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19 Pages·2008·0.31 MB·English

by  Akhil Mathew

Elliptic Curves
Akhil Mathew
Department ofMathematics
DrewUniversity
Math155,ProfessorAlanCandiotti
10 Dec. 2008
AkhilMathew(DepartmentofMathematics DrewUniversityEllMipatitchC1u5r5v,ePsrofessorAlanCandiotti) 10Dec.2008 1/19
What is an Elliptic Curve?
An elliptic curve is the locus of solutions of an equation of the form
y2 = x3+Ax +B,
where for nonsingularity
4A3+27B2 = 0.
6
There is also a point “at infinity” (not shown).
Figure: The elliptic curve y2 =x3 x
−
AkhilMathew(DepartmentofMathematics DrewUniversityEllMipatitchC1u5r5v,ePsrofessorAlanCandiotti) 10Dec.2008 2/19
The Addition Law
To add points M ,M , draw the line D through them. Find the third
1 2
intersection P of the line with the curve. Flip P over the x-axis to get
M = M +M .
3 1 2
Figure: The group law on an elliptic curve from [1]
AkhilMathew(DepartmentofMathematics DrewUniversityEllMipatitchC1u5r5v,ePsrofessorAlanCandiotti) 10Dec.2008 3/19
Properties of the Group Law
Theorem
Addition is commutative and associative. In fact under the addition law,
with the point at infinity as “zero,” an elliptic curve is an abelian group.
Proof.
Addition in an elliptic curve corresponds roughly to addition in the divisor
class group (i.e. using algebraic geometry, cf. [4]). In the complex case,
we will show another proof.
AkhilMathew(DepartmentofMathematics DrewUniversityEllMipatitchC1u5r5v,ePsrofessorAlanCandiotti) 10Dec.2008 4/19
Examples of the Addition Law
Example
The origin O is the point at infinity.
Example
If P is a point, then P is P flipped over the x-axis.
−
P
P0
Figure: An illustration of the preceding examples
AkhilMathew(DepartmentofMathematics DrewUniversityEllMipatitchC1u5r5v,ePsrofessorAlanCandiotti) 10Dec.2008 5/19
Elliptic Curves over the Complex Numbers
Let S1 = R/Z be the unit circle.
Theorem
An elliptic curve E over the complex numbers is group-isomorphic to the
torus
S1 S1,
×
cf. [3].
Figure: A torus from [2]
AkhilMathew(DepartmentofMathematics DrewUniversityEllMipatitchC1u5r5v,ePsrofessorAlanCandiotti) 10Dec.2008 6/19
Lattices
A lattice L C is a discrete free abelian subgroup of rank 2. Then C/L is
⊂
a torus and a complex Riemann surface.
Figure: A lattice
One can construct a lattice L C such that
⊂
E = C/L topologically, analytically, and group-theoretically.
AkhilMathew(DepartmentofMathematics DrewUniversityEllMipatitchC1u5r5v,ePsrofessorAlanCandiotti) 10Dec.2008 7/19
An Overview of the Proof
Overview of Proof.
The isomorphism is given by Weierstrass ℘-functions:
z (℘(z;L),℘0(z;L)).
→
The Weierstrass ℘-function is defined specifically as:
1 1 1
℘(z;L) = + .
z2 (z ω)2 − ω2
ω∈XL,ω6=0 −
℘ functions are doubly periodic and meromorphic (i.e. elliptic). Hence
they are defined as a map of C/L S2 (S2 being the Riemann sphere).
→
The addition law for Weierstrass-℘ functions is basically the theorem.
AkhilMathew(DepartmentofMathematics DrewUniversityEllMipatitchC1u5r5v,ePsrofessorAlanCandiotti) 10Dec.2008 8/19
Torsion Points over Elliptic Curves
E is an elliptic curve over C, E = C/L= S1 S1 as groups. We want
×
points P E of order m, i.e. such that
∈
mP = 0 and nP = 0 if 0 < n < m.
6
The points in S1 S1 of order dividing m are of the form (z ,z ) for z ,z
1 2 1 2
×
m-th roots of unity.
Hence:
Theorem
There are m2 points of order dividing m, and they form a group E[m]
isomorphic to Z/mZ Z/mZ.
×
In general, this is true over any algebraically closed field.
AkhilMathew(DepartmentofMathematics DrewUniversityEllMipatitchC1u5r5v,ePsrofessorAlanCandiotti) 10Dec.2008 9/19
Example: 2-Torsion Points
If 2P = 0, then P = P, so P has y-coordinate zero or P is the point at
−
infinity. This characterizes all 2-torsion points.
There are three ways for y = 0 in the equation
y2 = P(x),P a cubic polynomial (Fundamental Theorem of Algebra),
and we throw in the point at infinity to get:
Theorem
There are 4 points of order dividing 2.
AkhilMathew(DepartmentofMathematics DrewUniversityEllMipatitchC1u5r5v,ePsrofessorAlanCandiotti) 10Dec.2008 10/19