Szpiro's conjecture#Modified Szpiro conjecture

{{Infobox mathematical statement

| name = Modified Szpiro conjecture

| image =

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| field = Number theory

| conjectured by = Lucien Szpiro

| conjecture date = 1981

| first proof by =

| first proof date =

| open problem =

| known cases =

| implied by =

| equivalent to = abc conjecture

| generalizations =

| consequences = {{plainlist|

}}

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,{{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 }} 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.{{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 }}{{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 }}{{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 }}{{Cite journal |first=Andrzej |last=Dąbrowski |title=On the diophantine equation x! + A = y² | journal=Nieuw Archief voor Wiskunde, IV. |volume=14 |pages=321–324 |year=1996 | zbl=0876.11015 }}

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,{{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}}. and was then shown to be equivalent to the modified Szpiro's conjecture.{{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}}

Consequences

Szpiro's conjecture and its modified form are known to imply several important mathematical results and conjectures, including Roth's theorem,{{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}} Faltings's theorem,{{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}} Fermat–Catalan conjecture,{{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 }} and a negative solution to the Erdős–Ulam problem.

{{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

}}

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).{{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 }} However, the papers have not been accepted by the mathematical community as providing a proof of the conjecture,{{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}}{{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}}{{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 }} 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".

{{ 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 |author-link1= Peter Scholze

|first2= Jakob |last2= Stix |author-link2= Jakob Stix

|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]|){{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 }}{{ 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

References

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