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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
|
≤
C
(
ε
)
⋅
f
6
+
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
)
⋅
f
6
+
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