In mathematics, the modularity theorem (formerly
called the Taniyama–Shimura conjecture or
the Taniyama–Shimura–Weil conjecture)
states that elliptic curves over the field
of rational numbers are related to modular
forms. Andrew Wiles proved the modularity
theorem for semistable elliptic curves, which
was enough to imply Fermat's last theorem.
Later, Christophe Breuil, Brian Conrad, Fred
Diamond and Richard Taylor extended Wiles'
techniques to prove the full modularity theorem
in 2001. 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 (2015) proved that elliptic
curves defined over real quadratic fields
are modular.
== Statement ==
The theorem states that any elliptic curve
over Q can be obtained via a rational map
with integer coefficients from the classical
modular curve
X
0
(
N
)
{\displaystyle X_{0}(N)}
for some integer N; this is a curve with integer
coefficients with an explicit definition.
This mapping is called a modular parametrization
of level N. If N 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
N, a normalized newform with integer q-expansion,
followed if need be by an isogeny.
The modularity theorem implies a closely related
analytic statement: to an elliptic curve E
over Q we may attach a corresponding L-series.
The L-series is a Dirichlet series, commonly
written
L
(
E
,
s
)
=
∑
n
=
1
∞
a
n
n
s
.
{\displaystyle L(E,s)=\sum _{n=1}^{\infty
}{\frac {a_{n}}{n^{s}}}.}
The generating function of the coefficients
a
n
{\displaystyle a_{n}}
is then
f
(
E
,
q
)
=
∑
n
=
1
∞
a
n
q
n
.
{\displaystyle f(E,q)=\sum _{n=1}^{\infty
}a_{n}q^{n}.}
If we make the substitution
q
=
e
2
π
i
τ
{\displaystyle q=e^{2\pi i\tau }}
we see that we have written the Fourier expansion
of a function
f
(
E
,
τ
)
{\displaystyle f(E,\tau )}
of the complex variable τ, so the coefficients
of the q-series are also thought of as the
Fourier coefficients of
f
{\displaystyle f}
. The function obtained in this way is, remarkably,
a cusp form of weight two and level N 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 holomorphic differentials 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 isogenous
to the original curve (but not, in general,
isomorphic to it).
== History ==
Yutaka Taniyama (1956) stated a preliminary
(slightly incorrect) version of the conjecture
at the 1955 international symposium on algebraic
number theory in Tokyo and Nikkō. Goro Shimura
and Taniyama worked on improving its rigor
until 1957. André Weil (1967) rediscovered
the conjecture, and showed that it would follow
from the (conjectured) functional equations
for some twisted L-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.
The conjecture attracted considerable interest
when Gerhard Frey (1986) suggested that the
Taniyama–Shimura–Weil conjecture implies
Fermat's Last Theorem. He did this by attempting
to show that any counterexample to Fermat's
Last Theorem would imply the existence of
at least one non-modular elliptic curve. This
argument was completed when Jean-Pierre Serre
(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 (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 (Singh 1997, pp. 203–205, 223,
226). For example, Wiles' ex-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".
Wiles (1995), with some help from Richard
Taylor, proved the Taniyama–Shimura–Weil
conjecture for all semistable elliptic curves,
which he used to prove Fermat's Last Theorem,
and the full Taniyama–Shimura–Weil conjecture
was finally proved by Diamond (1996), Conrad,
Diamond & Taylor (1999), and Breuil et al.
(2001) who, building on Wiles' work, incrementally
chipped away at the remaining cases until
the full result was proved.
Once fully proven, the conjecture became known
as the modularity theorem.
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 coprime n-th powers, n ≥ 3.
(The case n = 3 was already known by Euler
