178C . 26450038. free .
14. Web site: Why abc is still a conjecture . Peter . Scholze . Peter Scholze . Jakob . Stix . Jakob Stix . https://web.archive.org/web/20200208075321/http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-08.pdf. February 8, 2020. dead. (updated version of their May report|)
15. Titans of Mathematics Clash Over Epic Proof of ABC Conjecture . . September 20, 2018 . Erica . Klarreich . Erica Klarreich .
16. Web site: March 2018 Discussions on IUTeich . October 2, 2018 . Web-page by Mochizuki describing discussions and linking consequent publications and supplementary material