Szpiro's conjecture
Conjecture in number theory
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 ,
|
Δ Δ -->
|
≤ ≤ -->
C
(
ε ε -->
)
⋅ ⋅ -->
f
6
+
ε ε -->
.
{\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 c 4 , c 6 and conductor f (using notation from Tate's algorithm ),
max
{
|
c
4
|
3
,
|
c
6
|
2
}
≤ ≤ -->
C
(
ε ε -->
)
⋅ ⋅ -->
f
6
+
ε ε -->
.
{\displaystyle \max\{\vert c_{4}\vert ^{3},\vert c_{6}\vert ^{2}\}\leq C(\varepsilon )\cdot 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.[ 7]
Consequences
Szpiro's conjecture and its modified form are known to imply several important mathematical results and conjectures, including Roth's theorem ,[ 8] Faltings's theorem ,[ 9] Fermat–Catalan conjecture ,[ 10] and a negative solution to the Erdős–Ulam problem .[ 11]
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).[ 12] However, the papers have not been accepted by the mathematical community as providing a proof of the conjecture,[ 13] [ 14] [ 15] 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".[ 16] [ 17] [ 18]
See also
References
^ Goldfeld, Dorian (1996). "Beyond the last theorem". Math Horizons . 4 (September): 26–34. doi :10.1080/10724117.1996.11974985 . JSTOR 25678079 .
^ 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 = y 2 ". Nieuw Archief voor Wiskunde, IV . 14 : 321–324. Zbl 0876.11015 .
^ Fesenko, Ivan (2015), "Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki" (PDF) , European Journal of Mathematics , 1 (3): 405–440, doi :10.1007/s40879-015-0066-0 .
^ Oesterlé, Joseph (1988), "Nouvelles approches du "théorème" de Fermat" , Astérisque , Séminaire Bourbaki exp 694 (161): 165–186, ISSN 0303-1179 , MR 0992208
^ Waldschmidt, Michel (2015). "Lecture on the abc Conjecture and Some of Its Consequences" (PDF) . Mathematics in the 21st Century . Springer Proceedings in Mathematics & Statistics. Vol. 98. pp. 211–230. doi :10.1007/978-3-0348-0859-0_13 . ISBN 978-3-0348-0858-3 .
^ 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.
^
Pasten, Hector (2017), "Definability of Frobenius orbits and a result on rational distance sets", Monatshefte für Mathematik , 182 (1): 99–126, doi :10.1007/s00605-016-0973-2 , MR 3592123 , S2CID 7805117
^ Ball, Peter (10 September 2012). "Proof claimed for deep connection between primes" . Nature . doi :10.1038/nature.2012.11378 . Retrieved 19 April 2020 .
^ Revell, Timothy (September 7, 2017). "Baffling ABC maths proof now has impenetrable 300-page 'summary' " . New Scientist .
^ Conrad, Brian (December 15, 2015). "Notes on the Oxford IUT workshop by Brian Conrad" . Retrieved March 18, 2018 .
^ 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 .
^
Scholze, Peter ; Stix, Jakob . "Why abc is still a conjecture" (PDF) . Archived from the original on February 8, 2020. (updated version of their May report |)
^ Klarreich, Erica (September 20, 2018). "Titans of Mathematics Clash Over Epic Proof of ABC Conjecture" . Quanta Magazine .
^ "March 2018 Discussions on IUTeich" . Retrieved October 2, 2018 . Web-page by Mochizuki describing discussions and linking consequent publications and supplementary material
Bibliography
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