The Complex -series
In Section 1.1 we defined a function
,
which encoded information about
for all but finitely
many primes . In this section we define the function ,
which includes information about all primes, and the function
that also includes information ``at infinity''.
Let be an elliptic curve over defined by a
minimal Weierstrass equation
|
(1.3.1) |
A minimal Weierstrass equation in one for which the
are all integers and the discriminant
is minimal
amongs all discriminants of Weierstrass equations for
(again, see [Sil92] for the definition of the
discriminant of a Weierstrass equation, and also for an
explicit description of the allowed transformations of
a Weierstrass equation).
For each prime number , the equation
(1.3.1) reduces modulo to define an elliptic
over the finite field . Let
For each prime , we use the following
recipe to define . If the singular curve
has a cuspidal singularity, e.g., is
, then let . If it has a a nodal
singularity, e.g., like , let
if the slope of the tangent line at the singular
point is in and let if the slope is
not in . Summarizing:
Even in the cases when , we still have
When has additive reduction, the nonsingular points
form a group isomorphic to
, and there is
one singular point, hence points,
so
When
has split multiplicative reduction, there is 1 singular
point plus the number of elements of a group isomorphic to
,
so points, and
When has non-split multiplicative
reduction, there is singular point plus the number
of elements of a group isomorphic
,
i.e., points, and
The definition of the full -function of is then
If in addition we add in a few more analytic factors to the -function
we obtain a function that satisfies a remarkably simple
functional equation.
Let
be the -function (e.g.,
),
which defines a meromorphic function on , with poles
at the non-positive integers.
Theorem 1.6 (Hecke, Wiles et al.)
There is a unique positive integer and sign
such that the function
extends to a complex analytic function on all that
satisfies the functional equation
|
(1.3.2) |
for all
.
The integer is called the conductor of
and
is called the sign in the functional
equation for or the root number of .
One can prove that the primes that divide are the same
as the primes that divide . Moreover, for ,
we have that
There is a geometric algorithm called Tate's algorithm that
computes in all cases and .
Example 1.7
Consider the elliptic curve
defined by
The above Weierstrass equation is minimal and
has discriminant
sage: e = EllipticCurve('1925d'); e
Elliptic Curve defined by y^2 + y = x^3 + 50*x + 31 over Rational Field
sage: e.is_minimal()
True
sage: factor(e.discriminant())
-1 * 5^6 * 7^2 * 11
At the curve has additive reduction so .
At the curve has split multiplicative reduction
so . At the curve has nonsplit multiplicative
reduction, so . Counting points for ,
we find that
sage: [e.ap(p) for p in primes(14)]
[0, 3, 0, 1, -1, 4]
Conjecture 1.9 (The Parity Conjecture)
Let be an elliptic curve over , let
be the
analytic rank and be the algebraic rank. Then
Jan Nekovar has done a huge amount of work toward
Conjecture 1.9; in particular, he proves it under the
(as yet unproved) hypothesis that
is finite (see
Section 2.2 below).
William
2007-05-25