Fermat’s Last Theorem - Elliptic Curves: Number Theory and Cryptography - page 447

This allows us to define a formal series
g = ∞ a n q n .

n =1

We refer to g as the potential modular form attached to ρ .Ofcourse,
some conditions must be imposed on the a r in order for this to represent a
complex function (for example, the numbers a n ∈O must be identified with
complex numbers), but we will not discuss this general problem here.
Let N be a positive integer. Recall that a modular form f of weight 2 and
level N is a function analytic in the upper half plane satisfying
f aτ + b
cτ + d

=( cτ + d ) 2 f ( τ )

(15.5)

for all

ab
cd

∈ Γ 0 ( N )

(where Γ 0 ( N ) is the group of integral matrices of determinant 1 such that
c ≡ 0(mod N )). There are also technical conditions that we won't discuss
for the behavior of f at the cusps. The cusp forms of weight 2 and level N ,
which we'll denote by S ( N ), are those modular forms that take the value 0 at
all the cusps. S ( N ) is a finite dimensional vector space over C . We represent
cusp forms by their Fourier expansions:

f ( τ )= ∞ b n q n ,

n =1

where q = e 2 πiτ .
If M Γ 0 ( M ), so a modular form of level M can be re-

garded as a modular form of level N . More generally, if d

| N ,thenΓ 0 ( N ) ⊆

( N/M )and f ( τ )
isacuspformoflevel M , then it can be shown that f ( dτ ) is a cusp form of
level N . The subspace of S ( N ) generated by such f ,where M ranges through
proper divisors of N and d ranges through divisors of N/M , is called the
subspace of oldforms . There is a naturally defined inner product on S ( N ),
called the Petersson inner product. The space of newforms of level N is the
perpendicular complement of the space of oldforms. Intuitively, the newforms
are those that do not come from levels lower than N .
We now need to introduce the Hecke operators .Let r be a prime. Define
T r ∞

|

⎧
⎨

n =1 b rn q n + n =1 rb n q rn ,

b n q n =

if r N

(15.6)

n =1 b rn q n ,

⎩

if r | N.

n =1

It can be shown that T r maps S ( N )into S ( N ) and that the T r 's commute
with each other. Define the Hecke algebra

T = T N ⊆ End( S ( N ))