Szpiro's conjecture
| Field | Number theory |
|---|---|
| Conjectured by | Lucien Szpiro |
| Conjectured in | 1981 |
| Equivalent to | abc conjecture |
| Consequences |
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, we have
- [math]\displaystyle{ \vert\Delta\vert \leq C(\varepsilon ) \cdot f^{6+\varepsilon }. }[/math]
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
- [math]\displaystyle{ \max\{\vert c_4\vert^3,\vert c_6\vert^2\} \leq C(\varepsilon )\cdot f^{6+\varepsilon }. }[/math]
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.[7]
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).[8] However, the papers have not been accepted by the mathematical community as providing a proof of the conjecture,[9][10][11] 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".<ref> {{ cite web | url=http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-08.pd | title=Why abc is still a conjecture
See also
References
- ↑ Goldfeld, Dorian (1996). "Beyond the last theorem". Math Horizons 4 (September): 26–34. doi:10.1080/10724117.1996.11974985.
- ↑ Bombieri, Enrico (1994). "Roth's theorem and the abc-conjecture". Preprint (ETH Zürich).
- ↑ Elkies, N. D. (1991). "ABC implies Mordell". International Mathematics Research Notices 1991 (7): 99–109. doi:10.1155/S1073792891000144.
- ↑ Pomerance, Carl (2008). "Computational Number Theory". The Princeton Companion to Mathematics. Princeton University Press. pp. 361–362.
- ↑ Dąbrowski, Andrzej (1996). "On the diophantine equation x! + A = y2". Nieuw Archief voor Wiskunde, IV. 14: 321–324.
- ↑ Fesenko, Ivan (2015), "Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki", European Journal of Mathematics 1 (3): 405–440, doi:10.1007/s40879-015-0066-0, https://www.maths.nottingham.ac.uk/personal/ibf/notesoniut.pdf.
- ↑ Oesterlé, Joseph (1988), "Nouvelles approches du "théorème" de Fermat", Astérisque, Séminaire Bourbaki exp 694 (161): 165–186, ISSN 0303-1179, http://www.numdam.org/item?id=SB_1987-1988__30__165_0
- ↑ Ball, Peter (10 September 2012). "Proof claimed for deep connection between primes". Nature. doi:10.1038/nature.2012.11378. https://www.nature.com/news/proof-claimed-for-deep-connection-between-primes-1.11378. Retrieved 19 April 2020.
- ↑ Revell, Timothy (September 7, 2017). "Baffling ABC maths proof now has impenetrable 300-page 'summary'". New Scientist. https://www.newscientist.com/article/2146647-baffling-abc-maths-proof-now-has-impenetrable-300-page-summary/.
- ↑ Conrad, Brian (December 15, 2015). "Notes on the Oxford IUT workshop by Brian Conrad". https://mathbabe.org/2015/12/15/notes-on-the-oxford-iut-workshop-by-brian-conrad/.
- ↑ Castelvecchi, Davide (8 October 2015). "The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof". Nature 526 (7572): 178–181. doi:10.1038/526178a. PMID 26450038. Bibcode: 2015Natur.526..178C.
Bibliography
- Lang, S. (1997), Survey of Diophantine geometry, Berlin: Springer-Verlag, p. 51, ISBN 3-540-61223-8, https://books.google.com/books?id=b9RqCQAAQBAJ&pg=PA51
- Szpiro, L. (1981). "Propriétés numériques du faisceau dualisant rélatif". Seminaire sur les pinceaux des courbes de genre au moins deux. 86. 44–78. http://www.numdam.org/item/AST_1981__86__R1_0.pdf.
- Szpiro, L. (1987), "Présentation de la théorie d'Arakelov", Contemp. Math., Contemporary Mathematics 67: 279–293, doi:10.1090/conm/067/902599, ISBN 9780821850749
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