Zeta Functions - Elliptic Curves: Number Theory and Cryptography - page 434

This series for L E ( s )convergesfor ( s ) > 3 / 2. It is natural to ask whether

L E ( s ) has an analytic continuation to all of C and a functional equation, as is
the case with the Riemann zeta function. As we'll discuss below, the answer
to these questions is yes. However, the proof is much too deep to be included
in this topic (but see Chapter 15 for a discussion of the proof).
To study the analytic properties of L E ( s ), we introduce a new function.
Let τ ∈H , the upper half of the complex plane, as in Chapter 9, and let
q = e 2 πiτ . (This is the standard notation; there should be no possibility of
confusion with the q for finite fields of Chapter 4.) Define

f E ( τ )= ∞ a n q n .

n =1

This is simply a generating function that encodes the number of points on E
mod the various primes. It converges for τ ∈H and satisfies some amazing

properties.
Let N be a positive integer and define
Γ 0 ( N )= ab ∈ SL 2 ( Z )

c ≡ 0(mod N ) .

cd

Then Γ 0 ( N ) is a subgroup of SL 2 ( Z ).
The following result was conjectured by Shimura and has been known by
various names, for example, the Weil conjecture ,the Taniyama-Shimura-
Weil conjecture , and the Taniyama-Shimura conjecture . All three

mathematicians played a role in its history.

THEOREM 14.4 (Breuil, Conrad, Diamond, Taylor, Wiles)
Let E be an elliptic curve defined over Q .Thereexistsaninteger N su ch
that,for all τ

∈H ,

1.

f E aτ + b

=( cτ + d ) 2 f E ( τ ) for all ab

∈ Γ 0 ( N )

cd

cτ + d

2.

Nτ 2 f E ( τ ) .

f E ( − 1 / ( Nτ )) = ±

For a sketch of the proof of this result, see Chapter 15. The theorem (if
we include statements about the behavior at cusps on the real axis) says that
f E ( τ ) is a modular form (in fact, a cusp form; see Section 15.2) of weight
2andlevel N . The smallest possible N is called the conductor of E .A
prime p divides this N if and only if E has bad reduction at p .When E has
multiplicative reduction, p divides N only to the first power. If E has additive
reduction and p> 3, then p 2 is the exact power of p dividing N . The formulas
for p = 2 and 3 are slightly more complicated in this case. See [117].