Download 1 Definition. First properties

Facultatea de Matematică
Anul II Master, Geometrie Algebrică
Curbe eliptice
These notes use as main source the text
J. S. Milne: Elliptic curves and algebraic geometry, Math679 U. Michigan
notes (1996)
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Definition. First properties
Definition 1.1. Let k be a field. An elliptic curve over k can be defined as:
• (a) A complete nonsingular curve E of genus 1 over k together with a
point O ∈ E(k).
• (b) A nonsingular plane projective E of degree 3 together with a point
O ∈ E(k).
• (c) A nonsingular plane projective curve E of the form
Y 2 Z + a1 XY Z + a3 Y Z 2 = X 3 + a2 X 2 Z + a4 XZ 2 + a6 Z 3 .
The definition makes sense once we prove
Proposition 1.1. Let E be an algebraic curve. The three properties listed
in the definition hereinbefore are equivalent.
Demonstraţie. Let E be as in (c). Then E(k) contains a canonical element
O = (0 : 1 : 0), and the pair (E; O) satisfies the other two properties. This
is obvious for (b), and (a) follows readily.
Let (E; O) be as in (a). Then there is an isomorphism from E onto a
curve as in (c) sending O to (0 : 1 : 0).
Let (E; O) be as in (b). Then there is a change of variables transforming
E into a curve as in (c) and O into (0 : 1 : 0) (if O is a point of inflection,
the change of variables can be taken to be linear).
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Let (E; O) be a nonsingular cubic curve in P2( k). We assume that O is a
point of inflection.
Proposition 1.2. (a) After a linear change of variables (with coefficients
in k), the point O will be (0 : 1 : 0) and its tangent line will be H∞ : Z = 0;
(b) If (0 : 1 : 0) ∈ E(k) and the tangent line to E at (0 : 1 : 0) is
H∞ : Z = 0, then the equation of E has the form
Y 2 Z + a1 XY Z + a3 Y Z 2 = X 3 + a2 X 2 Z + a4 XZ 2 + a6 Z 3 .
Thus, an elliptic curve is isomorphic to a curve of the form
E : Y 2 Z + a1 XY Z + a3 Y Z 2 = X 3 + a2 X 2 Z + a4 XZ 2 + a6 Z 3 .
and, conversely, every nonsingular such E is an elliptic curve. This is usually
referred to as the canonical or Weierstrass equation of the curve. One can
show that it is canonical up to a change of variables of the form:
X = u2 X 0 + r
Y = u3 Y 0 + su2 X 0 + t
with u; r; s; t ∈ k and u 6= 0.
Theorem 1.3. Assume char k 6= 2, 3.
(a) The curve
E(a, b) : Y 2 Z = X 3 + aXZ 2 + bZ 3
,
a, b ∈ k
is nonsingular, and hence (together with O = (0 : 1 : 0)) defines an elliptic
curve over k, if and only if 4a3 + 27b2 6= 0.
(b) Every elliptic curve over k is isomorphic to one of the form E(a, b).
(c) Two elliptic curves E(a, b) and E(a0 , b0 ) are isomorphic if and only if
there exists a c ∈ k ∗ such a0 = c4 a, b0 = c6 b; the isomorphism is then
(x : y : z) 7→ (c2 x : c3 y : z).
For an elliptic curve E, define
4a3
4a3 + 27b2
if E ' E(a, b). Since the expression on the right is unchanged when (a, b) is
replaced by (c4 a, c6 b), this is well-defined, and E ' E 0 iff j(E) = j(E 0 ) when
k is algebraically closed.
j(E) = 1728
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The group law
The point at infinity, O, is the zero for the group law. The group law is
determined by:
P + Q + R = O ⇔ P ; Q; Rlieonastraightline.
If P = (x : y : z), then −P = (x : −y : z).
In particular, −P = P , i.e., P has order 2 if and only if y = 0. The points
of order two are the points (x : 0 : 1) where x is a root of X 3 + aX + b.
Proposition 2.1.
• P + O = O + P = P for all P ∈ E(a, b) ( O is the
zero element of the group).
• Let P = (x, y) 6= O. Then
−P := (x, −y)
;
P + (−P ) = O
• Let P (x1 , y1 ) şi Q(x2 , y2 ) be points different to O, P 6= −Q. Then
P + Q = (x3 , y3 ), where
x3 = λ2 − x1 − x2
with
(
λ :=
,
y2 −y1
x2 −x1
3x21 +a
2y1
y3 = λ(x1 − x3 ) − y1 ,
dacă P 6= Q
dacă P = Q
Geometrically, the group law can be described as follows:
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For example:
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