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Report on the Birch and SwinnertonDyer conjecture
Massimo Bertolini
February 26, 2010
Contents
1 The basic problem and the MordellWeil theorem 2
2 The Lseries of an elliptic curve 7
3 Digression on conics 9
4 Statement of the Birch and SwinnertonDyer conjecture 10
5 Results on the Birch and SwinnertonDyer conjecture 12
5.1 Modularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.2 Heegner points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.3 The theorems of GrossZagierZhang and Kolyvagin . . . . . . . . . . . . . . 18
5.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.5 Reﬁnements and padic methods . . . . . . . . . . . . . . . . . . . . . . . . . 22
Abstract
This article begins with a description of the basic problems in the arithmetic theory
of elliptic curves, which motivate the statement of the Birch and SwinnertonDyer
conjecture. It then gives an account of recent results on the subject.
Keywords: Elliptic curves, Lseries, padic Lfunctions, ShafarevichTate groups, Main Con
jectures of Iwasawa theory.
Mathematics subject classiﬁcation: 11G40 (11G05, 11G15, 11F33, 11F67)
Introduction
This note grew out of a series of two lectures on the Birch and SwinnertonDyer conjecture,
delivered by the author at the “Riemann School” held at Verbania (Italy) in April 2009.
1
The Birch and SwinnertonDyer conjecture is considered one of the fundamental problems
in modern Mathematics. See for example the description of the “Millenium Problems”
formulated by the Clay Mathematics Institute, and in particular Wiles’ paper [Wi2].
The ﬁrst section of this paper reviews some basic material on the arithmetic theory of
elliptic curves, and includes details on the proof of the MordellWeil theorem on rational
points.
The second section recalls the deﬁnition and the various properties of the Lseries of an
elliptic curve, focusing for simplicity on the case of curves deﬁned over the rational numbers.
The third section explains the analogy with the simpler context of FermatPell equations,
in which diﬃcult questions on elliptic curves correspond to statements following directly from
classical results in algebraic number theory. This section provides some motivation for the
statement of the Birch and SwinnertonDyer conjecture (over the rationals), given in section
4.
The last section – somewhat more advanced than the previous ones – gives an uptodate
account of some of the main results on the Birch and SwinnertonDyer conjecture. It starts
by reviewing the theory of modular parametrisations of elliptic curves by Shimura curves,
and the construction of Heegner points arising from the theory of complex multiplication
of abelian varieties. It proceeds by stating the fundamental theorems of GrossZagier and
Zhang on derivatives of Rankin Lseries, and of Kolyvagin on bounding Selmer groups. A
proof of a special case of these results, following the methods of [BD3], is explained in some
detail. A brief discussion of padic methods, in connection with recent applications to the
Birch and SwinnertonDyer conjecture due to Kato and SkinnerUrban, concludes section 5.
1 The basic problem and the MordellWeil theorem
We begin by recalling some basic facts on elliptic curves. (See [Sil] for details.)
Deﬁnition 1.1. An elliptic curve E over a ﬁeld K is a nonsingular projective curve of genus
one, equipped with a Krational point.
By the RiemannRoch theorem, an elliptic curve can be described explicitly as the pro
jective nonsingular plane curve deﬁned by a cubic Weierstrass equation of the form
y
2
+ a
1
xy + a
3
y = x
3
+ a
2
x
2
+ a
4
x + a
6
. (1)
Here the coeﬃcients a
i
belong to K and the discriminant of the equation is nonzero. If the
characteristic of K is diﬀerent from 2 and 3, we may assume that equation (1) takes the
form
y
2
= x
3
+ ax + b, a, b ∈ K, with 4a
3
+ 27b
2
6= 0.
Equivalently, an elliptic curve can also be deﬁned as a projective algebraic group of
dimension one over K. It follows that the set E(K) of Krational points of E carries a
natural geometric structure of abelian group.
More concretely, assuming for simplicity that char(
K
)
6
= 2
,
3, the set
E
(
K
) is given by
E(K) = {P = (x
0
, y
0
) ∈ K × K : y
2
0
= x
3
0
+ ax
0
+ b} ∪ {O
E
},
2
where O
E
= [0, 1, 0] denotes the unique point at inﬁnity on E. Then E(K) is endowed with
the structure of abelian group with origin O
E
by the socalled “chord and tangent rule”:
given P , Q and R in E(K), then P + Q + R = O
E
if and only if P , Q and R are the points
of intersection (counting multiplicities) of E with a projective line.
Assume from now on that K is a number ﬁeld (i.e., a ﬁnite extension of Q). The funda
mental question in the arithmetic theory of elliptic curves is the following:
Problem 1.2. Describe the group E(K) of rational points on E.
The ﬁrst deep result on E(K) is the MordellWeil theorem, which brilliantly generalises
the descent method of Fermat.
Theorem 1.3 (MordellWeil). The abelian group E(K) is ﬁnitely generated.
Remark 1.4. Theorem 1.3 states that E(K) is isomorphic to a group of the form Z
r
E
⊕T ,
where T is a ﬁnite abelian group and r
E
is a nonnegative integer, called the rank of E.
The group T – the torsion of E – can be easily calculated for a given E. Furthermore,
a theorem of Merel [Me], extending previous work of Mazur [Maz1], states that the order
of T is uniformly bounded in terms of the degree [K : Q] of the ﬁeld of deﬁnition of E.
On the other hand, no eﬀective algorithm to determine r
E
is known, and the behaviour of
the function E 7→ r
E
– with E varying in the family of elliptic curves over K – is poorly
understood. A folklore conjecture states that this function is unbounded.
Proof. (Sketch) We sketch the main steps in the proof of Theorem 1.3, following the book
of Silverman [Sil], to which we refer the reader for more details.
Step 1. It consists in proving the following:
Theorem 1.5 (Descent theorem). If E(K)/mE(K) is ﬁnite for some m ≥ 2, then E(K) is
ﬁnitely generated.
The proof of Theorem 1.5 generalises the inﬁnite descent method, by showing the exis
tence of a height function h : E(K) −→ R satisfying the following properties:
1. for all Q ∈ E(K), there is a constant C
1
depending on Q (and on E) such that
h(P + Q) ≤ 2h(P ) + C
1
for all P ∈ E(K),
2. there is a constant C
2
(depending on E) such that h(mP ) ≥ m
2
h(P ) − C
2
for all
P ∈ E(K),
3. for any constant C
3
, the set {P ∈ E(K) : h(P ) ≤ C
3
} is ﬁnite.
Roughly speaking, the existence of a height function prevents the possibility of indeﬁnitely
dividing a point in E(K). This fact implies Theorem 1.5.
When K = Q, a height function can be deﬁned on aﬃne points P = (x, y) in E(Q), with
x = r/s such that (r, s) = 1, by the formula
h(P ) = log(max{r, s}).
3
One then extends h at the point at inﬁnity by setting h(O
E
) = 0.
The function h can be turned into a quadratic function, called the canonical N´eronTate
height, by the formula
ˆ
h
NT
(P ) := (1 /2) lim
n→∞
4
−n
h(2
n
P ), P ∈ E(Q).
The function
ˆ
h
NT
, which plays an important role in the statement of the Birch and
SwinnertonDyer conjecture, satisﬁes the following properties:
1. 2
ˆ
h
NT
(P ) − h(P ) = O(1),
2.
ˆ
h
NT
(P ) ≥ 0 for all P , and
ˆ
h
NT
(P ) = 0 if and only if P is a torsion p oint,
3.
ˆ
h
NT
(mP ) = m
2
ˆ
h
NT
(P ),
4. hP, Qi
NT
:=
ˆ
h
NT
(P + Q) −
ˆ
h
NT
(P ) −
ˆ
h
NT
(Q) is bilinear.
The above discussion generalises to any number ﬁeld K, and gives rise to a canonical N´eron
Tate nondegenerate pairing
h , i
NT
: E(K)/E(K)
tors
× E(K)/E(K)
tors
−→ R, (2)
where E(K)
tors
denotes the torsion subgroup of E(K).
Step 2. It consists in proving:
Theorem 1.6 (Weak MordellWeil theorem). E(K)/mE(K) is ﬁnite for m ≥ 2.
Write M
K
= {v} for the set of places of K, and K
v
for the completion of K at a place
v. (In the special case where K = Q, the set M
Q
is identiﬁed with the set rational primes
p together with the symbol ∞, so that Q
p
is the ﬁeld of padic numbers and Q
∞
is the ﬁeld
R of real numbers.) Set
G
K
:= Gal(
¯
K/K), G
K
v
= Gal(
¯
K
v
/K
v
).
Here G
K
v
is viewed as a subgroup of G
K
, by ﬁxing an extension of v to
¯
K and thus identifying
G
K
v
with a decomposition group at v. Recall that G
K
and G
K
v
are equipped with the
proﬁnite topology.
Let M denote the module E(
¯
K) or E
m
:= {P ∈ E(
¯
K) : mP = O
E
}, equipped with
the discrete topology. Note that M carries a natural structure of continuous G
K
module.
Let H
1
(G
K
, M) be the continuous Galois cohomology group attached to M, deﬁned as the
quotient the group of continuous 1cocycles from G
K
to M by the group of 1coboundaries.
Note that if the action of G
K
on M is the trivial one, then H
1
(G
K
, M) is identiﬁed with the
group Hom
cont
(G
K
, M) of continuous homomorphisms from G
K
to M. The local counterparts
H
1
(G
K
v
, M
v
) of H
1
(G
K
, M), with M
v
equal to E(
¯
K
v
) or to its submodule E
m
of mtorsion
points, are deﬁned in a similar way.
4
There is an exact sequence of discrete G
K
modules
0−→E
m
−→E(
¯
K)
[m]
−→ E(
¯
K)−→0,
where [m] is the morphism of multiplication by m on E. It induces a long exact sequence in
cohomology, from which one deduces the exact sequence
0 −→ E(K)/mE(K)
δ
−→ H
1
(G
K
, E
m
)−→H
1
(G
K
, E(
¯
K))
m
−→0, (3)
in which H
1
(G
K
, E(
¯
K))
m
stands for the mtorsion in H
1
(G
K
, E(
¯
K)). The coboundary map
δ is deﬁned as follows: given P ∈ E(K), choose Q ∈ E(
¯
K) such that [m]Q = P , and set
δ(P ) to be the class of the 1cocyle σ 7→ Q
σ
− Q.
The strategy of proof of the weak MordellWeil theorem amounts to showing that the im
age of E(K)/mE(K) by δ is contained in a ﬁnite subgroup of the torsion group H
1
(G
K
, E
m
)
(which is not ﬁnitely generated). To this purpose, deﬁne the mSelmer group of E/K to be
the kernel
Sel
m
(E/K) = ker
¡
H
1
(G
K
, E
m
)−→
Y
v
H
1
(G
K
v
, E(
¯
K
v
))
m
¢
(4)
of the natural morphism induced by restriction of cocycles. The ShafarevichTate group of
E/K is deﬁned to be
III(E/K) := ker
¡
H
1
(G
K
, E(
¯
K))−→
Y
v
H
1
(G
K
v
, E(
¯
K
v
))
¢
. (5)
In view of deﬁnitions (4) and (5), a diagram chase involving the exact sequence (3) and its
local counterparts, yields the exact descent sequence
0−→E(K)/mE( K)−→Sel
m
(E/K)−→III(E/K)
m
−→0. (6)
It now suﬃces to show the ﬁniteness of Sel
m
(E/K). Restriction of cocycles from G
K
to
Gal(
¯
K/K(E
m
)) deﬁnes a map
ρ : Sel
m
(E/K)−→Hom(Gal(
¯
K/K(E
m
)), E
m
),
whose target is a group of homomorphisms since E
m
is a trivial module for the action of
Gal(
¯
K/K(E
m
)).
One checks that:
1. Ker(ρ) is ﬁnite,
2. Im(ρ) ⊂ Hom(Gal(L/K(E
m
)), E
m
), where L is an abelian extension of K(E
m
) of
exponent m which is ramiﬁed at a ﬁnite set of primes.
(More precisely, the above extension is unramiﬁed at the nonarchimedean primes of K(E
m
)
which are of good reduction for E and do not divide m.)
Kummer theory, combined with Dirichlet’s unit theorem and the ﬁniteness of the class
group of K(E
m
), implies that L/K(E
m
) is ﬁnite and hence Sel
m
(E/K) is also ﬁnite.
5
Remark 1.7. Given a speciﬁc curve E/K, the mSelmer group Sel
m
(E/K) can in principle
be computed, as the proof of the weak MordellWeil theorem indicates. (For theoretical con
siderations on the Selmer group, it is convenient to refer to the duality techniques which are
described below.) However, this proof does not provide a method to describe the cohomology
classes in the Selmer group which are the image of points in E(K). As a consequence, one
cannot deduce an eﬀective algorithm for computing E(K). In this connection, we recall the
following fundamental:
Conjecture 1.8. III(E/K) is ﬁnite.
Duality in Galois cohomogy. We now elaborate on the cohomological study of the Selmer
and ShafarevichTate groups attached to E/K, and introduce the formalism which will b e
instrumental in proving the results on the Birch and SwinnertonDyer conjecture described
in Section 5.
Theorem 1.9 (Local Tate duality). For each place v, the cupproduct induces a non
degenerate pairing
h , i
v
: E(K
v
)/mE(K
v
) × H
1
(G
K
v
, E(
¯
K
v
))
m
−→Z/mZ.
See [Mi] for a proof of this result.
Theorem 1.10 (Global duality). Given s ∈ Sel
m
(E/K) and c ∈ H
1
(G
K
, E(
¯
K))
m
, write
res
v
(s) ∈ E(K
v
)/mE(K
v
) and ∂
v
(c) ∈ H
1
(G
K
v
, E(
¯
K
v
))
m
for the image of s and c, respec
tively, under the natural restriction maps. Then
X
v
hres
v
(s), ∂
v
(c)i
v
= 0.
Theorem 1.10 follows from the reciprocity law for the elements in the Brauer group of K.
See [CF] for a proof.
If S is any ﬁnite set of places, Theorem 1.9 implies the existence of a map
λ
S
: ⊕
v∈S
H
1
(K
v
, E(
¯
K
v
))
m
−→Sel
m
(E/K)
∗
:= Hom(Sel
m
(E/K), Z/mZ),
which is dual to the natural localisation map ⊕
v∈S
res
v
: Sel
m
(E/K)−→⊕
v∈S
E(K
v
)/mE(K
v
).
For suitable choices of S, the Chebotarev density theorem implies that λ
S
is surjective
(or equivalently, that the dual map is injective) . In this way, one obtains a system of
generators of Sel
m
(E/K)
∗
(and dually of Sel
m
(E/K)). Furthermore, given a set of classes
c in H
1
(G
K
, E(
¯
K))
m
such that ∂
v
(c) = 0 for v 6∈ S, Theorem 1.10 yields a set of relations
among the generators of Sel
m
(E/K)
∗
described above.
Question 1.11. Are there cases where one can produce as above enough generators and
relations to calculate Sel
m
(E/K) and, by letting m vary, prove that III(E/K) is ﬁnite?
Section 5 will provide an aﬃrmative answer to the above question.
6
Remark 1.12. The above discussion accounts for the state of the arithmetic theory of
elliptic curves at the beginning of the 1960’s. Speciﬁcally, at that time
1. not a single example of elliptic curve E/K such that III(E/K) is ﬁnite was known;
2. similarly, it was not known how to ﬁnd m such that III(E/K)
m
= 0 (although values
of m with this property could be calculated for speciﬁc curves);
3. on a related note, it was not known how to produce an eﬀective algorithm to compute
E(K);
4. on a simpler level, it was not known how to devise an eﬀective algorithm to determine
whether or not E(K) is inﬁnite.
At around that time, Birch and SwinnertonDyer introduced a fundamental new idea
in the theory, by formulating a conjectural relation between the arithmetic invariants of
E introduced above and the socalled HasseWeil Lseries L(E, s) of E. This Lseries is
described in Section 2.
2 The Lseries of an elliptic curve
In this section, we assume for simplicity that K = Q. Let E be an elliptic curve over Q,
and let N be the arithmetic conductor of E. Write E
p
for a minimal model of E over Z
p
(described by a Weierstrass equation with coeﬃcients in Z
p
whose discriminant has minimal
padic valuation), and
¯
E
p
for the special ﬁber of E
p
. Then:
1. a prime p does not divide N if and only if
¯
E
p
is smooth,
2. if p divides N exactly, then the (unique) singular point of
¯
E
p
is a node,
3. if p
2
divides N, then the singular point of
¯
E
p
is a cusp.
Write n
p
for the order of the group of F
p
rational nonsingular points of
¯
E
p
(all points in
¯
E
p
(F
p
) are nonsingular if p does not divide N). Then set a
p
to be equal to p + 1 − n
p
if p
does not divide N, and to p − n
p
otherwise. In the latter case, note that a
p
= 0 if
¯
E
p
has
a cusp, and a
p
= ±1 if
¯
E
p
has a node, with the plus sign holding when the tangents at the
node are deﬁned over F
p
.
Deﬁnition 2.1. The Lseries of E is the function of a complex variable s deﬁned by
L(E, s) =
Y
p6N
(1 − a
p
p
−s
+ p
1−2s
)
−1
Y
pN
(1 − a
p
p
−s
)
−1
.
7
Remark 2.2.
1) (Connection with torsion points) The Lseries of E can alternately be deﬁned in terms of
the torsion points of E (which played a crucial role in the deﬁnition of the Selmer group of
E, and in the proof of the MordellWeil theorem). Given a prime `, deﬁne
T
`
(E) = lim
←
E
`
n
, V
`
(E) = T
`
(E) ⊗
Z
`
Q
`
,
where the inverse limit is taken with respect to the natural projection maps. The modules
T
`
(E) and V
`
(E) have rank 2 over Z
`
and Q
`
, respectively, and are equipped with natu
ral continuous G
Q
actions. They are called the `adic Tate module and the `adic Galois
representation of E. When p  N`, then V
`
(E) is unramiﬁed at p, i.e., the action of the
decomposition group G
p
= G
Q
p
on V
`
(E) factors through G
p
/I
p
, where I
p
is the inertia
subgroup of G
p
. Moreover, the trace of the Frobenius morphism Frob
p
acting on V
`
(E) is
equal to the coeﬃcient a
p
deﬁned above. It follows that the factor at p in the deﬁnition of
L(E, s) can also be described as
det(1 − Frob
p
p
−s
; V
`
(E))
−1
.
This description, which similarly extends at the primes of bad reduction for E, shows that
L(E, s) can be deﬁned without resorting to the models E
p
of E over Z
p
. This point of view is
useful in the deﬁnition of the Lfunctions associated to the cohomology of algebraic varieties.
2) If E and E
0
are Qisogenous, then V
`
(E) and V
`
(E
0
) are isomorphic as G
Q
representations.
It follows from the remark above that L(E, s) is an invariant of the Qisogeny class of E.
Conversely, a theorem of Faltings [Fa] shows that the equality L(E, s) = L(E
0
, s) implies
that E and E
0
are related by a Qisogeny.
3) Deﬁnition 2.1 (and the above remarks) generalise directly to the case of an elliptic curve
E deﬁned over a number ﬁeld K. For example, one deﬁnes the Lseries of E over K by the
formula
L(E/K, s) =
Y
vN
(1 − a
v
Nv
−s
+ Nv
1−2s
)
−1
Y
vN
(1 − a
v
Nv
−s
)
−1
, (7)
where the product is taken over the nonarchimedean primes
v
of
K
, whose norm is denoted
Nv. Here, the ideal N of the ring of integers O
K
of K is the arithmetic conductor of E,
and the coeﬃcient a
v
is deﬁned similarly to a
p
. For example, if v  N is a prime of good
reduction, then a
v
= 1 + Nv − n
v
, where n
v
is the order of the group
¯
E
v
(F
v
) of F
v
rational
points of the special ﬁber of a minimal model E
v
of E over O
K
v
.
Properties of L(E, s):
i) The inﬁnite product deﬁning L(E, s) converges absolutely for <(s) > 3/2. This prop
erty is a direct consequence of the Hasse inequality a
p
 ≤ 2
√
p. This inequality follows from
the Riemann hypothesis for elliptic curves over ﬁnite ﬁelds, which states that a
p
= α
p
+ β
p
,
where the complex numbers α
p
and β
p
satisfy α
p
β
p
= p and α
p
 = β
p
 =
√
p.
8
ii) The Lseries L(E, s) admits an analytic continuation to the whole complex plane.
This deep property of the Lseries of E is a consequence of the mo dularity of E, discussed
in Section 5. It is worth observing that a similar property for L(E/K, s), with K a general
number ﬁeld, is unknown.
iii) The Lseries L(E, s ) satisﬁes a functional equation relative to s 7→ 2−s, whose center
of symmetry is the point s = 1. More precisely, set L
∞
(E, s) := N
s/2
(2π)
−s
Γ(s), where
Γ(s) denotes the Γfunction, and deﬁne the completed Lseries Λ( E, s) := L(E, s)L
∞
(E, s).
Then
Λ(E, s) = wΛ(E, 2 − s), with w = ±1. (8)
The existence of the functional equation again follows from the modularity of E. (Conversely,
a theorem of Weil shows that the existence of a functional equation for L(E, s) and its twists
by Dirichlet characters implies mo dularity.)
The sign w appearing in (8) is an “elementary” quantity, equal to a product of local
signs. For example, if N = p is prime, then w = w
p
w
∞
= −w
p
is equal to the coeﬃcient
a
p
. Thus, if E has nonsplit multiplicative reduction at p (i.e.,
¯
E
p
has a node with tangents
not deﬁned over F
p
), the Lseries of E vanishes to odd order at s = 1 (so that in particular
L(E, 1) = 0).
3 Digression on conics
As a way of motivating the relation between L(E, s) and the arithmetic invariants of E, which
will be discussed in Section 4, we recall the classical formulae connecting the FermatPell
equation to Dirichlet Lseries. See Darmon’s article [Da] for more details.
Let C
d
: x
2
− dy
2
= 1 denote the FermatPell equation, where d > 1 is a squarefree
integer, and let
C
d
(Z) = {(x
0
, y
0
) ∈ Z × Z : x
2
0
− dy
2
0
= 1}
be the set of integral points on the conic C
d
. Assume for simplicity that d is ≡ 2, 3 (mod 4).
Thus, the ring of integers O
K
of the real quadratic ﬁeld K = Q(
√
d) is equal to Z[
√
d]. The
map (x, y) 7→ x + y
√
d identiﬁes C
d
(Z) with the group O
×
K,1
of norm one units in O
K
. This
identiﬁcation induces a natural group structure on C
d
(Z).
Proposition 3.1 (Legendre). The group C
d
(Z) is isomorphic to Z ⊕Z/2Z. In other words,
all the solutions of the FermatPell equation are generated up to sign by a fundamental
solution (x
d
, y
d
), corresponding to a fundamental unit ²
d
= x
d
+ y
d
√
d of norm one.
For p 6 2d, set
n
p
:= #C
d
(F
p
) = #{(x
0
, y
0
) ∈ F
p
× F
p
: x
2
0
− dy
2
0
= 1}, a
p
:= p − n
p
.
When p2d, set a
p
:= 0. A direct calculation shows the following
Lemma 3.2. If p 6 2d, then a
p
is equal to the Legendre symbol
¡
d
p
¢
(i.e., a
p
= 1 if d is a
square modulo p, and a
p
= −1 otherwise).
9
Deﬁne the Lseries of C
d
by the formula
L(C
d
, s) :=
Y
p
(1 − a
p
p
−s
)
−1
.
(The reader should compare the deﬁnition of L(C
d
, s) with Deﬁnition 2.1.)
Writing χ
d
: (Z/4dZ)
×
−→{±1} for the Dirichlet character characterized by
χ
d
(p) =
µ
d
p
¶
for p 6 2d,
it follows that L(C
d
, s) is equal to the Dirichlet Lseries L(χ
d
, s). Gauss’ quadratic reciprocity
law shows that χ
d
is equal to the quadratic character attached to K.
Theorem 3.3 (The class number formula).
1) Let u
d
be the explicit unit of norm one
Q
k∈(Z/4dZ)
×
(1 −ζ
k
d
)
χ
d
(k)
in O
×
K
, where ζ
d
= e
2πi/4d
.
Then the equality
L(C
d
, 1) = (1/
√
4d) log(u
d
)
holds (up to sign).
2) Let h
d
= #Pic(O
K
) be the class number of K. Then u
d
is equal to ²
2h
d
d
, where ²
d
is a
fundamental unit in O
×
K
.
Remark 3.4.
1) Theorem 3.3 shows that L(C
d
, s) – deﬁned in terms of local arithmetic invariants of C
d
– encodes information about the global arithmetic invariants of C
d
, such as a generator
for (the free part of) the group of global integral points C
d
(Z). In particular, L(C
d
, 1) is
described in terms of an explicit global point, constructed from trigonometric functions. The
question arises whether similar prop erties hold for L(E, s). The Birch and SwinnertonDyer
conjecture, stated in Section 4, provides an indepth answer to this question. Moreover, as
explained in Section 5, in certain cases the values of L(E, s) can be related to explicit global
points on E, called Heegner points, arising from the theory of complex multiplication.
2) Since the functional equation of L(C
d
, s) relates s to 1 − s, it follows that Theorem 3.3
considers the value of L(C
d
, s) at an integer which is “just right” of the center of symmetry
for the functional equation. On the other hand, the Birch and SwinnertonDyer conjecture
involves the value of L(E, s) at the central critical point s = 1. Both cases can be viewed as
diﬀerent instances of the conjectures on critical values for the Lfunctions of motives due to
Deligne, Beilinson, BlochKato, and others. See for example [Ki].
4 Statement of the Birch and SwinnertonDyer conjec
ture
We attach the following arithmetic invariants to E (see [Sil] for more details).
10
1. The real period of E
Ω
E
:=
Z
E(R)
ω
E

where ω
E
= dx/(2y + a
1
x + a
3
) is an invariant diﬀerential on a global minimal Weier
strass equation (1).
Consider the comparison isomorphism of Rvector spaces
ι
+
: (H
1
B
(E) ⊗ R)
+
−→ (Fil
1
H
1
dR
(E) ⊗ R)
between the Betti and the deRham cohomology of E. Here (H
1
B
(E) ⊗ R)
+
denotes
the ﬁxed part by the automorphism induced by complex conjugation, and Fil
1
H
1
dR
(E)
is equal to H
0
(E, Ω
1
E
). Then the period Ω
E
can be deﬁned as the scalar comparing
the natural Zstructures underlying the above Rvector spaces. This point of view is
taken by Deligne in his generalisation of the concept of period to the context of motivic
Lfunctions [De].
2. The Tamagawa number at p is deﬁned to be
c
E
(p) := #(E(Q
p
)/E
0
(Q
p
))
where E
0
(Q
p
) is the subgroup of E(Q
p
) consisting of points which reduce to non
singular points of
¯
E
p
. (Thus, c
p
(E) = 1 if p does not divide N.)
The invariant c
E
(p) can also be deﬁned as the order of the group of connected com
ponents of the N´eron model of E over Z
p
. The arguments of Section 5.4 provide some
evidence for the relevance of the groups of connected components in the study of the
Birch and SwinnertonDyer conjecture.
3. The regulator R
E
of E is the discriminant of the canonical N´eronTate height pairing.
If P
1
, ··· , P
r
E
denotes a Zbasis for the freeabelian group E(Q)/E
tors
(Q), then
R
E
:= det(hP
i
, P
j
i
NT
),
where h , i
NT
is the pairing appearing in equation (2) (with K = Q).
4. The ShafarevichTate group III(E/Q) is deﬁned in equation (5).
As mentioned in Section 2, L(E, s) can be analytically continued to C, and hence one
may consider its behavior at the central critical point s = 1. (This deep fact will be discussed
at length in Section 5.1.) Write L
∗
(E, 1) for the ﬁrst nonvanishing coeﬃcient in the Taylor
series expansion of L(E, s) at s = 1.
Conjecture 4.1 (Birch and SwinnertonDyer).
1) (“Millenium problem”) The equality ord
s=1
L(E, s) = rank
Z
E(Q) holds.
2) (BSD(Q
×
)) The equality L
∗
(E, 1) = Ω
E
· R
E
(mod Q
×
) holds.
11
3) (BSD) The following equality holds
L
∗
(E, 1) = Ω
E
·
Y
p
c
E
(p) · R
E
· #E
tors
(Q)
−2
· #III(E/Q).
Remark 4.2.
1) Assuming that III(E/Q) is ﬁnite, it can be checked that the right hand side of the
equality (BSD) depends only on the Qisogeny class of E.
2) Note the analogy between Conjecture 4.1 and Theorem 3.3, in which the term R
E
cor
responds to log(²
d
), and #III(E/Q) to h
d
. When the order of vanishing of L(E, s) is ≤ 1,
the results explained in Section 5.3 – which involve explicit points on E – indicate that this
analogy becomes even more stringent.
3) The validity of part 1 of Conjecture 4.1 would provide an eﬀective criterion to determine
whether E(Q) is inﬁnite. In fact, the vanishing of the special value L(E, 1) can be checked
by using the theory of modular symbols [Man2] (or also, when the sign of the functional
equation is −1, by local calculations).
4) Manin [Man1] has shown that the validity of Conjecture 4.1 implies the existence of an
eﬀective algorithm to compute a basis for E(Q).
5) By setting s = 1 in Deﬁnition 2.1, one obtains that L(E, 1) is “formally equal” to the ex
pression
Q
p
p/n
p
. In [BSD], Birch and SwinnertonDyer formulated the following conjecture:
deﬁne
f(x) :=
Y
p≤x
n
p
/p ;
then, f(x) behaves asymptotically for x → ∞ as C
E
log(x)
r
E
, where r
E
is the rank of E
and C
E
is a constant depending on E. The reader is referred to [Co], [KM] and [Sa] for a
discussion of the relation between this conjectural statement and Conjecture 4.1.
6) Conjecture 4.1 generalises to an analogous statement for the Lseries L(E/K, s) of an
elliptic curve E over a number ﬁeld K.
5 Results on the Birch and SwinnertonDyer conjec
ture
5.1 Modularity
Let E b e an elliptic curve over Q of conductor N, and write its Lseries in the form
L(E, s) =
+∞
X
n=1
a
n
n
−s
, a
n
∈ Z,
where the coeﬃcients a
n
are deﬁned recursively in terms of the coeﬃcients a
p
introduced in
Section 2. For τ in the complex upper half plane H = {τ ∈ C : =(τ ) > 0}, set q = e
2πiτ
and
12
consider the inverse Mellin transform of L(E, s) deﬁned by the formula
f
E
(τ) :=
∞
X
n=1
a
n
q
n
.
The next result, proved for N squarefree by Wiles and Taylor [Wi], [TW] and generalised
to arbitrary N by Breuil, Conrad, Diamond and Taylor [BCDT], establishes the celebrated
ShimuraTaniyama conjecture.
Theorem 5.1 (ShimuraTaniyama conjecture). The function f
E
is a newform of weight 2
on Γ
0
(N).
Thus,
1. f
E
(τ) is holomorphic on H,
2. f
E
(
aτ+b
cτ+d
) = (cτ + d)
2
f
E
(τ) for all
µ
a b
c d
¶
belonging to the group Γ
0
(N) of matrices
in SL
2
(Z) such that Nc,
3. f
E
vanishes at the cusps in P
1
(Q), i.e., for all
µ
a b
c d
¶
∈ SL
2
(Z), the function
f
E
(
aτ+b
cτ+d
)(cτ + d)
−2
vanishes at the cusp i∞,
4. f
E
is an eigenform for the Hecke operators T
n
, and satisﬁes
T
n
f
E
= a
n
f
E
for all n ≥ 1. Furthermore, if w
N
is the AtkinLehner operator deﬁned by the rule
(w
N
f
E
)(τ) = f(−1/Nτ ), we have that w
N
f
E
= wf
E
with w = ±1.
Theorem 5.1 implies that L(E, s) is the Mellin transform of f
E
, i.e.,
L(E, s) = (2π)
s
/Γ(s)
Z
i∞
0
f
E
(iy)y
s−1
dy.
This implies the following:
Corollary 5.2. L(E, s) admits an analytic continuation to C and satisﬁes the functional
equation (8).
See for example [DDT] for details on the above concepts.
We now turn to describing some geometric consequences of Theorem 5.1.
Shimura curves. Assume there is an integer factorisation N = N
+
N
−
such that:
1. N
+
and N
−
are coprime;
13
2. N
−
is squarefree;
3. N
−
is divisible by an even number of prime factors.
Let B denote the indeﬁnite quaternion algebra over Q of discriminant N
−
. Fix an Eichler
Zorder R in B of level N
+
, and let R
max
be the maximal order containing R. (Note that
R is unique up to conjugation by elements of B
×
.)
A QM abelian surface A over a base scheme T is a triple (A, i, C) where:
1. A is an abelian scheme over T of relative dimension 2;
2. i : R
max
−→End
T
(A) is an injective homomorphism deﬁning an action of R
max
on A;
3. C is a levelN
+
structure, i.e., a subgroup scheme of A which is locally isomorphic to
Z/N
+
Z and is stable under the action of R.
Deﬁnition 5.3. The Shimura curve X = X
N
+
,N
−
over Q attached to the data (N
+
, N
−
)
is the coarse moduli space classifying isomorphism classes of QM abelian surfaces (A, i, C)
over the base T = Spec( Q).
Remark 5.4. The above mo duli deﬁnition can be extended in order to deﬁne a canonical
model of X over Z.
The curve X over C. Fix an embedding ι
∞
: B−→M
2
(R), and set
Γ
∞
:= ι
∞
(R
×
1
) ⊂ SL
2
(R),
where R
×
1
denotes the group of units in R having reduced norm equal to 1. Then the set
X(C) of complex points of X can be identiﬁed with the Riemann surface Γ
∞
\H. When
N
−
> 1, this surface is compact. When N
−
= 1, it can be compactiﬁed by adjoining
the ﬁnite set of cusps Γ
0
(N)\P
1
(Q). The resulting compact Riemann surface is called the
modular curve of level N, and is usually denoted by X
0
(N).
The curve X over C
p
for p dividing N
−
. This padic model of X will be useful in the
arguments of Section 5.3. Fix a prime p dividing N
−
, and denote by B the deﬁnite quaternion
algebra over Q of discriminant (N
−
/p)∞. Write R for the (unique, up to conjugation) Eichler
Z[1/p]order in B of level N
+
, and set
Γ
p
:= ι
p
(R
×
1
) ⊂ SL
2
(Q
p
),
where R
×
1
is the group of norm one units in R, and ι
p
denotes a ﬁxed embedding of B into
M
2
(Q
p
). Let
H
p
:= P
1
(C
p
) − P
1
(Q
p
)
be the padic upper half plane. Let Q
p
2
be the quadratic unramiﬁed extension of Q
p
, and
Z
p
2
the associated ring of integers.
14
Theorem 5.5 (CerednikDrinfeld). There is an identiﬁcation
X(C
p
) = Γ
p
\H
p
of rigidanalytic curves over Q
p
2
(and also of formal schemes over Z
p
2
).
See [BC] for more details on Shimura curves and their nonarchimedean uniformisation.
The JacquetLanglands correspondence. Let S
2
(Γ
0
(N), F ) denote the space of weight
2 cuspforms on Γ
0
(N) whose Fourier coeﬃcients a
n
belong to a ﬁeld F of characteristic 0.
Given f in S
2
(Γ
0
(N), F ), note that
ω
f
:= 2πif(τ)dτ
deﬁnes a regular diﬀerential in H
0
(X
0
(N)
F
, Ω
1
X
0
(N)/F
).
Theorem 5.6 (JacquetLanglands). Let f =
P
n≥1
a
n
q
n
be a newform in S
2
(Γ
0
(N), F ).
Given a Shimura curve X = X
N
+
,N
−
associated to a factorisation N = N
+
N
−
as above,
there exists a global section ω in H
0
(X
F
, Ω
1
X/F
) such that
T
n
ω = a
n
ω, (n, N) = 1,
where T
n
denotes the nth Fourier operator acting on X. The form ω is unique up to scaling
by nonzero constants in F
×
.
We call ω as in the statement of Theorem 5.6 a modular form of weight 2 on X deﬁned
over F .
Remark 5.7.
1) If ω is a weight two modular form on X/C, and π : H−→X(C) is the natural projection,
then π
∗
(ω) = f
ω
(τ)dτ, where f
ω
(τ) is a holomorphic function on H satisfying
f
ω
(γτ) = (cτ + d)
2
f
ω
(τ), (9)
for all γ =
µ
a b
c d
¶
in Γ
∞
.
2) Similarly, the pullback by the covering H
p
−→X(C
p
) of a weight two modular form ω on
X/C
p
yields a rigidanalytic function f
ω
: H
p
−→C
p
satisfying (9) for all γ in Γ
p
.
Modular parametrisations of E by Shimura curves. Let X = X
N
+
,N
−
, and let T =
T
N
+
,N
−
be the Hecke algebra attached to X. Thus, T is generated over Z by the Hecke
operators T
p
for p  N = N
+
N
−
and U
p
for p  N. If E is an elliptic curve of conductor
N, and f
E
=
P
a
n
q
n
is the newform attached to E by Theorem 5.1, then Theorem 5.6
guarantees the existence of an algebra homomorphism
f = f
N
+
,N
−
: T−→Z, T
n
7→ a
n
.
15
Writing E
f
for the abelian variety quotient
Pic(X)/ ker(f ) = Pic
0
(X)/ ker(f ),
we ﬁnd that E
f
is an elliptic curve over Q, whose Galois representation V
`
(E
f
) is isomor
phic to V
`
(E). By Faltings’ isogeny theorem [Fa], it follows that E
f
is Qisogenous to E.
Composing the natural projection of Pic(X) onto E
f
with such an isogeny yields a morphism
ϕ
X
: Pic(X)−→E
deﬁned over Q, called a modular parametrization of E by X.
Remark 5.8. Over the complex numbers, ϕ
X
can be described in terms of complex inte
gration by sending a divisor class [D] in Pic
0
(X) to the integral
R
D
ω, which gives rise to a
welldeﬁned element in the complex torus E(C) = C/Λ. A similar description can be ob
tained over C
p
, by invoking Coleman’s theory of padic integration. See for example [BDG]
for details.
5.2 Heegner points
Keeping notations as in Section 5.1, let K = Q(
√
−D) be an imaginary quadratic ﬁeld.
Assume for simplicity that O
×
K
= {±1}, and that K satisﬁes the following
Hypothesis 5.9 (Heegner hypothesis). The primes dividing N
+
are split in K, while those
dividing N
−
are inert in K.
Remark 5.10. Given an integer c ≥ 1, let H
c
denote the ring class ﬁeld of conductor c.
Thus, H
c
is an abelian extension of K which is (generalised) dihedral over Q. It can be
described as the ﬁeld generated over K by j(O
c
), where O
c
:= Z + cO
K
is the order of K of
conductor c and j denotes the mo dular jfunction. Let
χ : Gal(H
c
/K)−→C
×
be a complex character, and let L(E/K, χ, s) be the complex Lseries of E over K twisted
by χ. Rankin’s method implies that L(E/K, χ, s) admits an analytic continuation and a
functional equation relative to s 7→ 2 −s. (See for example [GZ] for details.) If (c, ND) = 1,
the sign of the functional equation of L(E/K, χ, s ) is equal to −ε
K
(N), where ε
K
is the
quadratic Dirichlet character attached to K. Hence, by the Heegner hypothesis, this sign is
equal to −1. In this case, the Birch and SwinnertonDyer conjecture (suitably generalised)
predicts the existence of a point of inﬁnite order in each χcomponent
E(H
c
)
χ
= {P ∈ E(H
c
) ⊗ C : σ(P ) = χ(σ)P, for all σ ∈ Gal(H
c
/K)}
of the MordellWeil group E(H
c
).
16
Under the current assumptions, there exists an embedding of Qalgebras
Ψ : K−→B, such that Ψ(O
c
) = Ψ(K) ∩ R.
Composing Ψ with ι
∞
yields an action of K
×
(and also of C
×
, by extension of scalars) on
H. Write P
Ψ
∈ X(C) for the image of the unique ﬁxed point τ
Ψ
∈ H for this action. By the
moduli interpretation of X, the point P
Ψ
is identiﬁed with the isomorphism class of a triple
(A, i, C) satisfying
End(A, i, C) = O
c
.
(An endomorphism of a triple as above is an endomorphism of A which commutes with
the quaternionic action i and preserves the level structure C.) By the theory of complex
multiplication for abelian varieties, the point P
Ψ
, called Heegner point of level c, is deﬁned
over H
c
. For σ ∈ Gal(H
c
/K), the formula
P
σ
Ψ
= P
Ψ
σ
expresses Shimura’s reciprocity law, where the action of Gal(H
c
/K) on (oriented optimal)
embeddings is deﬁned for example in [BDG]. Finally, by applying the modular parametri
sation ϕ
X
to P
Ψ
we obtain a point
α
Ψ
:= ϕ
X
(P
Ψ
) ∈ E(H
c
). (10)
Remark 5.11. When N
−
= 1, so that X is the modular curve X
0
(N) and all the primes
dividing N are split in K, there exists an O
c
ideal N such that O
c
/N = Z/NZ. In this
setting, the Heegner point P
Ψ
corresponds to a pair
(C/a, C/a[N] = N
−1
a/a),
consisting of an elliptic curve E
a
= C/a with complex multiplication by O
c
and a cyclic
subgroup C/a[N] of E
a
of order N.
Heegner points over C
p
for p  N
−
. Under the Heegner hypothesis, there exists an embedding
Ψ : K−→B, such that Ψ(O
c
[1/p]) = Ψ(K) ∩ R.
Via ι
p
, this embedding induces an action of K
×
p
on H
p
. Let τ
Ψ
denote the unique ﬁxed point
for this action satisfying the relation
ι
p
Ψ(α)
µ
τ
Ψ
1
¶
= α
µ
τ
Ψ
1
¶
for all α in K
×
p
. Finally, denote by P
Ψ
the image of τ
Ψ
in X(C
p
) = Γ
p
\H
p
. Note that τ
Ψ
belongs to H
p
∩ K
p
, and hence P
Ψ
is a point in X(K
p
). Using the moduli interpretation of
H
p
deﬁned by Drinfeld, it can be shown that P
Ψ
corresponds to a Heegner point in X(H
c
)
via the map X(H
c
)−→X(K
p
) induced by an embedding of H
c
in K
p
. See [BDG] for more
details.
17
5.3 The theorems of GrossZagierZhang and Kolyvagin
Let L(E/K, χ, s) be the Lseries of E/K twisted by an anticyclotomic character χ of con
ductor c, i.e., a complex character of Gal(H
c
/K). Assume for simplicity that c = 1, so that
H = H
c
is the Hilbert class ﬁeld of K and χ is unramiﬁed. Under the Heegner hypothesis,
the sign of the functional equation of L(E/K, χ, s) is −1. Recall the Heegner point α = α
Ψ
in E(H) deﬁned in equation (10) (with c = 1). Set
α
χ
:=
X
σ∈Gal(H/K)
χ(σ)α
σ
∈ E(H)
χ
.
The following result was proved by GrossZagier [GZ] in the case N
−
= 1, in which X is
the classical modular curve X
0
(N), and by Zhang [Zh1] for general X.
Theorem 5.12 (GrossZagier, Zhang). The equality
L
0
(E/K, χ, 1) = C · hα
χ
, α
χ
i
NT
holds, where C is an explicit nonzero constant.
Remark 5.13. If
L
0
(
E/K, χ,
1)
6
= 0
and
rank
C
E
(
H
)
χ
= 1, Theorem 5.12 combined with
the explicit deﬁnition of C implies that the Birch and SwinnertonDyer conjecture holds for
L(E/K, χ, s) up to Q
×
.
Assume to ﬁx ideas that N
−
= 1 and that χ = 1, and write
α
K
= Trace
H/K
α ∈ E(K).
The following result is proved in [Ko].
Theorem 5.14 (Kolyvagin). If α
K
is a point of inﬁnite order, then E(K) has rank 1 and
III(E/K) is ﬁnite.
Remark 5.15. More generally, it is shown in [BD1] that if α
χ
is nonzero, then rank
C
E(H)
χ
is equal to 1.
The next result combines Theorems 5.12 and 5.14 in order to deduce consequences on
the Birch and SwinnertonDyer conjecture for E over Q. Write r
an
E
:= ord
s=1
L(E, s) and
r
E
:= rank
Z
E(Q).
Theorem 5.16. If r
an
E
is ≤ 1, then r
E
= r
an
E
and III(E/Q) is ﬁnite.
Proof. (Sketch) Let K be an auxiliary quadratic imaginary ﬁeld. Note the factorisation of
Lseries
L(E/K, s) = L(E, s)L(E, ε
K
, s), (11)
where L(E, ε
K
, s) =
P
n≥1
a
n
ε
K
(n)n
−s
is the twist of L(E, s) by the quadratic Dirichlet
character ε
K
.
18
Assume that r
an
E
= 1. A result of Waldspurger [Wald] ensures the existence of K quadratic
imaginary in which all the primes dividing N are split, and such that L(E, ε
K
, 1) is non
zero. In view of (11), Theorem 5.12, applied in the setting of the classical modular curve
X
0
(N), implies that the Heegner point α
K
has inﬁnite order. By Theorem 5.14, it follows
that the rank of E(K) is one, and that III(E/K) is ﬁnite. This implies that III(E/Q) is
ﬁnite. In order to deduce that E(Q) has rank one, it suﬃces to note that, up to torsion,
complex conjugation acts on α
K
as multiplication by the opposite of the sign of the functional
equation of L(E, s). This sign is −1 is our case, and therefore α
K
gives rise to a point of
inﬁnite order in E(Q).
The case where r
an
E
= 0 can be treated in a similar way. In this setting one invokes
a result of BumpFriedbergHoﬀstein [BFH] or MurtyMurty [MM], which guarantees the
existence of a quadratic imaginary ﬁeld K is which all the primes dividing N are split, and
such that L(E, ε
K
, s) vanishes to order one at s = 1. It follows that L(E/K, s) vanishes to
order one at s = 1. Now the argument proceeds as above.
Remark 5.17.
1) When E is a CM elliptic curve, instances of the above results were established previously
thanks to the work of CoatesWiles [CW] and Rubin [Ru1].
2) The Birch and SwinnertonDyer conjecture remains wide open when r
an
E
> 1. For instance,
no example of E for which r
an
E
> 3 is known, although inﬁnitely many equations of elliptic
curves for which r
E
> 3 can be written down (and Elkies has produced an example showing
that r
E
can be at least 28).
5.4 Proofs
Assume for simplicity that E is semistable, i.e., N is squarefree. In order to illustrate the
methods of proof of the results above, we sketch a proof of the following (somewhat weaker)
variant, based on the techniques introduced in [BD3]. See also Longo’s article [L] in the
context of elliptic curves over totally real ﬁelds and Hilbert modular forms.
Theorem 5.18. If L(E, 1) is nonzero, then E(Q) is ﬁnite and III(E/Q)
p
= 0 for almost
all p.
Proof. (Sketch)
Step 1. Let K = Q(
√
−D) be an imaginary quadratic ﬁeld of discriminant −D such that:
1. (D, N) = 1,
2. L(E/K, 1) is nonzero.
The existence of K as above is guaranteed by a result of Waldspurger: see [Wald], or also
[IK]. Since the sign of the functional equation of L(E/K, s) is equal to −ε
K
(N) = +1, it
follows that N can be factored as N = N
+
N
−
, where the primes dividing N
+
, resp. N
−
are
split, resp. inert in K, and N
−
is divisible by an odd number of prime factors.
19
Remark 5.19. In the strategy of proof of Theorem 5.16 sketched before, the auxiliary
imaginary quadratic ﬁeld K was chosen so that L(E/K, s) has sign of the functional equation
equal to −1 and L
0
(E/K, 1) 6= 0.
Step 2. Fix a descent prime p ≥ 5 such that:
1. p  ND,
2. the Galois representation ρ
E,p
: G
Q
−→Aut(E
p
) is surjective,
3. p does not divide the minimal degree of a modular parametrisation ϕ
X
0
(N)
: X
0
(N)−→E,
4. p  L
alg
(E/K, 1), where L
alg
(E/K, 1) ∈ Z is the algebraic part of L(E/K, 1).
Informally, we may think of L
alg
(E/K, 1) – which is known to be the square of a rational
integer – as the quantity obtained by dividing L(E/K, 1) by the appropriate complex period.
An indirect description of L
alg
(E/K, 1) will be given in part 3 of Proposition 5.21.
By our assumptions E does not have complex multiplications, and hence Serre’s open
image theorem implies that the above conditions on p are satisﬁed by all but ﬁnitely many
primes (In fact, since E is semistable, a stronger result of Mazur [Maz2] shows that ρ
E,p
is
surjective for p ≥ 11.)
Step 3. Let ` denote an admissible prime (relative to p), i.e., a prime satisfying the following
conditions:
1. `  2pDN,
2. ` is inert in K,
3. p  `
2
− 1 and p  1 + ` − ²a
`
, where a
`
denotes the `th Fourier coeﬃcient of f
E
and
² = ±1 is a choice of sign.
By the Chebotarev density theorem, there are inﬁnitely many primes ` as above.
Step 4. Fix an admissible prime `. Write X for the Shimura curve X
N
+
,N
−
`
, J for its jacobian
Pic
0
(X), and Φ
`
= J
F
`
2
/J
0
F
`
2
for the group of connected components of the N´eron model of
J over Z
`
2
. Let T be the Hecke algebra acting on X. The following result is due to Ribet:
see Theorems 5.15 and 5.17 of [BD3].
Proposition 5.20 (Ribet). There exists a surjective homomorphism f
`
: T−→Z/pZ satis
fying the following properties:
1. f
`
(T
q
) = a
q
for all primes q  N`,
2. f
`
(U
q
) = a
q
for all primes q  N,
3. f
`
(U
`
) = ², where ² is the sign entering in the deﬁnition of the admissible prime `.
20
Furthermore, writing I
`
for the maximal ideal ker f
`
and T
p
(J) for the padic Tate module
of J, one has the identiﬁcations
Φ
`
/I
`
Φ
`
' Z/pZ, T
p
(J)/I
`
T
p
(J) ' E
p
,
the latter being an isomorphism of G
Q
modules.
Note that Proposition 5.20 states in particular the existence of a modular form f
`
of level
N` which is congruent modulo p to f
E
. Note also that, by the results of Section 5.2, the
Shimura curve X carries a family of Heegner points deﬁned over the ring class ﬁelds of K.
Step 5. (Specialisation of Heegner points to the group of connected components) Recall the
nonarchimedean description of Heegner points given in Section 5.2. Applied to the Shimura
curve X = X
N
+
,N
−
`
and to the prime `, it shows that a Heegner point P
Ψ
in X(H) ⊂ X(K
`
)
can be described in terms of an embedding
Ψ : O
K
[1/`]−→R,
where R is a ﬁxed Eichler Z[1/`]order of level N
+
in the deﬁnite quaternion algebra of
discriminant N
−
∞.
Let T
`
be the BruhatTits tree of PGL
2
(Q
`
), and let V(T
`
) denote the set of vertices of
T
`
. Write G
`
for the ﬁnite graph Γ
`
\T
`
, and V(G
`
) for the set of vertices of G
`
. The embedding
ι
`
Ψ induces an action of K
×
`
on T
`
ﬁxing a unique vertex, whose image in V(G
`
) we denote
by v
Ψ
. The vertex v
Ψ
can also be described as the image of P
Ψ
by the natural reduction
map from X(K
`
) to V(G
`
).
Write ∂
`
(P
Ψ
) for the specialisation of P
Ψ
to the group of connected components Φ
`
/I
`
,
deﬁned as the image of P
Ψ
by the composite of the natural maps
X(K
`
)−→Pic(X)(K
`
)/I
`
= J(K
`
)/I
`
−→Φ
`
/I
`
.
Let P
K
be the divisor Trace
H/K
P
Ψ
in Pic(X)(K). Let Z[V(G
`
)] denote the module of
formal divisors with Zcoeﬃcients supported on V(G
`
); it is equipped with an action of the
algebra T.
Proposition 5.21.
1) There is a canonical identiﬁcation Φ
`
/I
`
= Z[V(G
`
)]/I
`
.
2) Under the above identiﬁcation, the equality ∂
`
(P
Ψ
) = v
Ψ
holds modulo I
`
.
3) The equality ∂
`
(P
K
) = L
alg
(E/K, 1)
1/2
(mod p) holds in Φ
`
/I
`
' Z/pZ.
For part 1 and 2, see Edixhoven’s appendix to [BD2]. In view of part 1 and 2, the third
part follows from a description of the special value L(E/K, 1) due to Gross [Gr] and Zhang
[Zh2]. The reader is referred to Chapter 5 of [BD3] for more details.
Step 6. (Controlling Sel
p
(E/K)) As noted above, the divisor P
K
∈ Pic(X)(K) gives rise to
a natural element in J(K)/I
`
, and hence to a cohomology class in H
1
(K, T
p
(J)/I
`
) via the
Kummer map. Fix an isomorphism ψ of G
Q
modules from T
p
(J)/I
`
to E
p
as in part 3 of
Proposition 5.20. Since E
p
is irreducible by our assumptions, Schur’s lemma implies that
21
ψ is determined up to multiplication by nonzero scalars in Z/pZ. It follows that P
K
gives
rise to a cohomology class c in H
1
(K, E
p
), welldeﬁned up to multiplication by elements of
(Z/pZ)
×
.
Write H
1
(K, E
p
)
±
for the eigenspace of H
1
(K, E
p
) on which complex conjugation acts
via multiplication by ±1. Given a rational prime q, set K
q
= K ⊗ Q
q
and let %
q
denote the
natural localisation map from H
1
(K, E
p
) to H
1
(K
q
, E)
p
. The following lemma is proved in
[BD3], Section 5.
Lemma 5.22.
1) The class c belongs H
1
(K, E
p
)
²
.
2) There is a natural isomorphism
κ : Φ
`
/I
`
−→H
1
(K
`
, E)
²
p
such that the equality κ(∂
`
P
K
) = %
`
c holds.
3) For q 6= `, %
q
c = 0.
Choosing ² to be +1, we ﬁnd that c belongs to H
1
(K, E
p
)
+
= H
1
(Q, E
p
). By combining
part 3 of Lemma 5.22 with Theorem 1.10, we obtain that the equality
hres
`
s, %
`
ci
`
= 0
holds for all s ∈ Sel
p
(E/Q). Our assumption that p does not divide L
alg
(E/K, 1) and part
3 of Proposition 5.21 imply that ∂
`
(P
K
) is nonzero. Using the nondegeneracy of the local
Tate pairing together with part 2 of Lemma 5.22, we deduce that res
`
s = 0 for all admissible
` for which ² = +1. It follows from the Chebotarev density theorem that s = 0. Hence
Sel
p
(E/Q) = 0, so that E(Q) is ﬁnite and III(E/Q)
p
is zero. This concludes our proof.
Remark 5.23.
1) Let χ be a complex character of Gal( H
c
/K). The methods of the above proof can be used
to show that the nonvanishing of L(E/K, χ, 1) implies that E(H
c
)
χ
and III(E/H
c
)
χ
are
both ﬁnite. The original methods of Kolyvagin, based on the GrossZagier formula, allow to
prove a similar statement only when χ is quadratic.
2) The methods of [BD3] – notably the second explicit reciprocity law of Section 9 – can
also be used in order to reduce the proof of the analogue of Theorem 5.18 for E of analytic
rank one to the arguments sketched above.
5.5 Reﬁnements and padic methods
Continuing to assume that E is semistable and that the analytic rank r
an
E
is ≤ 1, we begin
by brieﬂy elaborating on the possibility of proving the precise relation between L(E, s) and
the arithmetic invariants of E, such as for instance the order of the ShafarevichTate group,
which is predicted by the Birch and SwinnertonDyer conjecture.
Assume that r
an
E
= 0. Theorem 5.16 and the theory of modular symbols [Man2] imply
the Birch and SwinnertonDyer conjecture for E up to Q
×
. Furthermore, a recent result of
22
SkinnerUrban [SU], combined with prior work of Kato [Ka], shows the ppart of the Birch
and SwinnertonDyer conjecture for primes p ≥ 11 of good ordinary reduction for E. In
other words, for such p’s the equality
ord
p
(L(E, 1)/Ω
E
) = ord
p
(#III(E/Q) ·
Y
`N
c
E
(`)) (12)
holds. A similar result was established before for elliptic curves with CM thanks to the work
of Rubin [Ru2]. The proof of (12) is conditional on the validity of a conjecture formulated
in [SU], stating the existence of the fourdimensional padic Galois representation associated
with certain cuspidal automorphic representations of unitary groups of fourdimensional
hermitian spaces. (Work in progress of several authors is expected to lead to a proof of this
conjecture.) Note that under the assumptions stated above the Galois representation ρ
E,p
is
surjective [Maz2], so that in particular E
p
(Q) is trivial.
Now assume that r
an
E
= 1. By following the argument in the proof of Theorem 5.16 –
which chooses an imaginary quadratic ﬁeld K satisfying the Heegner hypothesis and such
that L
0
(E/K, 1) is nonzero – and invoking Theorems 5.12 and 5.14, one checks that the
Birch and SwinnertonDyer holds up to Q
×
. The question arises whether the analogue of
equation (12) holds (under similar assumptions), i.e.,
ord
p
(L
0
(E, 1)/Ω
E
hP
E
, P
E
i
NT
)) = ord
p
(#III(E/Q) ·
Y
`N
c
E
(`)), (13)
where P
E
is a generator for E(Q) modulo torsion. The methods of Kolyvagin, together
with formula (12) applied to a suitable quadratic twist of E, can be used to deal with the
inequality ≥. Building on the methods of [BD3] it may be possible to obtain results on the
opposite inequality. We plan to address this question in another paper.
It is worth observing that the works [Ka], [Ru2], [SU] and [BD3] cited above ﬁnd their
motivation in the problem of establishing a socalled Main Conjecture of Iwasawa theory
for the elliptic curve E. The main step in the formulation of such a problem consists in
the construction of a padic Lfunction attached to E, which may be viewed as a padic
analogue of the complex Lfunction L(E, s). Let K
∞
be a Z
p
extension of K, i.e., an abelian
extension with Galois group G
∞
:= Gal(K
∞
/K) isomorphic to the additive group Z
p
of the
padic integers. If p is an ordinary prime for E, the padic Lfunction L
p
(E, K
∞
/K) attached
to the triple (E, p, K
∞
/K) is an element of the Iwasawa algebra
Λ = Z
p
[[G
∞
]] := lim
←
Z
p
[G
n
], with G
n
:= G
∞
/p
n
G
∞
,
where the inverse limit is taken with respect to the natural projection maps on group rings.
Note that L
p
(E, K
∞
/K) can naturally be viewed as a function on the set of ﬁnite order
characters χ of G
∞
. It can often be deﬁned by an interpolation property of the kind
L
p
(E, K
∞
/K)(χ) = L
alg
(E/K, χ, 1), (14)
where L(E/K, χ, s) denotes the twist of L(E/K, s) by χ, and L
alg
(E/K, χ, 1) is an “algebraic
part” obtained by dividing L(E/K, χ, 1) by appropriate periods. It should be stressed that
23
the existence of L
p
(E, K
∞
/K) is known only in speciﬁc cases, as a consequence of techniques
based on the concept of modularity. It is known, for example, when K = Q, so that Q
∞
is
the cyclotomic Z
p
extension of Q. The resulting padic Lfunction, constructed by Mazur
SwinnertonDyer [MSw] (see also [MTT]), is the one considered in the work of Kato [Ka]
and SkinnerUrban [SU] mentioned above. In this setting, the Main Conjecture of Iwasawa
theory states that L
p
(E, Q
∞
/Q) generates the characteristic ideal I
∞
of the Pontrjagin dual
of the Λmodule
Sel
p
∞
(E/Q
∞
) := lim
→
Sel
p
n
(E/Q
n
),
where Q
n
denotes the subﬁeld of Q
∞
of degree p
n
over Q and the direct limit is taken with re
spect to the natural restriction maps. The article [Ka] shows that I
∞
divides ΛL
p
(E, Q
∞
/Q),
while [SU] shows the opposite divisibility. This establishes the cyclotomic Main Conjecture,
which implies equation (12). More generally, without any assumption on the analytic rank
r
an
E
, one has the following padic analogue of the Birch and SwinnertonDyer conjecture.
Noting that Λ can be identiﬁed with the Z
p
algebra of formal power series in one variable
T , a welldeﬁned order of vanishing ord(L
p
(E, Q
∞
/Q)) (at T = 0) can be attached to the
cyclotomic padic Lfunction. Then the inequality
ord(L
p
(E, Q
∞
/Q)) ≥ rank
Z
E(Q). (15)
holds, as a consequence of [Ka]. As for the opposite inequality, it would follow from [SU],
combined with the conjectural nondegeneracy of the cyclotomic padic height pairing and
ﬁniteness of III(E/Q).
Similar considerations can be repeated for the padic Lfunction L
p
(E, K
∞
/K) attached
to the anticyclotomic Z
p
extension K
∞
of a quadratic imaginary ﬁeld K, i.e., the unique Z
p

extension of K which is prodihedral over Q. In this context, the analogue of Kato’s theorem
is proved in [BD3], using the techniques sketched in Section 5.4. The opposite divisibility can
again be attacked by the techniques of [SU], which apply to a padic Lfunction associated
to the composite of the cyclotomic and the anticyclotomic Z
p
extension of K.
As observed before, the analogue of equation (15) for the complex Lfunction L(E, s)
seems out of reach at the moment. This marked diﬀerence between the complex and the p
adic setting may be explained on the grounds that the special values L(E/K, χ, 1) appearing
in equation (14) admit (at least in some cases) a cohomological interpretation. (Consider,
for example, the formula appearing in part 3 of Proposition 5.21.) This brings them more
in line with the deﬁnition of the Selmer group and makes it easier to establish connections
between the two objects. It follows, by its very deﬁnition, that the padic Lfunction is
amenable to a cohomological description. On the other hand, no such an interpretation is
currently known for the whole complex Lfunction L(E, s).
Remark 5.24. The theory of complex multiplication, described in Section 5.2, provides the
only known metho d to construct a systematic supply of global points on modular elliptic
curves. By deﬁnition, a Heegner point on an elliptic curve E is the image of a CM point on a
Shimura curve X by a modular parametrisation ϕ
X
. Recall that the existence of ϕ
X
follows
from Faltings’ isogeny theorem, which establishes the Tate conjecture for abelian varieties
over number ﬁelds, combined with the fact that the Galois representation of E arises as a
24
constituent of the ﬁrst ´etale cohomology group of X, as a consequence of the modularity of
E.
These considerations prompt the question of whether it is possible to enlarge the reper
toire of techniques for constructing algebraic points on elliptic curves. This goal may be
achieved by replacing the Shimura curve X by a suitable Shimura variety, equipped with a
systematic family of algebraic cycles, whose ´etale cohomology contains the Galois representa
tion of E. In some cases, the image of these cycles by the socalled higher padic AbelJacobi
maps give rise to elements in the Selmer group of E, which should arise from global points
on E in view of the Tate conjecture for algebraic varieties over number ﬁelds.
In the context of CM elliptic curves and generalised KugaSato varieties, this program is
carried out in [BDP1] and [BDP2]. These papers show that the padic AbelJacobi image
of certain CM cycles gives rise to global points, without assuming the Tate conjecture for
higher dimensional varieties. The proof relates the abovementioned AbelJacobi images to
values of padic Lfunctions, arising from the interpolation of special values associated to
padic analytic families of CM modular forms.
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