Modularity theorem: Difference between revisions

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{{Short description|Relates rational elliptic curves to modular forms}}

*[[{{Distinguish|Serre's modularity conjecture]]}}

<!--{{no footnotes|date=January 2021}}Harvard style footnotes converted to short footnotes-->

{{Infobox mathematical statement

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TheIn '''modularity[[number theorem''' (formerly calledtheory]], the '''Taniyama–Shimura conjecture''', '''Taniyama-Shimura-Weil conjecture''' or '''modularity conjecture for elliptic curvestheorem''') states that [[elliptic curve]]s over the field of [[rational number]]s are related to [[modular form]]s in a particular way. [[Andrew Wiles]] and [[Richard Taylor (mathematician)|Richard Taylor]] proved the modularity theorem for [[semistable elliptic curve]]s, which was enough to imply [[Fermat's Last Theorem]] (FLT). Later, a series of papers by Wiles's former students [[Brian Conrad]], [[Fred Diamond]] and [[Richard Taylor (mathematician)|Richard Taylor]], culminating in a joint paper with [[Christophe Breuil]], extended Wiles's techniques to prove the full modularity theorem in 2001. Before that, the statement was known as the '''Taniyama–Shimura conjecture''', '''Taniyama–Shimura–Weil conjecture''', or the '''modularity conjecture for elliptic curves'''.

==Statement==

{{More citations needed|section|date=March 2021}}<!--added after harvard style citations converted to short foot notes-->

The [[theorem]] states that any [[elliptic curve]] over <math>\mathbf{Q}</math> can be obtained via a [[rational map]] with [[integer]] [[coefficient]]s from the [[classical modular curve]] <{{math|''X''<sub>X_0(N)0</mathsub>(''N'')}} for some integer <math>{{mvar|N</math>}}; this is a curve with integer coefficients with an explicit definition. This mapping is called a modular parametrization of level <math>{{mvar|N</math>}}. If <math>{{mvar|N</math>}} is the smallest integer for which such a parametrization can be found (which by the modularity theorem itself is now known to be a number called the ''[[conductor of an elliptic curve|conductor]]''), then the parametrization may be defined in terms of a mapping generated by a particular kind of modular form of weight two and level <math>{{mvar|N</math>}}, a normalized [[newform]] with integer <math>{{mvar|q</math>}}-expansion, followed if need be by an [[Elliptic curve#Isogeny|isogeny]].

===Related statements===

The modularity theorem implies a closely related analytic statement:

To each elliptic curve ''{{mvar|E''}} over <math>\mathbf{Q}</math> we may attach a corresponding [[L-series of an elliptic curve|{{mvar|L}}-series]]. The <math>{{mvar|L</math>}}-series is a [[Dirichlet series]], commonly written

:<math>L(E, s) = \sum_{n=1}^\infty \frac{a_n}{n^s}.</math>

The [[generating function]] of the coefficients {{math|''a''<mathsub>a_n''n''</mathsub>}} is then

:<math>f(E, q) = \sum_{n=1}^\infty a_n q^n.</math>

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:<math>q = e^{2 \pi i \tau}</math>

we see that we have written the [[Fourier series|Fourier expansion]] of a function <{{math>|''f''(''E'', \tau''τ'')</math>}} of the complex variable <math>\tau</math>{{mvar|τ}}, so the coefficients of the <math>{{mvar|q</math>}}-series are also thought of as the Fourier coefficients of <math>{{mvar|f</math>}}. The function obtained in this way is, remarkably, a [[cusp form]] of weight two and level <math>{{mvar|N</math>}} and is also an eigenform (an eigenvector of all [[Hecke operator]]s); this is the '''Hasse–Weil conjecture''', which follows from the modularity theorem.

Some modular forms of weight two, in turn, correspond to [[holomorphic differential]]s for an elliptic curve. The Jacobian of the modular curve can (up to isogeny) be written as a product of irreducible [[Abelian varieties]], corresponding to Hecke eigenforms of weight 2. The 1-dimensional factors are elliptic curves (there can also be higher-dimensional factors, so not all Hecke eigenforms correspond to rational elliptic curves). The curve obtained by finding the corresponding cusp form, and then constructing a curve from it, is [[Elliptic curve#Isogeny|isogenous]] to the original curve (but not, in general, isomorphic to it).

==History==

{{see also|Taniyama's problems}}

[[Yutaka Taniyama]]{{sfn|Taniyama|1956}}<!--{{harvs|txt|authorlink=Yutaka Taniyama|last=Taniyama|first=Yutaka|year=1956}}--> stated a preliminary (slightly incorrect) version of the conjecture at the 1955 international symposium on algebraic number theory in [[Tokyo]] and [[Nikkō, Tochigi|Nikkō]]. [[Goro Shimura]] and Taniyama worked on improving its rigor until 1957. André Weil{{sfn|Weil|1967}}<!--{{harvs|txt|authorlink=André Weil|last=Weil|first=André|year= 1967}}--> rediscovered the conjecture, and showed in 1967 that it would follow from the (conjectured) functional equations for some twisted <math>L</math>-series of the elliptic curve; this was the first serious evidence that the conjecture might be true. Weil also showed that the conductor of the elliptic curve should be the level of the corresponding modular form. The Taniyama–Shimura–Weil conjecture became a part of the [[Langlands program]].<ref name="Harris Virtues of Priority">{{cite arXiv | last=Harris | first=Michael | title=Virtues of Priority | year=2020 | eprint=2003.08242 | page=| class=math.HO }}</ref><ref>{{cite journal

{{further|Fermat's Last Theorem|Wiles's proof of Fermat's Last Theorem}}

[[Yutaka Taniyama]]{{sfn|Taniyama|1956}}<!--{{harvs|txt|authorlink=Yutaka Taniyama|last=Taniyama|first=Yutaka|year=1956}}--> stated a preliminary (slightly incorrect) version of the conjecture at the 1955 international symposium on [[algebraic number theory]] in [[Tokyo]] and [[Nikkō, Tochigi|Nikkō]] as the twelfth of his [[Taniyama's problems|set of 36 unsolved problems]]. [[Goro Shimura]] and Taniyama worked on improving its rigor until 1957. [[André Weil]]{{sfn|Weil|1967}}<!--{{harvs|txt|authorlink=André Weil|last=Weil|first=André|year= 1967}}--> rediscovered the conjecture, and showed in 1967 that it would follow from the (conjectured) functional equations for some twisted <math>{{mvar|L</math>}}-series of the elliptic curve; this was the first serious evidence that the conjecture might be true. Weil also showed that the conductor of the elliptic curve should be the level of the corresponding modular form. The Taniyama–Shimura–Weil conjecture became a part of the [[Langlands program]].<ref name="Harris Virtues of Priority">{{cite arXiv | last=Harris | first=Michael | title=Virtues of Priority | year=2020 | eprint=2003.08242 | page=| class=math.HO }}</ref><ref>{{cite journal

| last = Lang

| first = Serge

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}}</ref>

The conjecture attracted considerable interest when [[Gerhard Frey (mathematician)|Gerhard Frey]]{{sfn|Frey|1986}}<!--{{harvs|txt|authorlink=Gerhard Frey (mathematician)|last=Frey|first=Gerhard|year=1986}}--> suggested in 1986 that it implies [[Fermat's Last Theorem|FLT]]. He did this by attempting to show that any counterexample to Fermat's Last TheoremFLT would imply the existence of at least one non-modular elliptic curve. This argument was completedmoved closer to its goal in 1987 when Jean-Pierre Serre{{sfn|Serre|1987}}<!--{{harvs|txt|authorlink=Jean-Pierre Serre|last=Serre|first=Jean-Pierre|year=1987}}--> identified a missing link (now known as the [[epsilon conjecture]] or Ribet's theorem) in Frey's original work, followed two years later by Ken Ribet's completion of a proof of the epsilon conjecture.{{sfn|Ribet|1990}}<!--{{harvs|txt|authorlink=Ken Ribet|last=Ribet|first=Ken|year=1990}}-->'s completion of a proof of the epsilon conjecture.

Even after gaining serious attention, the Taniyama–Shimura–Weil conjecture was seen by contemporary mathematicians as extraordinarily difficult to prove or perhaps even inaccessible to proof<!--{{harv|Singh|1997|pp=203–205, 223, 226}}-->.{{sfn|Singh|1997|pp=203–205, 223, 226}} For example, Wiles's Ph.D. supervisor [[John H. Coates|John Coates]] states that it seemed "impossible to actually prove", and Ken Ribet considered himself "one of the vast majority of people who believed [it] was completely inaccessible".

With Ribet’s proof of the epsilon conjecture, Andrew Wiles saw an opportunity: Fermat’s Last Theorem was a respectable research project because it was now a corollary of the TSW conjecture. He had expertise in [[Iwasawa theory]]; maybe there was a path from Iwasawa theory to Taniyama–Shimura–Weil.

In 1995 Andrew Wiles, with some help from [[Richard Taylor (mathematician)|Richard Taylor]], proved the Taniyama–Shimura–Weil conjecture for all [[semistable elliptic curve]]s, which he used to prove Fermat's Last Theorem,{{sfnm|Wiles|1995a|Wiles|1995b}}<!--{{harvs|txt|authorlink=Andrew Wiles|last=Wiles|year=1995}}--> and the full Taniyama–Shimura–Weil conjecture was finally proved by Diamond,{{sfn|Diamond|1996}}<!--{{harvtxt|Diamond|1996}}--> Conrad, Diamond & Taylor; and Breuil, Conrad, Diamond & Taylor; building on Wiles's work, they incrementally chipped away at the remaining cases until the full result was proved in 1999.{{sfn|Conrad|Diamond|Taylor|1999}}<!--{{harvtxt|Conrad|Diamond|Taylor|1999}}-->{{sfn|Breuil|Conrad|Diamond|Taylor|2001}}<!--{{harvtxt|Breuil|Conrad|Diamond|Taylor|2001}}-->

{{further|Fermat's Last Theorem|Wiles's proof of Fermat's Last Theorem}}

In 1995, Andrew Wiles, with some help from [[Richard Taylor (mathematician)|Richard Taylor]], proved the Taniyama–Shimura–Weil conjecture for all [[semistable elliptic curve]]s,. which heWiles used this to prove Fermat's Last TheoremFLT,{{sfnm|Wiles|1995a|Wiles|1995b}}<!--{{harvs|txt|authorlink=Andrew Wiles|last=Wiles|year=1995}}--> and the full Taniyama–Shimura–Weil conjecture was finally proved by Diamond,{{sfn|Diamond|1996}}<!--{{harvtxt|Diamond|1996}}--> Conrad, Diamond & Taylor; and Breuil, Conrad, Diamond & Taylor; building on Wiles's work, they incrementally chipped away at the remaining cases until the full result was proved in 1999.{{sfn|Conrad|Diamond|Taylor|1999}}<!--{{harvtxt|Conrad|Diamond|Taylor|1999}}-->{{sfn|Breuil|Conrad|Diamond|Taylor|2001}}<!--{{harvtxt|Breuil|Conrad|Diamond|Taylor|2001}}--> Once fully proven, the conjecture became known as the modularity theorem.

Once fully proven, the conjecture became known as the modularity theorem.

Several theorems in number theory similar to Fermat's Last TheoremFLT follow from the modularity theorem. For example: no cube can be written as a sum of two [[coprime]] <math>{{mvar|n</math>-}}th powers, <{{math>|''n'' \geq≥ 3</math>}}. ({{efn|The case <{{math>|''n'' {{=}} 3</math>}} was already known by [[Euler]].)}}

In 2025, modularity was extended to over 10% of [[abelian surfaces]].<ref>{{Cite web |last=Howlett |first=Joseph |date=2025-06-02 |title=The Core of Fermat’s Last Theorem Just Got Superpowered |url=https://www.quantamagazine.org/the-core-of-fermats-last-theorem-just-got-superpowered-20250602/ |access-date=2025-08-06 |website=Quanta Magazine |language=en}}</ref>

==Generalizations==

The modularity theorem is a special case of more general conjectures due to [[Robert Langlands]]. The [[Langlands program]] seeks to attach an [[automorphic form]] or [[automorphic representation]] (a suitable generalization of a modular form) to more general objects of [[arithmetic algebraic geometry]], such as to every elliptic curve over a [[number field]]. Most cases of these extended conjectures have not yet been proved.

In However2013, Freitas, Le Hung, &and Siksek{{sfn|Freitas|Le Hung|Siksek|2015}}<!--{harvtxt|Freitas|Le Hung|Siksek|2015}}--> proved that elliptic curves defined over real [[quadratic fieldsfield]]s are modular.{{sfn|Freitas|Le Hung|Siksek|2015}}

==Example==

For example,<ref>For the calculations, see for example {{harvnb|Zagier|1985|pp=225–248}}</ref><ref>LMFDB: http://www.lmfdb.org/EllipticCurve/Q/37/a/1</ref><ref>OEIS: https://oeis.org/A007653</ref> the elliptic curve <{{math>|''y^''² -− ''y'' {{=}} ''x^''<sup>3 -x</mathsup> − ''x''}}, with discriminant (and conductor) 37, is associated to the form

:<math>f(z) = q - 2q^2 - 3q^3 + 2q^4 - 2q^5 + 6q^6 + \cdots, \qquad q = e^{2 \pi i z}</math>

For prime numbers ℓ{{mvar|l}} not equal to 37, one can verify the property about the coefficients. Thus, for {{math|''ℓl''&nbsp; {{=}} 3}}, there are 6 solutions of the equation modulo 3: {{nowrapmath|(0, 0)}}, {{nowrapmath|(0, 1)}}, {{nowrapmath|(1, 0)}}, {{nowrapmath|(1, 1)}}, {{nowrapmath|(2, 0)}}, {{nowrapmath|(2, 1)}}; thus {{nowrapmath|1=''a''(3) = 3 − 6 = −3}}.

The conjecture, going back to the 1950s, was completely proven by 1999 using the ideas of [[Andrew Wiles]], who proved it in 1994 for a large family of elliptic curves.<ref>A synthetic presentation (in French) of the main ideas can be found in [http://www.numdam.org/item/SB_1994-1995__37__319_0/ this] [[Nicolas Bourbaki|Bourbaki]] article of [[Jean-Pierre Serre]]. For more details see {{Harvard citations |last=Hellegouarch |year=2001 |nb=yes}}</ref>

There are several formulations of the conjecture. Showing that they are equivalent was a main challenge of number theory in the second half of the 20th century. The modularity of an elliptic curve ''{{mvar|E''}} of conductor ''{{mvar|N''}} can be expressed also by saying that there is a non-constant [[rational map]] defined over '''Q'''{{math|ℚ}}, from the modular curve {{math|''X''₀(''N'')}} to ''{{mvar|E''}}. In particular, the points of ''{{mvar|E''}} can be parametrized by [[modular function]]s.

For example, a modular parametrization of the curve <{{math>|''y^''² -− ''y'' {{=}} ''x^''<sup>3 - x</mathsup> − ''x''}} is given by<ref>{{cite book |first=D. |last=Zagier |chapter=Modular points, modular curves, modular surfaces and modular forms |series=Lecture Notes in Mathematics |volume=1111 |publisher=Springer |year=1985 |pages=225–248 |doi=10.1007/BFb0084592 |isbn=978-3-540-39298-9 |title=Arbeitstagung Bonn 1984 }}</ref>

:<math>\begin{align}

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\end{align}</math>

where, as above, {{math|''q'' {{=}} exp(2π''ize'')^2''πiz''}}. The functions {{math|''x''(''z'')}} and {{math|''y''(''z'')}} are modular of weight 0 and level 37; in other words they are [[meromorphic]], defined on the [[upper half-plane]] {{math|Im(''z'') > 0}} and satisfy

:<math>x\!\left(\frac{az + b}{cz + d}\right) = x(z)</math>

and likewise for {{math|''y''(''z'')}}, for all integers {{math|''a'', ''b'', ''c'', ''d''}} with {{math|''ad'' − ''bc'' {{=}} 1}} and 37{{math|37 {{!}} ''c''}}.

Another formulation depends on the comparison of [[Galois representation]]s attached on the one hand to elliptic curves, and on the other hand to modular forms. The latter formulation has been used in the proof of the conjecture. Dealing with the level of the forms (and the connection to the conductor of the curve) is particularly delicate.

The most spectacular application of the conjecture is the proof of [[Fermat's Last Theorem|FLT]] (FLT). Suppose that for a prime {{math|''p'' ≥ 5}}, the Fermat equation

:<math>a^p + b^p = c^p</math>

has a solution with non-zero integers, hence a counter-example to FLT. Then as {{ill|Yves Hellegouarch|fr|Yves Hellegouarch|lt=Yves Hellegouarch}} was the first to notice,<ref>{{Cite journal | last1=Hellegouarch | first1=Yves | title=Points d'ordre 2p2''p''^''h'' sur les courbes elliptiques | url=http://matwbn.icm.edu.pl/ksiazki/aa/aa26/aa2636.pdf| mr=0379507 | year=1974 | journal=Acta Arithmetica | issn=0065-1036 | volume=26 | issue=3 | pages=253–263| doi=10.4064/aa-26-3-253-263 | doi-access=free }}</ref> the elliptic curve

:<math>y^2 = x(x - a^p)(x + b^p)</math>

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:<math>\Delta = \frac{1}{256}(abc)^{2p}</math>

cannot be modular.{{sfn|Ribet|1990}} Thus, the proof of the Taniyama–Shimura–Weil conjecture for this family of elliptic curves (called Hellegouarch–Frey curves) implies FLT. The proof of the link between these two statements, based on an idea of [[Gerhard Frey (mathematician)|Gerhard Frey]] (1985), is difficult and technical. It was established by [[Kenneth Ribet]] in 1987.<ref>See the survey of {{cite journal |first=K. |last=Ribet |title=From the Taniyama–Shimura conjecture to Fermat's Last Theorem |journal=Annales de la Faculté des Sciences de Toulouse |volume=11 |year=1990b |pages=116–139 |doi= 10.5802/afst.698|url=http://www.numdam.org/item?id=AFST_1990_5_11_1_116_0 |doi-access=free }}</ref>

== See also ==

*[[Serre's modularity conjecture]]

==Notes==

{{notelist}}

==References==

{{reflist|20em}}

=== ReferencesBibliography ===

{{refbegin|20em}}

*{{Citation | last1=Breuil | first1=Christophe | last2=Conrad | first2=Brian | last3=Diamond | first3=Fred | last4=Taylor | first4=Richard | title=On the modularity of elliptic curves over '''Q''': wild 3-adic exercises | doi=10.1090/S0894-0347-01-00370-8 | mr=1839918 | year=2001 | journal=[[Journal of the American Mathematical Society]] | issn=0894-0347 | volume=14 | issue=4 | pages=843–939| doi-access=free }}

*{{Citation | last1=Conrad | first1=Brian | last2=Diamond | first2=Fred | last3=Taylor | first3=Richard | title=Modularity of certain potentially Barsotti–Tate Galois representations | doi=10.1090/S0894-0347-99-00287-8 | mr=1639612 | year=1999 | journal=[[Journal of the American Mathematical Society]] | issn=0894-0347 | volume=12 | issue=2 | pages=521–567| doi-access=free }}

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*{{Citation | last1=Frey | first1=Gerhard | title=Links between stable elliptic curves and certain Diophantine equations | mr=853387 | year=1986 | journal=Annales Universitatis Saraviensis. Series Mathematicae | issn=0933-8268 | volume=1 | issue=1 | pages=iv+40}}

*{{Citation | last1=Mazur | first1=Barry | author1-link=Barry Mazur | title=Number theory as gadfly | doi=10.2307/2324924 | mr=1121312 | year=1991 | journal=[[American Mathematical Monthly|The American Mathematical Monthly]] | issn=0002-9890 | volume=98 | issue=7 | pages=593–610| jstor=2324924 }} Discusses the Taniyama–Shimura–Weil conjecture 3 years before it was proven for infinitely many cases.

*{{Citation | last1=Ribet | first1=Kenneth A. | title=On modular representations of Gal({{overline|'''Q'''}}/'''Q''') arising from modular forms | doi=10.1007/BF01231195 | mr=1047143 | year=1990 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=100 | issue=2 | pages=431–476| bibcode=1990InMat.100..431R | hdl=10338.dmlcz/147454 | s2cid=120614740 | hdl-access=free }}

*{{Citation | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Sur les représentations modulaires de degré 2 de Gal({{overline|Q}}/Q) | doi=10.1215/S0012-7094-87-05413-5 | mr=885783 | year=1987 | journal=[[Duke Mathematical Journal]] | issn=0012-7094 | volume=54 | issue=1 | pages=179–230}}

*{{Citation | last1=Shimura | first1=Goro | title=Yutaka Taniyama and his time. Very personal recollections | doi=10.1112/blms/21.2.186 | mr=976064 | year=1989 | journal=The Bulletin of the London Mathematical Society | issn=0024-6093 | volume=21 | issue=2 | pages=186–196| doi-access=free }}

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*{{Citation | last1=Wiles | first1=Andrew | author1-link=Andrew Wiles | title=Modular elliptic curves and Fermat's last theorem | jstor=2118559 | mr=1333035 | year=1995a | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=141 | issue=3 | pages=443–551 | doi=10.2307/2118559| citeseerx=10.1.1.169.9076 }}

*{{Citation | last1=Wiles | first1=Andrew | author1-link=Andrew Wiles | title=Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) | publisher=Birkhäuser | ___location=Basel, Boston, Berlin | mr=1403925 | year=1995b | chapter=Modular forms, elliptic curves, and Fermat's last theorem | pages=243–245}}

{{refend}}

==External links==