Modularity theorem

In number theory, the modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms in a particular way. Andrew Wiles and Richard Taylor proved the modularity theorem for semistable elliptic curves, 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, 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

The theorem states that any elliptic curve over

Q

{\displaystyle \mathbb {Q} }

can be obtained via a rational map with integer coefficients from the classical modular curve X0(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.

Related statements

The modularity theorem implies a closely related analytic statement:

To each elliptic curve E over

Q

{\displaystyle \mathbb {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 an 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) of the complex variable τ, so the coefficients of the q-series are also thought of as the Fourier coefficients of f.