Szpiro's conjecture: Difference between revisions

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;

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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.

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

\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

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

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

^ 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). Retrieved September 23, 2018. (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

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

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[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

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 {{Main|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).<ref>{{cite journal |last1=Ball |first1=Peter |date= 10 September 2012|title=Proof claimed for deep connection between primes |url=https://www.nature.com/news/proof-claimed-for-deep-connection-between-primes-1.11378 |journal=Nature |volume= |issue= |pages= |doi=10.1038/nature.2012.11378 |access-date=19 April 2020|doi-access=free }}</ref> However, the papers have not been accepted by the mathematical community as providing a proof of the conjecture,<ref>{{cite magazine|magazine=[[New Scientist]]|title=Baffling ABC maths proof now has impenetrable 300-page 'summary'|url=https://www.newscientist.com/article/2146647-baffling-abc-maths-proof-now-has-impenetrable-300-page-summary/|first=Timothy|last=Revell|date=September 7, 2017}}</ref><ref>{{cite web | url=https://mathbabe.org/2015/12/15/notes-on-the-oxford-iut-workshop-by-brian-conrad/ |first = Brian |last=Conrad |authorlink=Brian Conrad| date=December 15, 2015 | title=Notes on the Oxford IUT workshop by Brian Conrad | access-date=March 18, 2018}}</ref><ref>{{cite journal |last1=Castelvecchi |first1=Davide |date=8 October 2015 |title=The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof |journal=Nature |volume=526 |issue= 7572|pages=178–181 |doi=10.1038/526178a |bibcode=2015Natur.526..178C |pmid=26450038|doi-access=free }}</ref> 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>
+In August 2012, [[Shinichi Mochizuki]] claimed a proof of Szpiro's conjecture by developing a new theory called [[inter-universal Teichmüller theory]] (IUTT).<ref>{{cite journal |last1=Ball |first1=Peter |date= 10 September 2012|title=Proof claimed for deep connection between primes |url=https://www.nature.com/news/proof-claimed-for-deep-connection-between-primes-1.11378 |journal=Nature |doi=10.1038/nature.2012.11378 |access-date=19 April 2020|doi-access=free }}</ref> However, the papers have not been accepted by the mathematical community as providing a proof of the conjecture,<ref>{{cite magazine|magazine=[[New Scientist]]|title=Baffling ABC maths proof now has impenetrable 300-page 'summary'|url=https://www.newscientist.com/article/2146647-baffling-abc-maths-proof-now-has-impenetrable-300-page-summary/|first=Timothy|last=Revell|date=September 7, 2017}}</ref><ref>{{cite web | url=https://mathbabe.org/2015/12/15/notes-on-the-oxford-iut-workshop-by-brian-conrad/ |first = Brian |last=Conrad |author-link=Brian Conrad| date=December 15, 2015 | title=Notes on the Oxford IUT workshop by Brian Conrad | access-date=March 18, 2018}}</ref><ref>{{cite journal |last1=Castelvecchi |first1=Davide |date=8 October 2015 |title=The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof |journal=Nature |volume=526 |issue= 7572|pages=178–181 |doi=10.1038/526178a |bibcode=2015Natur.526..178C |pmid=26450038|doi-access=free }}</ref> 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.pdf | title=Why abc is still a conjecture
- |first1= Peter |last1= Scholze |authorlink1= Peter Scholze
+ |first1= Peter |last1= Scholze |author-link1= Peter Scholze
- |first2= Jakob |last2= Stix    |authorlink2= Jakob Stix
+ |first2= Jakob |last2= Stix    |author-link2= Jakob Stix
- | access-date=September 23, 2018 }} (updated version of their [http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-05.pdf May report])</ref><ref>{{cite magazine|url=https://www.quantamagazine.org/titans-of-mathematics-clash-over-epic-proof-of-abc-conjecture-20180920/ |title=Titans of Mathematics Clash Over Epic Proof of ABC Conjecture |magazine= [[Quanta Magazine]] |date=September 20, 2018 |first= Erica |last= Klarreich |authorlink= Erica Klarreich }}</ref><ref>{{ cite web | url=http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTch-discussions-2018-03.html | title=March 2018 Discussions on IUTeich | access-date=October 2, 2018 }} Web-page by Mochizuki describing discussions and linking consequent publications and supplementary material</ref>
+ | access-date=September 23, 2018 }} (updated version of their [http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-05.pdf May report])</ref><ref>{{cite magazine|url=https://www.quantamagazine.org/titans-of-mathematics-clash-over-epic-proof-of-abc-conjecture-20180920/ |title=Titans of Mathematics Clash Over Epic Proof of ABC Conjecture |magazine= [[Quanta Magazine]] |date=September 20, 2018 |first= Erica |last= Klarreich |author-link= Erica Klarreich }}</ref><ref>{{ cite web | url=http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTch-discussions-2018-03.html | title=March 2018 Discussions on IUTeich | access-date=October 2, 2018 }} Web-page by Mochizuki describing discussions and linking consequent publications and supplementary material</ref>
 ==References==
 {{Reflist}}
 {{more footnotes|date=January 2016}}
-* {{citation |first=S. |last=Lang |authorlink=Serge Lang |title=Survey of Diophantine geometry |publisher=[[Springer-Verlag]] |location=Berlin |year=1997 |isbn=3-540-61223-8 | zbl=0869.11051 | page=51 }}
+* {{citation |first=S. |last=Lang |author-link=Serge Lang |title=Survey of Diophantine geometry |publisher=[[Springer-Verlag]] |location=Berlin |year=1997 |isbn=3-540-61223-8 | zbl=0869.11051 | page=51 }}
 * {{citation |first=L. |last=Szpiro |title=Seminaire sur les pinceaux des courbes de genre au moins deux |journal=Astérisque |volume=86 |issue=3 |year=1981 | zbl=0463.00009 | pages=44–78 }}
 * {{citation |first=L. |last=Szpiro |title=Présentation de la théorie d'Arakelov |journal=Contemp. Math. |volume=67 |year=1987 | zbl=0634.14012 | pages=279–293 |doi=10.1090/conm/067/902599}}