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 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 )\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),

\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]

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 = y²". 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

This number theory-related article is a stub. You can help Wikipedia by expanding it.

[Goldfeld-1996-1] Goldfeld, Dorian (1996). "Beyond the last theorem". Math Horizons. 4 (September): 26–34. doi:10.1080/10724117.1996.11974985. JSTOR 25678079.

[2] Bombieri, Enrico (1994). "Roth's theorem and the abc-conjecture". Preprint. ETH Zürich.

[3] Elkies, N. D. (1991). "ABC implies Mordell". International Mathematics Research Notices. 1991 (7): 99–109. doi:10.1155/S1073792891000144.

[4] Pomerance, Carl (2008). "Computational Number Theory". The Princeton Companion to Mathematics. Princeton University Press. pp. 361–362.

[5] Dąbrowski, Andrzej (1996). "On the diophantine equation x! + A = y²". Nieuw Archief voor Wiskunde, IV. 14: 321–324. Zbl 0876.11015.

[6] 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.

[7] Oesterlé, Joseph (1988), "Nouvelles approches du "théorème" de Fermat", Astérisque, Séminaire Bourbaki exp 694 (161): 165–186, ISSN 0303-1179, MR 0992208

[8] 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.

[9] Elkies, N. D. (1991). "ABC implies Mordell". International Mathematics Research Notices. 1991 (7): 99–109. doi:10.1155/S1073792891000144.

[10] Pomerance, Carl (2008). "Computational Number Theory". The Princeton Companion to Mathematics. Princeton University Press. pp. 361–362.

[11] 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

[12] Ball, Peter (10 September 2012). "Proof claimed for deep connection between primes". Nature. doi:10.1038/nature.2012.11378. Retrieved 19 April 2020.

[13] Revell, Timothy (September 7, 2017). "Baffling ABC maths proof now has impenetrable 300-page 'summary'". New Scientist.

[14] Conrad, Brian (December 15, 2015). "Notes on the Oxford IUT workshop by Brian Conrad". Retrieved March 18, 2018.

[15] 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.

[16] 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|)

[17] Klarreich, Erica (September 20, 2018). "Titans of Mathematics Clash Over Epic Proof of ABC Conjecture". Quanta Magazine.

[18] "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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@@ Line 1: / Line 1: @@
+{{Short description|Conjecture in number theory}}
 {{Infobox mathematical statement
 | name = Modified Szpiro conjecture
@@ Line 10: / Line 11: @@
 | open problem =
 | known cases =
-| implied by
+| implied by =
 | equivalent to = [[abc conjecture]]
 | generalizations =
@@ Line 22: / Line 23: @@
 }}
+In [[number theory]], '''Szpiro's conjecture''' relates to the [[conductor of an elliptic curve|conductor]] and the discriminant of an [[elliptic curve]].  In a slightly modified form, it is equivalent to the well-known [[abc conjecture|''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]],<ref name="Goldfeld-1996">{{cite journal |last=Goldfeld |first=Dorian | author-link=Dorian M. Goldfeld |year=1996 |title=Beyond the last theorem |journal=[[Math Horizons]] |volume=4 |issue=September |pages=26–34 |jstor= 25678079 |doi=10.1080/10724117.1996.11974985 }}</ref> in part to its large number of consequences in number theory including [[Roth's theorem]], the [[Faltings's theorem|Mordell conjecture]], the [[Fermat–Catalan conjecture]], and [[Brocard's problem]].<ref>{{cite journal | first1=Enrico | last1=Bombieri|author-link=Enrico Bombieri | title=Roth's theorem and the abc-conjecture | journal=Preprint | year=1994 | publisher=ETH Zürich }}</ref><ref>{{Cite journal |last=Elkies |first=N. D. |author-link=Noam Elkies |title=ABC implies Mordell |journal= International Mathematics Research Notices|volume=1991 |year=1991 |pages=99–109 |doi=10.1155/S1073792891000144 |issue=7 |doi-access=free }}</ref><ref>{{Cite book |last=Pomerance |first=Carl |author-link=Carl Pomerance |chapter=Computational Number Theory |title=The Princeton Companion to Mathematics |publisher=[[Princeton University Press]] |year=2008 |pages=361–362 }}</ref><ref>{{Cite journal |first=Andrzej |last=Dąbrowski |title=On the diophantine equation ''x''! + ''A'' = ''y''<sup>2</sup> | journal=Nieuw Archief voor Wiskunde, IV. |volume=14 |pages=321–324 |year=1996 | zbl=0876.11015 }}</ref>
-In [[number theory]], '''Szpiro's conjecture''' relates the [[conductor of an elliptic curve|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 &epsilon; &gt; 0, there exists a constant ''C''(&epsilon;) such that for any elliptic curve ''E'' defined over '''Q''' with minimal discriminant &Delta; and conductor ''f'', we have
+The conjecture states that: given &epsilon; &gt; 0, there exists a constant ''C''(&epsilon;) such that for any elliptic curve ''E'' defined over '''Q''' with minimal discriminant &Delta; and conductor ''f'',
 :<math> \vert\Delta\vert \leq C(\varepsilon ) \cdot f^{6+\varepsilon }. </math>
 ==Modified Szpiro conjecture==
-The '''modified Szpiro conjecture''' states that: given &epsilon; &gt; 0, there exists a constant ''C''(&epsilon;) such that for any elliptic curve ''E'' defined over '''Q''' with invariants ''c''<sub>4</sub>, ''c''<sub>6</sub> and conductor ''f'' (using [[Tate's algorithm#Notation|notation from Tate's algorithm]]), we have
+The '''modified Szpiro conjecture''' states that: given &epsilon; &gt; 0, there exists a constant ''C''(&epsilon;) such that for any elliptic curve ''E'' defined over '''Q''' with invariants ''c''<sub>4</sub>, ''c''<sub>6</sub> and conductor ''f'' (using [[Tate's algorithm#Notation|notation from Tate's algorithm]]),
 :<math> \max\{\vert c_4\vert^3,\vert c_6\vert^2\} \leq C(\varepsilon )\cdot f^{6+\varepsilon }. </math>
+===''abc'' conjecture===
+The [[Abc conjecture|''abc'' conjecture]] originated as the outcome of attempts by [[Joseph Oesterlé]] and [[David Masser]] to understand Szpiro's conjecture,<ref>{{citation|title=Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki|journal= European Journal of Mathematics|first=Ivan|last=Fesenko|volume=1 |issue=3| pages=405–440 | year=2015 |url=https://www.maths.nottingham.ac.uk/personal/ibf/notesoniut.pdf |doi=10.1007/s40879-015-0066-0|doi-access=free}}.</ref> and was then shown to be equivalent to the modified Szpiro's conjecture.<ref>{{Citation | last1=Oesterlé | first1=Joseph | author-link=Joseph Oesterlé | title=Nouvelles approches du "théorème" de Fermat | url= http://www.numdam.org/item?id=SB_1987-1988__30__165_0 | series=Séminaire Bourbaki exp 694 |mr=992208 | year=1988 | journal=Astérisque | issn=0303-1179 | issue=161 | pages=165–186}}</ref>
+==Consequences==
+{{further|abc conjecture#Some consequences}}
+Szpiro's conjecture and its modified form are known to imply several important mathematical results and conjectures, including [[Roth's theorem]],<ref>{{cite book |doi=10.1007/978-3-0348-0859-0_13|chapter=Lecture on the abc Conjecture and Some of Its Consequences|title=Mathematics in the 21st Century|series=Springer Proceedings in Mathematics & Statistics|year=2015|last1=Waldschmidt|first1=Michel|volume=98|pages=211–230|isbn=978-3-0348-0858-3|chapter-url=https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/abcLahoreProceedings.pdf}}</ref> [[Faltings's theorem]],<ref>{{Cite journal |last=Elkies |first=N. D. |author-link=Noam Elkies |title=ABC implies Mordell |journal=  International Mathematics Research Notices|volume=1991 |year=1991 |pages=99–109 |doi=10.1155/S1073792891000144 |issue=7 |doi-access= free}}</ref> [[Fermat–Catalan conjecture]],<ref>{{Cite book |last=Pomerance |first=Carl |author-link=Carl Pomerance |chapter=Computational Number Theory |title=The Princeton Companion to Mathematics |publisher=Princeton University Press |year=2008 |pages=361–362 }}</ref> and a negative solution to the [[Erdős–Ulam problem]].<ref>
+{{citation
+ | last1 = Pasten | first1 = Hector
+ | doi = 10.1007/s00605-016-0973-2
+ | issue = 1
+ | journal = [[Monatshefte für Mathematik]]
+ | mr = 3592123
+ | pages = 99–126
+ | title = Definability of Frobenius orbits and a result on rational distance sets
+ | volume = 182
+ | year = 2017| s2cid = 7805117
+ }}
+</ref>
 ==Claimed proofs==
-{{Main|Abc conjecture#Claimed proofs}}
+{{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
+{{ cite web | url=http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-08.pd | 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>
+ |archive-url=https://web.archive.org/web/20200208075321/http://www.kurims.kyoto-u.ac.jp/~motizuki/SS2018-08.pdf|archive-date=February 8, 2020|url-status=dead}} (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>
+==See also==
+* [[Arakelov theory]]
 ==References==
+{{Reflist}}
-{{no 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=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}}
-[[Category:Conjectures]]
-[[Category:Unsolved problems in mathematics]]
-[[Category:Number theory]]
+==Bibliography==
+* {{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| url={{Google books|title=Encyclopedia of Mathematical Science vol.60|b9RqCQAAQBAJ|page=51|plainurl=yes}}}}
+* {{Cite book|first=L. |last=Szpiro |title=Seminaire sur les pinceaux des courbes de genre au moins deux|chapter=Propriétés numériques du faisceau dualisant rélatif|series=Astérisque |volume=86 |issue=3 |year=1981 | zbl=0517.14006 | pages=44–78|url=http://www.numdam.org/item/AST_1981__86__R1_0.pdf }}
+* {{citation |first=L. |last=Szpiro |title=Présentation de la théorie d'Arakelov |journal=Contemp. Math. |series=Contemporary Mathematics |volume=67 |year=1987 | zbl=0634.14012 | pages=279–293 |doi=10.1090/conm/067/902599|isbn=9780821850749 }}
+[[Category:Conjectures]]
+[[Category:Unsolved problems in number theory]]
+[[Category:Abc conjecture]]
 {{numtheory-stub}}