Modularity theorem: Difference between revisions

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==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. However, Freitas, Le Hung & Siksek{{sfn|Freitas|Le Hung|Siksek|2015}}<!--{harvtxt|Freitas|Le Hung|Siksek|2015}}--> proved that elliptic curves defined over real quadratic fields are modular.

==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^3 -x</math>, 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 not equal to 37, one can verify the property about the coefficients. Thus, for ''ℓ''&nbsp;= 3, there are 6 solutions of the equation modulo 3: {{nowrap|(0, 0)}}, {{nowrap|(0, 1)}}, {{nowrap|(1, 0)}}, {{nowrap|(1, 1)}}, {{nowrap|(2, 0)}}, {{nowrap|(2, 1)}}; thus {{nowrap|1=''a''(3) = 3 − 6 = −3}}.

The conjecture, going back to the 1950s, was completely proven by 1999 using 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 ''E'' of conductor ''N'' can be expressed also by saying that there is a non-constant [[rational map]] defined over '''Q''', from the modular curve ''X''₀(''N'') to ''E''. In particular, the points of ''E'' can be parametrized by [[modular function]]s.

For example, a modular parametrization of the curve <math>y^2 - y = x^3 - x</math> 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}

x(z) &= q^{-2} + 2q^{-1} + 5 + 9q + 18q^2 + 29q^3 + 51q^4 +\cdots\\

y(z) &= q^{-3} + 3q^{-2} + 9q^{-1} + 21 + 46q + 92q^2 + 180q^3 +\cdots

\end{align}</math>

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

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

and likewise for ''y''(''z''), for all integers ''a, b, c, d'' with ''ad'' − ''bc'' = 1 and 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). Suppose that for a prime ''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 [[Yves Hellegouarch]] was the first to notice,<ref>{{Cite journal | last1=Hellegouarch | first1=Yves | title=Points d'ordre 2p^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>

of discriminant

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

cannot be modular.<ref name="ribet">{{cite journal |last=Ribet |first=Ken |author-link=Ken Ribet |title=On modular representations of Gal({{overline|'''Q'''}}/'''Q''') arising from modular forms |journal=Inventiones Mathematicae |volume=100 |year=1990 |issue=2 |pages=431–476 |doi=10.1007/BF01231195 |mr=1047143 |url=https://math.berkeley.edu/~ribet/Articles/invent_100.pdf |hdl=10338.dmlcz/147454 |bibcode=1990InMat.100..431R |s2cid=120614740 }}</ref> 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]] (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=1990 |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>

==Notes==