Zeta Functions

14.1 Elliptic Curves over Finite Fields

Let E be an elliptic curve over a ļ¬nite ļ¬eld Fq . Let

Nn = #E(Fqn )

be the number of points on E over the ļ¬eld Fqn . The Z-function of E is

deļ¬ned to be

ā

Nn n

ZE (T ) = exp T .

n

n=1

tn /n! is the usual exponential function. The Z-function

Here exp(t) =

encodes certain arithmetic information about E as the coeļ¬cients of a gen-

erating function. The presence of the exponential function is justiļ¬ed by the

simple form for ZE (T ) in the following result.

PROPOSITION 14.1

Let E be an elliptic curve deļ¬ned over Fq , and let #E(Fq ) = q + 1 ā’ a. Then

qT 2 ā’ aT + 1

ZE (T ) = .

(1 ā’ T )(1 ā’ qT )

Factor X 2 ā’ aX + q = (X ā’ Ī±)(X ā’ Ī²). Theorem 4.12 says that

PROOF

Nn = q n + 1 ā’ Ī±n ā’ Ī² n .

429

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430 CHAPTER 14 ZETA FUNCTIONS

tn /n, we have

Therefore, using the expansion ā’ log(1 ā’ t) =

ā

Nn n

ZE (T ) = exp T

n

n=1

ā

Tn

(q n + 1 ā’ Ī±n ā’ Ī² n )

= exp

n

n=1

= exp (ā’ log(1 ā’ qT ) ā’ log(1 ā’ T ) + log(1 ā’ Ī±T ) + log(1 ā’ Ī²T ))

(1 ā’ Ī±T )(1 ā’ Ī²T )

=

(1 ā’ T )(1 ā’ qT )

qT 2 ā’ aT + 1

= .

(1 ā’ T )(1 ā’ qT )

Note that the numerator of ZE (T ) is the characteristic polynomial of the

Frobenius endomorphism, as in Chapter 4, with the coeļ¬cients in reverse

order.

A function ZC (T ) can be deļ¬ned in a similar way for any curve C over a

ļ¬nite ļ¬eld, and, more generally, for any variety over a ļ¬nite ļ¬eld. It is always

a rational function (proved by E. Artin and F. K. Schmidt for curves and by

Dwork for varieties).

The zeta function of E is deļ¬ned to be

Ī¶E (s) = ZE (q ā’s ),

where s is a complex variable. As weā™ll see below, Ī¶E (s) can be regarded as

an analogue of the classical Riemann zeta function

ā

1

Ī¶(s) = .

ns

n=1

One of the important properties of the Riemann zeta function is that it sat-

isļ¬es a functional equation relating the values at s and 1 ā’ s:

Ļ ā’s/2 Ī“(s/2)Ī¶(s) = Ļ ā’(1ā’s)/2 Ī“((1 ā’ s)/2)Ī¶(1 ā’ s).

A famous conjecture for Ī¶(s) is the Riemann Hypothesis, which predicts that

if Ī¶(s) = 0 with 0 ā¤ (s) ā¤ 1 then (s) = 1/2 (there are also the ātrivialā

zeros at the negative even integers). The elliptic curve zeta function Ī¶E (s) also

satisļ¬es a functional equation, and the analogue of the Riemann Hypothesis

holds.

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431

SECTION 14.1 ELLIPTIC CURVES OVER FINITE FIELDS

THEOREM 14.2

Let E be an elliptic curve deļ¬ned over a ļ¬nite ļ¬eld.

1. Ī¶E (s) = Ī¶E (1 ā’ s)

2. If Ī¶E (s) = 0, then (s) = 1/2.

PROOF The proof of the ļ¬rst statement follows easily from Proposi-

tion 14.1:

q 1ā’2s ā’ aq ā’s + 1

Ī¶E (s) =

(1 ā’ q ā’s )(1 ā’ q 1ā’s )

1 ā’ aq sā’1 + q ā’1+2s

=

(q s ā’ 1)(q sā’1 ā’ 1)

= Ī¶E (1 ā’ s).

Since the numerator of ZE (T ) is (1 ā’ Ī±T )(1 ā’ Ī²T ), we have

Ī¶E (s) = 0 āā’ q s = Ī± or Ī².

By the quadratic formula,

aĀ± a2 ā’ 4q

Ī±, Ī² = .

2

Hasseā™s theorem (Theorem 4.2) says that

ā

|a| ā¤ 2 q,

hence a2 ā’ 4q ā¤ 0. Therefore, Ī± and Ī² are complex conjugates of each other,

and

ā

|Ī±| = |Ī²| = q.

If q s = Ī± or Ī², then

ā

= |q s | =

(s)

q q.

Therefore, (s) = 1/2.

There are inļ¬nitely many solutions to q s = Ī±. However, if s0 is one such

solution, all others are of the form s0 + 2Ļin/ log q with n ā Z. A similar

situation holds for Ī².

If C is a curve, or a variety, over a ļ¬nite ļ¬eld, then an analogue of Theo-

rem 14.2 holds. For curves, the functional equation was proved by E. Artin

and F. K. Schmidt, and the Riemann Hypothesis was proved by Weil in the

1940s. In 1949, Weil announced what became known as the Weil conjectures,

which predicted that analogues of Proposition 14.1 and Theorem 14.2 hold for

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432 CHAPTER 14 ZETA FUNCTIONS

varieties over ļ¬nite ļ¬elds. The functional equation was proved in the 1960s

by M. Artin, Grothendieck, and Verdier, and the analogue of the Riemann

Hypothesis was proved by Deligne in 1973. Much of Grothendieckā™s algebraic

geometry was developed for the purpose of proving these conjectures.

Finally, we show how Ī¶E (s) can be deļ¬ned in a way similar to the Riemann

zeta function. Recall that the Riemann zeta function has the Euler product

expansion

ā’1

1

1ā’ s

Ī¶(s) =

p

p

when (s) > 1. The product is over the prime numbers. We obtain Ī¶E (s) if

we replace the primes p by points on E. Consider a point P ā E(Fq ). Deļ¬ne

deg(P ) to be the smallest n such that P ā E(Fqn ). The Frobenius map Ļq

acts on P , and it is not diļ¬cult to show that the set

SP = {P, Ļq (P ), Ļ2 (P ), . . . , Ļnā’1 (P )}

q q

has exactly n = deg(P ) elements and that Ļn (P ) = P . Each of the points in

q

SP also has degree n.

PROPOSITION 14.3

Let E be an elliptic curve over Fq . Then

ā’1

1

1ā’

Ī¶E (s) = ,

q s deg(P )

SP

where the product is over the points P ā E(Fq ), but we take only one point

from each set SP .

PROOF If deg(P ) = m, then P and all the other points in SP have

coordinates in Fqm . Since Fqm ā Fqn if and only if m|n, we see that SP

contributes m points to Nn = #E(Fqn ) if and only if m|n, and otherwise it

contributes no points to Nn . Therefore,

Nn = m.

m|n SP

deg(P )=m

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433

SECTION 14.2 ELLIPTIC CURVES OVER Q

Substituting this into the deļ¬nition of Z(T ), we obtain

ā

Nn n

log Z(T ) = T

n

n=1

ā

1n

= T m

n

n=1 m|n SP

deg(P )=m

ā ā

1

mT mj

= (where mj = n)

mj

j=1 m=1 SP

deg(P )=m

ā

1j deg(P )

= T

j

j=1 SP

=ā’ log(1 ā’ T deg(P ) ).

SP

Let T = q ā’s and exponentiate to obtain the result.

14.2 Elliptic Curves over Q

Let E be an elliptic curve deļ¬ned over Q. By changing variables if necessary,

we may assume that E is deļ¬ned by y 2 = x3 + Ax + B with A, B ā Z. For

a prime p, we can reduce the equation y 2 = x3 + Ax + B mod p. If E mod

p is an elliptic curve, then we say that E has good reduction mod p. This

happens for all but ļ¬nitely many primes. For each such p, we have

#E(Fp ) = p + 1 ā’ ap ,

as in Section 14.1. The L-function of E is deļ¬ned to be approximately the

Euler product

ā’1

1 ā’ ap pā’s + p1ā’2s .

good p

This deļ¬nition is good enough for many purposes. However, for completeness,

we say a few words below about what happens at the primes of bad reduction.

The factor 1ā’ap pā’s +p1ā’2s perhaps seems to be rather artiļ¬cially constructed.

However, it is just the numerator of the zeta function for E mod p, as in

Section 14.1. It might seem more natural to use the whole mod p zeta function,

but the factors arising from the denominator yield the Riemann zeta function

(with a few factors removed) evaluated at s and at s + 1. Since the presence

of the zeta function would complicate matters, the denominators are omitted

in the deļ¬nition of LE (s).

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434 CHAPTER 14 ZETA FUNCTIONS

For the primes where there is bad reduction, the cubic x3 + Ax + B has

multiple roots mod p. If it has a triple root, we say that E has additive

reduction mod p. If it has a double root mod p, it has multiplicative re-

duction. Moreover, if the slopes of the tangent lines at the singular point (see

Theorem 2.31) are in Fp , we say that E has split multiplicative reduction

mod p. Otherwise, it has nonsplit multiplicative reduction.

To treat the primes p = 2 and p = 3, we need to use the general Weierstrass

form for E. For simplicity, we have ignored these primes in the preceding

discussion. However, in the example below, weā™ll include them.

There are many possible equations for E with A, B ā Z. We assume that

A, B are chosen so that the reduction properties of E are as good as possible.

In other words, we assume that A and B are chosen so that the cubic has the

largest obtainable number of distinct roots mod p, and the power of p in the

discriminant 4A3 + 27B 2 is as small as possible, for each p. It can be shown

that there is such a choice of A, B. Such an equation is called a minimal

Weierstrass equation for E.

Example 14.1

Suppose we start with E given by the equation

y 2 = x3 ā’ 270000x + 128250000.

The discriminant of the cubic is ā’28 312 512 11, so E has good reduction except

possibly at 2, 3, 5, 11. The change of variables

x = 25x1 , y = 125y1

transforms the equation into

y1 = x3 ā’ 432x1 + 8208.

2

1

The discriminant of the cubic is ā’28 312 11, so E also has good reduction at

5. This is as far as we can go with the standard Weierstrass model. To treat

2 and 3 we need to allow generalized Weierstrass equations. The change of

variables

x1 = 9x2 ā’ 12, y1 = 27y2

changes the equation to

y2 = x3 ā’ 4x2 + 16.

2

2 2

The discriminant of the cubic is ā’28 11, so E has good reduction at 3. Since

any change of variables can be shown to change the discriminant by a square,

this is the best we can do, except possibly at the prime 2. The change of

variables

x2 = 4x3 , y2 = 8y3 + 4

changes the equation of E to

y3 + y3 = x3 ā’ x2 .

2

3 3

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435

SECTION 14.2 ELLIPTIC CURVES OVER Q

This is nonsingular at 2 (since the partial derivative with respect to y is

2y + 1 ā” 1 ā” 0 (mod 2)). Therefore, E has good reduction at 2. We conclude

that E has good reduction at all primes except p = 11, where it has bad

reduction. The equation y3 +y 3 = x3 ā’x2 is the minimal Weierstrass equation

2

3 3

for E.

Letā™s analyze the situation at 11 more closely. The polynomial in x2 factors

as

x3 ā’ 4x2 + 16 = (x2 + 1)2 (x2 + 5).

2 2

Therefore, E has multiplicative reduction at 11. The method of Section 2.10

shows that the slopes of the tangent lines at the singular point (x2 , y2 ) =

(ā’1, 0) are Ā±2, which lie in F11 . Therefore, E has split multiplicative reduc-

tion at 11.

We now give the full deļ¬nition of the L-series of E. For a prime p of bad

reduction, deļ¬ne

ā§

āØ 0 if E has additive reduction at p

1 if E has split multiplicative reduction at p

ap =

ā©

ā’1 if E has nonsplit multiplicative reduction at p.

The numbers ap for primes of good reduction are those given above: ap =

p + 1 ā’ #E(Fp ). Then the L-function of E is the Euler product

ā’1

ā’1

1 ā’ ap pā’s 1 ā’ ap pā’s + p1ā’2s

LE (s) = .

bad p good p

ā

The estimate |ap | < 2 p easily implies that the product converges for (s) >

3/2 (see Exercise 14.3).

Each good factor can be expanded in the form

(1 ā’ ap pā’s + p1ā’2s )ā’1 = 1 + ap pā’s + ap2 pā’2s + Ā· Ā· Ā· ,

where the ap on the left equals the ap on the right (so this is not bad notation)

and

ap2 = a2 ā’ p. (14.1)

p

The product over all p yields an expression

ā

an nā’s .

LE (s) =

n=1

e

pj j , then

If n = j

an = apej . (14.2)

j

j

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436 CHAPTER 14 ZETA FUNCTIONS

This series for LE (s) converges for (s) > 3/2. It is natural to ask whether

LE (s) has an analytic continuation to all of C and a functional equation, as is

the case with the Riemann zeta function. As weā™ll discuss below, the answer

to these questions is yes. However, the proof is much too deep to be included

in this book (but see Chapter 15 for a discussion of the proof).

To study the analytic properties of LE (s), we introduce a new function.

Let Ļ„ ā H, the upper half of the complex plane, as in Chapter 9, and let

q = e2ĻiĻ„ . (This is the standard notation; there should be no possibility of

confusion with the q for ļ¬nite ļ¬elds of Chapter 4.) Deļ¬ne

ā

an q n .

fE (Ļ„ ) =

n=1

This is simply a generating function that encodes the number of points on E

mod the various primes. It converges for Ļ„ ā H and satisļ¬es some amazing

properties.

Let N be a positive integer and deļ¬ne

ab

ā SL2 (Z) c ā” 0

Ī“0 (N ) = (mod N ) .

cd

Then Ī“0 (N ) is a subgroup of SL2 (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 deļ¬ned over Q. There exists an integer N such

that, for all Ļ„ ā H,

1.

aĻ„ + b ab

ā Ī“0 (N )

= (cĻ„ + d)2 fE (Ļ„ ) for all

fE

cd

cĻ„ + d

2.

fE (ā’1/(N Ļ„ )) = Ā±N Ļ„ 2 fE (Ļ„ ).

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

fE (Ļ„ ) is a modular form (in fact, a cusp form; see Section 15.2) of weight

2 and level 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 ļ¬rst power. If E has additive

reduction and p > 3, then p2 is the exact power of p dividing N . The formulas

for p = 2 and 3 are slightly more complicated in this case. See [117].

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437

SECTION 14.2 ELLIPTIC CURVES OVER Q

The transformation law in (1) can be rewritten as

aĻ„ + b aĻ„ + b

fE d = fE (Ļ„ ) dĻ„

cĻ„ + d cĻ„ + d

(this is bad notation: d represents both an integer and the diļ¬erentiation

operator; it should be clear which is which). Therefore,

fE (Ļ„ ) dĻ„

is a diļ¬erential that is invariant under the action of Ī“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 fE lies in either the plus space or the minus

space.

Taniyama ļ¬rst 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 coeļ¬cients

an are integers, then there is an elliptic curve E with fE = f . This is the

converse of the theorem, but it gave the ļ¬rst real evidence that Taniyamaā™s

suggestion was reasonable. In 1967, Weil made precise what the integer N

must be for any given elliptic curve. Since there are only ļ¬nitely many modu-

lar forms f of a given level N that could arise from elliptic curves, this meant

that the conjecture (Taniyamaā™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 coeļ¬cients to see that

fE was not on the ļ¬nite list of possibilities. Moreover, Weil showed that if

functions like LE (s) (namely LE and its twists) have analytic continuations

and functional equations such as the one given in Corollary 14.5 below, then

fE must be a modular form. Since most people believe that naturally deļ¬ned

L-functions should have analytic continuations and functional equations, this

gave the conjecture more credence. Around 1990, Wiles proved that there are

inļ¬nitely 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.

Letā™s assume Theorem 14.4 and show that LE (s) analytically continues and

satisļ¬es a functional equation. Recall that the gamma function is deļ¬ned

for (s) > 0 by

ā

tsā’1 eā’t dt.

Ī“(s) =

0

Integration by parts yields the relation sĪ“(s) = Ī“(s + 1), which yields the

meromorphic continuation of Ī“(s) to the complex plane, with poles at the

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438 CHAPTER 14 ZETA FUNCTIONS

nonpositive integers. It also yields the relation Ī“(n) = (n ā’ 1)! for positive

integers n.

COROLLARY 14.5

Let E and N be as in Theorem 14.4. Then

ā ā

( N /2Ļ)s Ī“(s)LE (s) = ā“( N /2Ļ)2ā’s Ī“(2 ā’ s)LE (2 ā’ s)

for all s ā C (and both sides continue analytically to all of C). The sign here

is the opposite of the sign in (2) of Theorem 14.4.

PROOF Using the deļ¬nition of the gamma function, we have

ā

ā ā ā

s s

tsā’1 eā’t dt

( N /2Ļ) Ī“(s)LE (s) = an ( N /2Ļn)

0

n=1

ā

ā

ā

du

(u N )s eā’2Ļnu

= an (let t = 2Ļnu)

u

0

n=1

ā

ā

du

(u N )s fE (iu)

=

u

0

ā

ā ā

1/ N ā

du du

(u N )s fE (iu) (u N )s fE (iu) .

+

= ā

u u

1/ N

0

(The interchange of summation and integration to obtain the third equality

is justiļ¬ed since the sum for f (iu) converges very quickly near ā.) Let be

the sign in part (2) of Theorem 14.4. Then

fE (i/(N u)) = (iu)2 fE (iu) = ā’ u2 fE (iu).

Therefore, let u = 1/N v to obtain

ā

ā ā

1/ N ā

du dv

s

=ā’ (v N )2ā’s fE (iv) .

(u N ) fE (iu) ā

u v

1/ N

0

This implies that

ā

( N /2Ļ)s Ī“(s)LE (s) =

ā ā

ā ā

du dv

s

ā’ (v N )2ā’s fE (iv) .

ā (u N ) fE (iu) ā

u v

1/ N 1/ N

Since f (iu) ā’ 0 exponentially as u ā’ ā, it follows easily that both integrals

converge and deļ¬ne analytic functions of s. Under s ā’ 2 ā’ s, the right side,

hence the left side, is multiplied by ā’ . This is precisely what the functional

equation claims.

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439

SECTION 14.2 ELLIPTIC CURVES OVER Q

Example 14.2

Let E be the elliptic curve y 2 +y = x3 ā’x2 considered in the previous example.

If we compute the number Np of points on E mod p for various primes, we

obtain, with ap = p + 1 ā’ Np ,

a2 = ā’2, a3 = ā’1, a7 = ā’2,

a5 = 1, a13 = 4, . . .

(except for p = 2, 3, 5, the numbers ap can be calculated using any of the

equations in the previous example). The value

a11 = 1

is speciļ¬ed by the formulas for bad primes. We then calculate the coeļ¬cients

for composite indices. For example,

a4 = a2 ā’ 2 = 2

a6 = a2 a3 = 2, 2

(see (14.2) and (14.1)). Therefore,

fE (Ļ„ ) = q ā’ 2q 2 ā’ q 3 + 2q 4 + q 5 + 2q 6 ā’ 2q 7 + Ā· Ā· Ā· .

It can be shown that

ā

(1 ā’ q j )2 (1 ā’ q 11j )2

f (Ļ„ ) = q

j=1

is a cusp form of weight 2 and level N = 11. In fact, it is the only such form,

up to scalar multiples. The product for f can be expanded into an inļ¬nite

series

f (Ļ„ ) = q ā’ 2q 2 ā’ q 3 + 2q 4 + q 5 + 2q 6 ā’ 2q 7 + Ā· Ā· Ā· .

It can be shown that f = fE (see [61]).

The L-series for E satisļ¬es the functional equation

ā ā

( 11/2Ļ)s Ī“(s)LE (s) = +( 11/2Ļ)2ā’s Ī“(2 ā’ s)LE (2 ā’ s).

In the early 1960s, Birch and Swinnerton-Dyer performed computer exper-

iments to try to understand the relation between the number of points on

an elliptic curve mod p as p ranges through the primes and the number of

rational points on the curve. Ignoring the fact that the product for LE (s)

doesnā™t converge at s = 1, letā™s substitute s = 1 into the product (weā™ll ignore

the ļ¬nitely many bad primes):

ā’1

p ā’ ap + 1 p

ā’1 ā’1

ā’1

1 ā’ ap p +p = = .

p Np

p p p

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440 CHAPTER 14 ZETA FUNCTIONS

If E has a lot of points mod p for many p, then many factors in the product are

small, so we expect that LE (1) might be small. In fact, the data that Birch

and Swinnerton-Dyer obtained led them to make the following conjecture.

CONJECTURE 14.6 (Conjecture of Birch and Swinnerton-Dyer,

Weak Form)

Let E be an elliptic curve deļ¬ned over Q. The order of vanishing of LE (s)

at s = 1 is the rank r of E(Q). In other words, if E(Q) torsion ā• Zr , then

LE (s) = (s ā’ 1)r g(s), with g(1) = 0, ā.

One consequence of the conjecture is that E(Q) is inļ¬nite if and only if

LE (1) = 0. This statement remains unproved, although there has been some

progress. In 1977, Coates and Wiles showed that if E has complex multipli-

cation and has a point of inļ¬nite order, then LE (1) = 0. The results of Gross

and Zagier on Heegner points (1983) imply that if E is an elliptic curve over

Q such that LE (s) vanishes to order exactly 1 at s = 1, then there is a point

of inļ¬nite order. However, if LE (s) vanishes to order higher than 1, nothing

has been proved, even though there is conjecturally an abundance of points

of inļ¬nite order. This is a common situation in mathematics. It seems that a

solution is often easier to ļ¬nd when it is essentially unique than when there

are many choices.

Soon, Conjecture 14.6 was reļ¬ned to give not only the order of vanishing,

but also the leading coeļ¬cient of the expansion at s = 1. To state the

conjecture, we need to introduce some notation. If P1 , . . . , Pr form a basis for

the free part of E(Q), then

E(Q) = E(Q)torsion ā• Z P1 ā• Ā· Ā· Ā· ā• Z Pr .

Recall the height pairing P, Q deļ¬ned in Section 8.5. We can form the r Ć— r

matrix Pi , Pj and compute its determinant to obtain what is known as the

elliptic regulator for E. If r = 0, deļ¬ne this determinant to equal 1. Let

Ļ1 , Ļ2 be a basis of a lattice in C that corresponds to E by Theorem 9.21.

We may assume that Ļ2 ā R, by Exercise 9.5. If E[2] ā‚ E(R), let ā„¦ = 2Ļ2 .

Otherwise, let ā„¦ = Ļ2 . For each prime p, there are integers cp that we wonā™t

deļ¬ne, except to say that if p is a prime of good reduction then cp = 1.

A formula for computing them is given [117]. Finally, recall that is the

(conjecturally ļ¬nite) Shafarevich-Tate group of E.

CONJECTURE 14.7 (Conjecture of Birch and Swinnerton-Dyer)

Let E be an elliptic curve deļ¬ned over Q. Let r be the rank of E(Q). Then

ā„¦ p cp (# E ) det Pi , Pj

r

+ (s ā’ 1)r+1 (br+1 + Ā· Ā· Ā· ).

LE (s) = (s ā’ 1)

#E(Q)2

torsion

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441

SECTION 14.2 ELLIPTIC CURVES OVER Q

This important conjecture combines most of the important information

about E into one equation. When it was ļ¬rst made, there were no exam-

ples. As Tate pointed out in 1974 ([116, p. 198]),

This remarkable conjecture relates the behavior of a function L at

a point where it is not at present known to be deļ¬ned to the order

which is not known to be ļ¬nite!

of a group

In 1986, Rubin gave the ļ¬rst examples of curves with ļ¬nite , and was able to

compute the exact order of in several examples. Since they were complex

multiplication curves, LE (1) could be computed explicitly by known formulas

(these had been used by Birch and Swinnerton-Dyer in their calculations), and

this allowed the conjecture to be veriļ¬ed for these curves. Soon thereafter,

Kolyvagin obtained similar results for elliptic curves satisfying Theorem 14.4

(which was not yet proved) such that LE (s) vanishes to order at most 1 at

s = 1. Therefore, the conjecture is mostly proved (up to small rational factors)

when LE (s) vanishes to order at most one at s = 1. In general, nothing is

known when LE (s) vanishes to higher order. In fact, it is not ruled out (but

most people believe itā™s very unlikely) that LE (s) could vanish at s = 1 to

very high order even though E(Q) has rank 0 or 1.

In 2000, the Clay Mathematics Institute listed the Conjecture of Birch and

Swinnerton-Dyer as one of its million dollar problems. There are surely easier

(but certainly less satisfying) ways to earn a million dollars.

For those who know some algebraic number theory, the conjecture is very

similar to the analytic class number formula. For an imaginary quadratic ļ¬eld

K, the zeta function of K satisļ¬es

2Ļh

Ī¶K (s) = (s ā’ 1)ā’1 + Ā·Ā·Ā· ,

|d|

w

where h is the class number of K, d is the discriminant of K, and w is the

number of roots of unity in K. Conjecture 14.7 for a curve of rank r = 0

predicts that

ā„¦ p cp # E

+ Ā·Ā·Ā· .

LE (s) =

#E(Q)2torsion

The group E can be regarded as the analogue of the ideal class group, the

number ā„¦ p cp plays the role of 2Ļ/ |d|, and #E(Q)torsion is the analogue

of w. Except for the square on the order of the torsion group, the two formulas

for the leading coeļ¬cients have very similar forms.

Now letā™s look at real quadratic ļ¬elds K. The class number formula says

that

4h log(Ī·)

Ī¶K (s) = (s ā’ 1)ā’1 ā + Ā·Ā·Ā· ,

2d

where h is the class number of K, d is the discriminant, and Ī· is the fun-

damental unit. The Conjecture of Birch and Swinnerton-Dyer for a curve of

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442 CHAPTER 14 ZETA FUNCTIONS

rank r = 1, with generator P , predicts that

Ė

ā„¦ p cp (# E ) h(P )

LE (s) = (s ā’ 1) + Ā·Ā·Ā· .

#E(Q)2

torsion

ā

In this case, ā„¦ is the analogue of 4/ d and #E(Q)torsion plays the role of 2,

Ė

which is the number of roots of unity in K. The height h(P ) gives the size of

P . Similarly, log(Ī·) gives the size of Ī·.

In general, we can write down a dictionary between elliptic curves and

number ļ¬elds:

elliptic curves āā’ number ļ¬elds

points āā’ units

torsion points āā’ roots of unity

Shafarevich-Tate group āā’ ideal class group

This is not an exact dictionary, but it helps to interpret results in one area

in terms of the other. For example, the Dirichlet unit theorem in algebraic

number theory, which describes the group of units in a number ļ¬eld, is the

analogue of the Mordell-Weil theorem, which describes the group of rational

points on an elliptic curve. The ļ¬niteness of the ideal class group in algebraic

number theory is the analogue of the conjectured ļ¬niteness of the Shafarevich-

Tate group.

Exercises

14.1 Let P1 be one-dimensional projective space.

(a) Show that the number of points in P1 (Fq ) is q + 1.

(b) Let Nn = #P1 (Fqn ). Deļ¬ne the Z-function for P1 by

ā

Nn n

ZP1 (T ) = exp T .

n

n=1

Show that

1

ZP1 (T ) = .

(1 ā’ T )(1 ā’ qT )

ab

ā GL2 (R) with det(M ) > 0. Deļ¬ne an action of M

14.2 Let M =

cd

on functions on H by

(f |M )(z) = det(M )(cz + d)ā’2 f (M z),

az+b

where M z = cz+d .

Ā© 2008 by Taylor & Francis Group, LLC

443

EXERCISES

(a) Show that (f |M1 )|M2 = f |(M1 M2 ).

0 ā’1

. Show that W Ī“0 (N ) W ā’1 = Ī“0 (N ).

(b) Let W =

N0

(c) Suppose that f is a function with f |M = f for all M ā Ī“0 (N ). Let

g(z) = (f |W )(z). Show that g|M = g for all M ā Ī“0 (N ). (Hint:

Combine parts (a) and (b).)

(d) Suppose that f is a function with f |M = f for all M ā Ī“0 (N ). Let

f + = 1 (f + f |W ) and f ā’ = 1 (f ā’ f |W ). Show that f + |W = f +

2 2

and f ā’ |W = ā’f ā’ . This gives a decomposition f = f + + f ā’ in

which f is written as a sum of two eigenfunctions for W .

|bn | converges.

14.3 It is well known that a product (1 + bn ) converges if

Use this fact, plus Hasseā™s theorem, to show that the Euler product

deļ¬ning LE (s) converges for (s) > 3/2.

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