L-Series, the Conjecture of
Birch and Swinnerton-Dyer,
and a Million Dollar Prize
L-Series, BirchSwinnerton-Dyer, and $1,000,000
The L-Series of an Elliptic Curve
Let E be an elliptic curve given as usual by an equation
y 2 = x3 + Ax + B
with A, B Z.
For
Canonical Heights on Elliptic Curves
Using Heights To Compute Relations
Here is an initially plausible way to solve ECDLP via
lifting. Let S, T E(Fp).
(1) Lift E and T to a curve E/Q and point T E(Q).
(2) Do this in such a way that E(Q) has rank 1. (Thi
74
CHAPTER 6. VARIETIES, MORPHISMS, AND RATIONAL MAPS
also assume Z is afne, since Z is a union of open afne subvarieties. But this case
is trivial, since the product of polynomial maps is certainly a polynomial map.
(3): Apply (2) to the morphism ( f pr1
6.4. PRODUCTS AND GRAPHS
73
Problem 1.17(c).)
6.24. Let X be a variety, P,Q two distinct points of X . Show that there is an f k(X )
that is dened at P and at Q, with f (P ) = 0, f (Q) = 0 (Problems 6.23, 1.17). So f
mP (X ), 1/ f OQ (X ). The local ring
72
CHAPTER 6. VARIETIES, MORPHISMS, AND RATIONAL MAPS
Corollary. Let X be a variety, U a neighborhood of a point P in X . Then there is a
neighborhood V of P , V U , such that V is an afne variety.
Proof. If X is open in a projective variety X Pn , and P
Canonical Heights on Elliptic Curves
Canonical Heights
Taking a limit gets rid of those pesky O(1)s.
Theorem. (Nron, Tate). The limit
e
) = lim 1 h([n]P )
h(P
n n2
exists and has the following properties:
h(P ) = h(P ) + O(1).
h([n]P ) = n2h(P ).
h(P
6.3. MORPHISMS OF VARIETIES
71
Proposition 4. Any closed subvariety of Pn1 Pnr is a projective variety. Any
variety is isomorphic to an open subvariety of a projective variety.
Proof. The second statement follows from the rst, since Pn1 Pnr Am is isomorph
Height Functions on Elliptic Curves
The Quadratic Growth of the Height on Elliptic Curves
We illustrate with the elliptic curve and point
E : y 2 = x3 + x + 1
and
P = (0, 1).
Here is a table of H(xnP ) for n = 1, 2, . . . , 25.
1
1
2
2
3
13
4
36
5
685
6
7
70
CHAPTER 6. VARIETIES, MORPHISMS, AND RATIONAL MAPS
6.10. Let U be an open subvariety of a variety X , Y a closed subvariety of U . Let Z
be the closure of Y in X . Show that
(a) Z is a closed subvariety of X .
(b) Y is an open subvariety of Z .
6.11. (
Height Functions on Elliptic Curves
Height Functions
In order to understand lifting (and for many other purposes), it is important to answer the question:
How complicated are the points in E(Q)?
The answer is provided by the Theory of Heights.
The Height
6.5. ALGEBRAIC FUNCTION FIELDS AND DIMENSION OF VARIETIES
75
(d) GLn (k) = cfw_invertible n n matrices is an afne open subvariety of M n (k),
and a group under multiplication.
(e) C a nonsingular plane cubic, O C , the resulting addition (see Problem
5.38
76
CHAPTER 6. VARIETIES, MORPHISMS, AND RATIONAL MAPS
(2) If V is the projective closure of an afne variety V , then dimV = dimV .
(3) A variety has dimension zero if and only if it is a point.
(4) Every proper closed subvariety of a curve is a point.
(5)
82
CHAPTER 7. RESOLUTION OF SINGULARITIES
Corollary 1. Let f be a rational map from a curve C to a projective curve C . Then the
domain of f includes every simple point of C . If C is nonsingular, f is a morphism.
Proof. If F is not dominating, it is cons
Chapter 7
Resolution of Singularities
7.1 Rational Maps of Curves
A point P on an arbitrary curve C is called a simple point if O P (C ) is a discrete
valuation ring. If C is a plane curve, this agrees with our original denition (Theorem1 of 3.2). We let
Factorization Using Elliptic Curves
Using Elliptic Curves for Factorization
Hendrik Lenstra observed that one can replace the multiplicative group F with an elliptic curve group E(Fp).
p
More precisely, choose an elliptic curve modulo N and
a point on the
6.6. RATIONAL MAPS
79
6.41. Every n-dimensional variety is birationally equivalent to a hypersurface in
An+1 (or Pn+1 ).
6.42. Suppose X , Y varieties, P X , Q Y , with O P (X ) isomorphic (over k) to
OQ (Y ). Then there are neighborhoods U of P on X , V
Factorization Using
Elliptic Curves
Factorization Using Elliptic Curves
Pollards p 1 Factorization Algorithm
Let N = p1p2 pr be a number to be factored.
Pollards p 1 Factorization Algorithm works
if one of the primes pk dividing N has the property
that pk
78
CHAPTER 6. VARIETIES, MORPHISMS, AND RATIONAL MAPS
(3) We may assume X and Y are afne. Then, as in (2), if : k(Y ) k(X ),
(Y ) (X b ) for some b (X ), so is induced by a morphism f : X b Y .
Therefore f (X b ) is dense in Y since f is one-to-one (Probl
Canonical Heights on Elliptic Curves
Descent and the Mordell-Weil Theorem
Recall that Mordells Theorem (which was later generalized by Andr Weil) says:
e
Mordell-Weil Theorem. The group of rational
points E(Q) is a nitely generated abelian group.
The proo
6.6. RATIONAL MAPS
77
6.34. Show that dim An = dim Pn = n.
6.35. Let Y be a closed subvariety of a variety X . Then dim Y dim X , with equality
if and only if Y = X .
6.36. Let K = k(x 1 , . . . , x n ) be a function eld in r variables over k. (a) Show th
6.2. VARIETIES
69
6.8. Let U be an open subset of a variety V , z k(V ). Suppose z O P (V ) for all
P U . Show that U z = cfw_P U | z(P ) = 0 is open, and that the mapping from U to
k = A1 dened by P z(P ) is continuous.
6.2 Varieties
Let V be a nonempty
68
CHAPTER 6. VARIETIES, MORPHISMS, AND RATIONAL MAPS
For any subset Y of a topological space X , the closure of Y in X is the intersection
of all closed subsets of X that contain Y . The set Y is said to be dense in X if X is the
closure of Y in X ; equi
60
CHAPTER 5. PROJECTIVE PLANE CURVES
Theorem 2. If F is an irreducible curve of degree n, then
m P (m P 1)
2
(n1)(n2)
.
2
Proof. Since r := (n1)(n1+3) (mP 1)(mP ) (n1)n (mP 1)mP 0, we may choose
2
2
2
2
simple points Q 1 , . . . ,Q r F . Then Theorem 1 o
5.4. MULTIPLE POINTS
59
5.21. Show that every nonsingular projective plane curve is irreducible. Is this true
for afne curves?
5.22. Let F be an irreducible curve of degree n. Assume F = 0. Apply Corollary 1
X
to F and F X , and conclude that m P (F )(m P
The Elliptic Curve Discrete Logarithm Problem
The Tate Pairing
The Weil pairing is alternating. It is often more ecient
to use the Tate pairing, which is symmetric.
The Tate Pairing is a non-degenerate symmetric
bilinear form
F
E(Fq )
q
E(Fq )N
N , (P,
58
CHAPTER 5. PROJECTIVE PLANE CURVES
If we restrict these maps to the forms of various degrees, we get the following exact
sequences:
0 R d mn R d m R d n R d d 0.
Since dim R d = (d +1)(d +2) , it follows from Proposition 7 of 2.10 (with a calculation)
5.3. BZOUTS THEOREM
57
Let W0 = V (d 1; (r 2)P, r 2 P 2 , . . . , r n P n ). For F W0 , F = a i X i Y r 2i + . Set
Wi = cfw_F W0 | a j = 0 for j < i . By induction,
W0
W1
Wr 1 = V (d 1; (r 1)P, r 2 P 2 , . . . , r n P n ).
If F i Wi , F i Wi +1 , then Y F
The Elliptic Curve Discrete Logarithm Problem
The Weil Pairing
An important tool for studying elliptic curves E over
any eld is the Weil Pairing. Let
EN = cfw_P E : N P = O and N = cfw_ : N = 1.
The Weil pairing is a non-degenerate alternating bilinear
fo
5.2. LINEAR SYSTEMS OF CURVES
55
equivalence, there is only one irreducible conic: Y Z = X 2 . Any irreducible conic is
nonsingular.
5.10. Let F be an irreducible cubic, P = [0 : 0 : 1] a cusp on F , Y = 0 the tangent line
to F at P . Show that F = aY 2 Z