Modularity theorem
Field | Number theory |
---|---|
Conjectured by | Yutaka Taniyama Goro Shimura |
Conjectured in | 1957 |
First proof by | Christophe Breuil Brian Conrad Fred Diamond Richard Taylor |
First proof in | 2001 |
Consequences | Fermat's Last Theorem |
The modularity theorem (formerly called the Taniyama–Shimura conjecture, Taniyama-Shimura-Weil conjecture or modularity conjecture for elliptic curves) states that
Statement
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The theorem states that any elliptic curve over can be obtained via a
for some integer ; this is a curve with integer coefficients with an explicit definition. This mapping is called a modular parametrization of level . If 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), then the parametrization may be defined in terms of a mapping generated by a particular kind of modular form of weight two and level , a normalizedRelated statements
The modularity theorem implies a closely related analytic statement:
To each elliptic curve E over we may attach a corresponding
The generating function of the coefficients is then
If we make the substitution
we see that we have written the Fourier expansion of a function of the complex variable , so the coefficients of the -series are also thought of as the Fourier coefficients of . The function obtained in this way is, remarkably, a cusp form of weight two and level and is also an eigenform (an eigenvector of all Hecke operators); this is the Hasse–Weil conjecture, which follows from the modularity theorem.
Some modular forms of weight two, in turn, correspond to
History
The conjecture attracted considerable interest when Gerhard Frey
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.[8] For example, Wiles's Ph.D. supervisor 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".
In 1995, Andrew Wiles, with some help from
Several theorems in number theory similar to Fermat's Last Theorem follow from the modularity theorem. For example: no cube can be written as a sum of two
Generalizations
The modularity theorem is a special case of more general conjectures due to
In 2013, Freitas, Le Hung, and Siksek proved that elliptic curves defined over real quadratic fields are modular.[13]
Example
For example,[14][15][16] the elliptic curve , with discriminant (and conductor) 37, is associated to the form
For prime numbers ℓ not equal to 37, one can verify the property about the coefficients. Thus, for ℓ = 3, there are 6 solutions of the equation modulo 3: (0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1); thus 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.[17]
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
For example, a modular parametrization of the curve is given by[18]
where, as above, q = exp(2πiz). The functions x(z) and y(z) are modular of weight 0 and level 37; in other words they are
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
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
has a solution with non-zero integers, hence a counter-example to FLT. Then as Yves Hellegouarch was the first to notice,[19] the elliptic curve
of discriminant
cannot be modular.
Notes
- ^ The case was already known by Euler.
References
- ^ Taniyama 1956.
- ^ Weil 1967.
- arXiv:2003.08242 [math.HO].
- ^ Lang, Serge (November 1995). "Some History of the Shimura-Taniyama Conjecture" (PDF). Notices of the American Mathematical Society. 42 (11): 1301–1307. Retrieved 2022-11-08.
- ^ Frey 1986.
- ^ Serre 1987.
- ^ a b Ribet 1990.
- ^ Singh 1997, pp. 203–205, 223, 226.
- ^ Wiles 1995a; Wiles 1995b.
- ^ Diamond 1996.
- ^ Conrad, Diamond & Taylor 1999.
- ^ Breuil et al. 2001.
- ^ Freitas, Le Hung & Siksek 2015.
- ^ For the calculations, see for example Zagier 1985, pp. 225–248
- ^ LMFDB: http://www.lmfdb.org/EllipticCurve/Q/37/a/1
- ^ OEIS: https://oeis.org/A007653
- ^ A synthetic presentation (in French) of the main ideas can be found in this Bourbaki article of Jean-Pierre Serre. For more details see Hellegouarch 2001
- ISBN 978-3-540-39298-9.
- MR 0379507.
- doi:10.5802/afst.698.
Bibliography
- Breuil, Christophe; Conrad, Brian; Diamond, Fred; Taylor, Richard (2001), "On the modularity of elliptic curves over Q: wild 3-adic exercises", MR 1839918
- Conrad, Brian; Diamond, Fred; Taylor, Richard (1999), "Modularity of certain potentially Barsotti–Tate Galois representations", MR 1639612
- Cornell, Gary; MR 1638473
- Darmon, Henri (1999), "A proof of the full Shimura–Taniyama–Weil conjecture is announced" (PDF), MR 1723249Contains a gentle introduction to the theorem and an outline of the proof.
- Diamond, Fred (1996), "On deformation rings and Hecke rings", MR 1405946
- Freitas, Nuno; Le Hung, Bao V.; Siksek, Samir (2015), "Elliptic curves over real quadratic fields are modular", S2CID 119132800
- Frey, Gerhard (1986), "Links between stable elliptic curves and certain Diophantine equations", Annales Universitatis Saraviensis. Series Mathematicae, 1 (1): iv+40, MR 0853387
- MR 1121312Discusses the Taniyama–Shimura–Weil conjecture 3 years before it was proven for infinitely many cases.
- Ribet, Kenneth A. (1990), "On modular representations of Gal(Q/Q) arising from modular forms", S2CID 120614740
- MR 0885783
- Shimura, Goro (1989), "Yutaka Taniyama and his time. Very personal recollections", The Bulletin of the London Mathematical Society, 21 (2): 186–196, MR 0976064
- ISBN 978-1-85702-521-7
- Taniyama, Yutaka (1956), "Problem 12", Sugaku (in Japanese), 7: 269 English translation in (Shimura 1989, p. 194)
- Taylor, Richard; MR 1333036
- S2CID 120553723
- MR 1333035
- MR 1403925
External links
- Darmon, H. (2001) [1994], "Shimura–Taniyama conjecture", Encyclopedia of Mathematics, EMS Press
- Weisstein, Eric W. "Taniyama–Shimura Conjecture". MathWorld.