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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 Swinnerton-Dyer conjecture. It then gives an account of recent results on the subject. Mathematics Subject Classification (2010)11G40-11G05-11G15-11F33-11F67 KeywordsElliptic curves- L-series- p-adic L-functions-Shafarevich-Tate groups-main Conjectures of Iwasawa theory
Report on the Birch and Swinnerton-Dyer conjecture
Massimo Bertolini
February 26, 2010
Contents
1 The basic problem and the Mordell-Weil theorem 2
2 The L-series of an elliptic curve 7
3 Digression on conics 9
4 Statement of the Birch and Swinnerton-Dyer conjecture 10
5 Results on the Birch and Swinnerton-Dyer conjecture 12
5.1 Modularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.2 Heegner points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.3 The theorems of Gross-Zagier-Zhang and Kolyvagin . . . . . . . . . . . . . . 18
5.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.5 Refinements and p-adic 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 Swinnerton-Dyer
conjecture. It then gives an account of recent results on the subject.
Keywords: Elliptic curves, L-series, p-adic L-functions, Shafarevich-Tate 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 Swinnerton-Dyer conjecture,
delivered by the author at the “Riemann School” held at Verbania (Italy) in April 2009.
1
The Birch and Swinnerton-Dyer 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 Mordell-Weil theorem on rational
points.
The second section recalls the definition and the various properties of the L-series 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 Fermat-Pell 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 Swinnerton-Dyer conjecture (over the rationals), given in section
4.
The last section somewhat more advanced than the previous ones gives an up-to-date
account of some of the main results on the Birch and Swinnerton-Dyer 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 Gross-Zagier and
Zhang on derivatives of Rankin L-series, 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 p-adic methods, in connection with recent applications to the
Birch and Swinnerton-Dyer conjecture due to Kato and Skinner-Urban, concludes section 5.
1 The basic problem and the Mordell-Weil 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 non-singular projective curve of genus
one, equipped with a K-rational point.
By the Riemann-Roch theorem, an elliptic curve can be described explicitly as the pro-
jective non-singular 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 non-zero. 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 K-rational 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 so-called “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 Mordell-Weil theorem, which brilliantly generalises
the descent method of Fermat.
Theorem 1.3 (Mordell-Weil). 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 non-negative 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´eron-Tate
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
Swinnerton-Dyer 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 eron-
Tate non-degenerate 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 Mordell-Weil 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 p-adic 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 1-cocycles from G
K
to M by the group of 1-coboundaries.
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 m-torsion
points, are defined in a similar way.
4
There is an exact sequence of discrete G
K
-modules
0E
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 m-torsion 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 1-cocyle σ 7→ Q
σ
Q.
The strategy of proof of the weak Mordell-Weil 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 m-Selmer 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 Shafarevich-Tate 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
0E(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 non-archimedean 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 m-Selmer group Sel
m
(E/K) can in principle
be computed, as the proof of the weak Mordell-Weil 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 Shafarevich-Tate groups attached to E/K, and introduce the formalism which will b e
instrumental in proving the results on the Birch and Swinnerton-Dyer conjecture described
in Section 5.
Theorem 1.9 (Local Tate duality). For each place v, the cup-product 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
:
vS
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
vS
res
v
: Sel
m
(E/K)→⊕
vS
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 Swinnerton-Dyer introduced a fundamental new idea
in the theory, by formulating a conjectural relation between the arithmetic invariants of
E introduced above and the so-called Hasse-Weil L-series L(E, s) of E. This L-series is
described in Section 2.
2 The L-series 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
p-adic 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 non-singular points of
¯
E
p
(all points in
¯
E
p
(F
p
) are non-singular 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 L-series of E is the function of a complex variable s defined by
L(E, s) =
Y
p6|N
(1 a
p
p
s
+ p
12s
)
1
Y
p|N
(1 a
p
p
s
)
1
.
7
Remark 2.2.
1) (Connection with torsion points) The L-series 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 Mordell-Weil 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 L-functions associated to the cohomology of algebraic varieties.
2) If E and E
0
are Q-isogenous, 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 Q-isogeny 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 Q-isogeny.
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 L-series of E over K by the
formula
L(E/K, s) =
Y
v-N
(1 a
v
Nv
s
+ Nv
12s
)
1
Y
v|N
(1 a
v
Nv
s
)
1
, (7)
where the product is taken over the non-archimedean 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 L-series L(E, s) admits an analytic continuation to the whole complex plane.
This deep property of the L-series 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 L-series L(E, s ) satisfies a functional equation relative to s 7→ 2s, 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 Γ-function, and define the completed L-series 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 non-split multiplicative reduction at p (i.e.,
¯
E
p
has a node with tangents
not defined over F
p
), the L-series 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 Fermat-Pell
equation to Dirichlet L-series. See Darmon’s article [Da] for more details.
Let C
d
: x
2
dy
2
= 1 denote the Fermat-Pell 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 Fermat-Pell 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 p|2d, 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 L-series 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 L-series 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 Swinnerton-Dyer
conjecture, stated in Section 4, provides an in-depth 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 Swinnerton-Dyer 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 L-functions of motives due to
Deligne, Beilinson, Bloch-Kato, and others. See for example [Ki].
4 Statement of the Birch and Swinnerton-Dyer conjec-
ture
We attach the following arithmetic invariants to E (see [Sil] for more details).
10
1. The real period of E
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 R-vector 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,
1
E
). Then the period
E
can be defined as the scalar comparing
the natural Z-structures underlying the above R-vector spaces. This point of view is
taken by Deligne in his generalisation of the concept of period to the context of motivic
L-functions [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 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 Swinnerton-Dyer conjecture.
3. The regulator R
E
of E is the discriminant of the canonical N´eron-Tate height pairing.
If P
1
, ··· , P
r
E
denotes a Z-basis for the free-abelian 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 Shafarevich-Tate 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 non-vanishing coefficient in the Taylor
series expansion of L(E, s) at s = 1.
Conjecture 4.1 (Birch and Swinnerton-Dyer).
1) (“Millenium problem”) The equality ord
s=1
L(E, s) = rank
Z
E(Q) holds.
2) (BSD(Q
×
)) The equality L
(E, 1) =
E
· R
E
(mod Q
×
) holds.
11
3) (BSD) The following equality holds
L
(E, 1) =
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 Q-isogeny 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 Swinnerton-Dyer formulated the following conjecture:
define
f(x) :=
Y
px
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 L-series L(E/K, s) of an
elliptic curve E over a number field K.
5 Results on the Birch and Swinnerton-Dyer conjec-
ture
5.1 Modularity
Let E b e an elliptic curve over Q of conductor N, and write its L-series 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
2π
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
Shimura-Taniyama conjecture.
Theorem 5.1 (Shimura-Taniyama 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
(
+b
+d
) = ( + 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 N|c,
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
(
+b
+d
)( + 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 Atkin-Lehner 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
s1
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
Z-order 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 level-N
+
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
: BM
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 p-adic 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 p-adic 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 (Cerednik-Drinfeld). There is an identification
X(C
p
) = Γ
p
\H
p
of rigid-analytic curves over Q
p
2
(and also of formal schemes over Z
p
2
).
See [BC] for more details on Shimura curves and their non-archimedean uniformisation.
The Jacquet-Langlands correspondence. Let S
2
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
0
(N), F ), note that
ω
f
:= 2πif(τ)
defines a regular differential in H
0
(X
0
(N)
F
,
1
X
0
(N)/F
).
Theorem 5.6 (Jacquet-Langlands). Let f =
P
n1
a
n
q
n
be a newform in S
2
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
,
1
X/F
) such that
T
n
ω = a
n
ω, (n, N) = 1,
where T
n
denotes the n-th Fourier operator acting on X. The form ω is unique up to scaling
by non-zero 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), where f
ω
(τ) is a holomorphic function on H satisfying
f
ω
(γτ) = ( + d)
2
f
ω
(τ), (9)
for all γ =
µ
a b
c d
in Γ
.
2) Similarly, the pull-back by the covering H
p
X(C
p
) of a weight two modular form ω on
X/C
p
yields a rigid-analytic 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
: TZ, 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 Q-isogenous 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
well-defined 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 p-adic 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 j-function. Let
χ : Gal(H
c
/K)C
×
be a complex character, and let L(E/K, χ, s) be the complex L-series 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 Swinnerton-Dyer conjecture (suitably generalised)
predicts the existence of a point of infinite order in each χ-component
E(H
c
)
χ
= {P E(H
c
) C : σ(P ) = χ(σ)P, for all σ Gal(H
c
/K)}
of the Mordell-Weil group E(H
c
).
16
Under the current assumptions, there exists an embedding of Q-algebras
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 : KB, 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 Gross-Zagier-Zhang and Kolyvagin
Let L(E/K, χ, s) be the L-series 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
σGal(H/K)
χ(σ)α
σ
E(H)
χ
.
The following result was proved by Gross-Zagier [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 (Gross-Zagier, Zhang). The equality
L
0
(E/K, χ, 1) = C · hα
χ
, α
χ
i
NT
holds, where C is an explicit non-zero 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 Swinnerton-Dyer 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 non-zero, 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 Swinnerton-Dyer 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
L-series
L(E/K, s) = L(E, s)L(E, ε
K
, s), (11)
where L(E, ε
K
, s) =
P
n1
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 Bump-Friedberg-Hoffstein [BFH] or Murty-Murty [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 Coates-Wiles [CW] and Rubin [Ru1].
2) The Birch and Swinnerton-Dyer 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 non-zero, 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 non-zero.
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
`
: TZ/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 p-adic 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
non-archimedean 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 Bruhat-Tits 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 Z-coefficients 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 non-zero scalars in Z/pZ. It follows that P
K
gives
rise to a cohomology class c in H
1
(K, E
p
), well-defined 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 non-zero. Using the non-degeneracy 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 non-vanishing 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 Gross-Zagier 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 p-adic 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 Shafarevich-Tate group,
which is predicted by the Birch and Swinnerton-Dyer conjecture.
Assume that r
an
E
= 0. Theorem 5.16 and the theory of modular symbols [Man2] imply
the Birch and Swinnerton-Dyer conjecture for E up to Q
×
. Furthermore, a recent result of
22
Skinner-Urban [SU], combined with prior work of Kato [Ka], shows the p-part of the Birch
and Swinnerton-Dyer 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)/
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 four-dimensional p-adic Galois representation associated
with certain cuspidal automorphic representations of unitary groups of four-dimensional
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 non-zero and invoking Theorems 5.12 and 5.14, one checks that the
Birch and Swinnerton-Dyer 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)/
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 so-called 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 p-adic L-function attached to E, which may be viewed as a p-adic
analogue of the complex L-function 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
p-adic integers. If p is an ordinary prime for E, the p-adic L-function 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 p-adic L-function, constructed by Mazur-
Swinnerton-Dyer [MSw] (see also [MTT]), is the one considered in the work of Kato [Ka]
and Skinner-Urban [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 Λ-module
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 p-adic analogue of the Birch and Swinnerton-Dyer conjecture.
Noting that Λ can be identified with the Z
p
-algebra of formal power series in one variable
T , a well-defined order of vanishing ord(L
p
(E, Q
/Q)) (at T = 0) can be attached to the
cyclotomic p-adic L-function. 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 non-degeneracy of the cyclotomic p-adic height pairing and
finiteness of III(E/Q).
Similar considerations can be repeated for the p-adic L-function 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 pro-dihedral 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 p-adic L-function 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 L-function 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 p-adic L-function is
amenable to a cohomological description. On the other hand, no such an interpretation is
currently known for the whole complex L-function 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 so-called higher p-adic Abel-Jacobi
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 Kuga-Sato varieties, this program is
carried out in [BDP1] and [BDP2]. These papers show that the p-adic Abel-Jacobi image
of certain CM cycles gives rise to global points, without assuming the Tate conjecture for
higher dimensional varieties. The proof relates the above-mentioned Abel-Jacobi images to
values of p-adic L-functions, arising from the interpolation of special values associated to
p-adic analytic families of CM modular forms.
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... We then prove that our conjecture (equations (5) and (6)) is equivalent to the Birch and Swinnerton-Dyer conjecture for the Jacobian of X F , provided that our intersection pairing agrees with the Arakelov intersection pairing. This result was shown for f smooth in [6,Theorem 5.27] but only rather indirectly, via compatibility of both conjectures with the Tamagawa number conjecture. ...
... For the statement of the Birch and Swinnerton-Dyer conjecture for abelian varieties over number fields, we refer to seminar notes of B. Conrad [5]. Our formula (91) is equivalent to [5, L1, Conjecture 1.4.2], ...
Article
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We prove that the special-value conjecture for the zeta function of a proper, regular, flat arithmetic surface formulated in [6] at $s=1$ is equivalent to the Birch and Swinnerton-Dyer conjecture for the Jacobian of the generic fibre. There are two key results in the proof. The first is the triviality of the correction factor of [6, Conjecture 5.12], which we show for arbitrary regular proper arithmetic schemes. In the proof we need to develop some results for the eh-topology on schemes over finite fields which might be of independent interest. The second result is a different proof of a formula due to Geisser, relating the cardinalities of the Brauer and the Tate–Shafarevich group, which applies to arbitrary rather than only totally imaginary base fields.
... This p-adic variant of the "easy half" (11) of the Birch Swinnerton-Dyer conjecture has recently been proved in [BD6]. ...
... 2. Is it possible to replace the rigid analytic L-functions by classical ones in the proof of theorem BD? The proof in [BD6] is based on congruences in an essential way and breaks down entirely when the prime p is replaced by the "place at ∞". In this sense, it sheds no light on the original Birch and Swinnerton-Dyer conjecture, even on the "easy inequality". ...
... As in (i), we conclude that L(E/M 1 ) = 0 which implies L(E/F, 1) = 0. Now note that in part (iii) of Theorem 1, the only groups with a possible three dimensional irreducible representation, are those given in the statement of the theorem. This completes the proof. 2 Remark 4. If M/Q is a dihedral extension of degree 2n such that the fixed field C of the cyclic subgroup of order n of Gal(M/Q) is imaginary quadratic and of discriminant prime to the conductor of E, and (E(M ) ⊗ C) χ = 0 is infinite (χ is a two dimensional character of Gal(M/Q)), then by recent work of Bertolini and Darmon [2], L(E/Q ⊗ χ, 1) = 0. Applying this with the factorization of the L-function of E over M (see the paragraph before Proposition 4) and part (ii) of Theorem 1, we deduce that if F is a finite solvable extension of Q such that any quadratic subfield is imaginary and of discriminant prime to the conductor of E, and rank(E(F )) = 2 then L(E/F, 1) = 0. ...
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In this paper, we study the Mordell-Weil group of an elliptic curve as a Galois module. We consider an elliptic curve E dened over a number eld K whose Mordell-Weil rank over a Galois extensionF is 1, 2 or 3. We show thatE acquires a point (points) of innite order over a eld whose Galois group is one of Cn Cm (n = 1; 2; 3; 4; 6; m = 1; 2), Dn Cm (n = 2; 3; 4; 6; m = 1; 2), A4 Cm (m = 1; 2), S4 Cm (m = 1; 2). Next, we consider the case where E has complex multiplication by the ring of integersO of an imaginary quadratic eld K contained in K. Suppose that theO-rank over a Galois extension F is 1 or 2. If K6=Q( p 1) andQ( p 3) and hK (class number of K) is odd, we show that E acquires positiveO-rank over a cyclic extension of K or over a eld whose Galois group is one of SL2(Z=3Z), an extension of SL2(Z=3Z) by Z=2Z, or a central extension by the dihedral group. Finally, we discuss the relation of the above results to the vanishing of L-functions.
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This article is a revised version of the text of the plenary conference I gave at the XIX Congress of "Unione Matematica Italiana", held in Bologna in September 2011. It discusses the arithmetic significance of the values at integers of the complex and p-adic L-functions associated to Dirichlet characters and to elliptic curves.
Article
Let E be an elliptic curve over QQ and let K be a quadratic imaginary field that satisfies the Heegner hypothesis. We study the arithmetic of E over ring class extensions of K, with particular focus on the case when E has analytic rank at least 2 over QQ. We also point out an issue in the literature regarding generalizing the Gross–Zagier formula, and offer a conjecturally correct formula.
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Contents 1 Elliptic curves and modular forms 5 1.1 Elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 The Birch and Swinnerton-Dyer conjecture . . . . . . . . . . . 7 1.3 Modularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Cyclotomic p-adic L-functions 12 2.1 The Mazur{Swinnerton-Dyer p-adic L-function . . . . . . . . . 13 2.2 The Mazur-Tate-Teitelbaum conjecture . . . . . . . . . . . . . 16 2.3 Results on the Mazur-Tate-Teitelbaum conjecture . . . . . . . 17 3 Schneider's approach 18 3.1 Rigid analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 Shimura Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.4 Schneider's distribution . . . . . . . . . . . . . . . . . . . . . . 25 3.5 The Jacquet-Langlands correspondence . . . . . . . . . . .
Chapter
The seminal formula of Gross and Zagier relating heights of Heegner points to derivatives of the associated Rankin L-series has led to many generalisations and extensions in a variety of different directions, spawning a fertile area of study that remains active to this day. This volume, based on a workshop on Special Values of Rankin L-series held at the MSRI in December 2001, is a collection of thirteen articles written by many of the leading contributors in the field, having the Gross-Zagier formula and its avatars as a common unifying theme. It serves as a valuable reference for mathematicians wishing to become further acquainted with the theory of complex multiplication, automorphic forms, the Rankin-Selberg method, arithmetic intersection theory, Iwasawa theory, and other topics related to the Gross-Zagier formula.
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We outline a new construction of rational points on CM elliptic curves, using cycles on higher-dimensional varieties, contingent on certain cases of the Tate conjecture. This construction admits of complex and p-adic analogs that are defined independently of the Tate conjecture. In the p-adic case, using p-adic Rankin L-functions and a p-adic Gross–Zagier type formula proved in our articles [2, 3], we show unconditionally that the points so constructed are in fact rational. In the complex case, we are unable to prove rationality (or even algebraicity) but we can verify it numerically in several cases.
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Let E/F be a modular elliptic curve defined over a totally real number field F and let φ be its associated eigenform. This article presents a new method, inspired by a recent work of Bertolini and Darmon, to control the rank of E over suitable quadratic imaginary extensions K/F. In particular, this argument can also be applied to the cases not covered by the work of Kolyvagin and Logachëv, that is, when [F : ℚ] is even and φ not new at any prime.
Book
The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic theory of elliptic curves in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. The book begins with a brief discussion of the necessary algebro-geometric results, and proceeds with an exposition of the geometry of elliptic curves, the formal group of an elliptic curve, and elliptic curves over finite fields, the complex numbers, local fields, and global fields. Included are proofs of the Mordell–Weil theorem giving finite generation of the group of rational points and Siegel's theorem on finiteness of integral points. For this second edition of The Arithmetic of Elliptic Curves, there is a new chapter entitled Algorithmic Aspects of Elliptic Curves, with an emphasis on algorithms over finite fields which have cryptographic applications. These include Lenstra's factorization algorithm, Schoof's point counting algorithm, Miller's algorithm to compute the Tate and Weil pairings, and a description of aspects of elliptic curve cryptography. There is also a new section on Szpiro's conjecture and ABC, as well as expanded and updated accounts of recent developments and numerous new exercises. The book contains three appendices: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and a third appendix giving an overview of more advanced topics.