Zeta Functions - Elliptic Curves: Number Theory and Cryptography - page 435

The transformation law in (1) can be rewritten as
f E aτ + b

d aτ + b
cτ + d

= f E ( τ ) dτ

cτ + d

(this is bad notation: d represents both an integer and the differentiation
operator; it should be clear which is which). Therefore,

f E ( τ ) dτ

is a differential that is invariant under the action of Γ 0 ( N ).
Once we have the relation (1), the second relation of the theorem is perhaps
not as surprising. Every function satisfying (1) is a sum of two functions
satisfying (2), one with a plus sign and one with a minus sign (see Exercise
14.2). Therefore (2) says that f E lies in either the plus space or the minus
space.
Taniyama first suggested the existence of a result of this form in the 1950s.
Eichler and Shimura then showed that if f is a cusp form (more precisely, a
newform) of weight 2 (and level N for some N ) such that all the coecients
a n are integers, then there is an elliptic curve E with f E = f . Thisisthe
converse of the theorem, but it gave the first real evidence that Taniyama's
suggestion was reasonable. In 1967, Weil made precise what the integer N
must be for any given elliptic curve. Since there are only finitely many modu-
lar forms f of a given level N that could arise from elliptic curves, this meant
that the conjecture (Taniyama's suggestion evolved into a conjecture) could
be investigated numerically. If the conjecture had been false for some explicit
E , it could have been disproved by computing enough coe cients to see that
f E was not on the finite list of possibilities. Moreover, Weil showed that if
functions like L E ( s )(namely L E and its twists) have analytic continuations
and functional equations such as the one given in Corollary 14.5 below, then
f E must be a modular form. Since most people believe that naturally defined
L -functions should have analytic continuations and functional equations, this
gave the conjecture more credence. Around 1990, Wiles proved that there are
infinitely many distinct E (that is, with distinct j -invariants) satisfying the
theorem. In 1994, with the help of Taylor, he showed that the theorem is true
for all E such that there is no additive reduction at any prime (but multi-
plicative reduction is allowed). Such curves are called semistable . Finally,
in 2001, Breuil, Conrad, Diamond, and Taylor [20] proved the full theorem.
Let's assume Theorem 14.4 and show that L E ( s ) analytically continues and
satisfies a functional equation. Recall that the gamma function is defined
for ( s ) > 0by

Γ( s )= ∞
0

t s− 1 e −t dt.

Integration by parts yields the relation s Γ( s )=Γ( s + 1), which yields the
meromorphic continuation of Γ( s ) to the complex plane, with poles at the