ELLIPTIC CURVES
P. Stevenhagen
Universiteit Leiden
2008
,Date of this online version: March 11, 2008
Mail address of the author:
P. Stevenhagen
Mathematisch Instituut
Universiteit Leiden
Postbus 9512
2300 RA Leiden
Netherlands
,CONTENTS
1. Elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2. Elliptic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3. Complex elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . 22
, Elliptic curves - §1 version March 11, 2008, 22:25
1. Elliptic integrals
The subject of elliptic curves has its roots in the differential and integral calculus, which
was developed in the 17th and 18th century and became the main subject of what is nowa-
days a ‘basic mathematical education’. In calculus, one tries to integrate the differentials
f (t)dt associated with, say, a real-valued function f on the real line. As is well known,
such integrals are related to the area of certain surfaces bounded by the graph of f . Ex-
plicit integration of the differential f (t)dt, which amounts to finding an anti-derivative F
satisfying dF/dt = f , can only be performed for a very limited number of ‘standard inte-
grals’. These include the integrals of polynomial differentials tk dt with k ∈ Z≥0 , rational
differentials as (t − α)−k with k ∈ Z>0 and a few ‘exponential differentials’ as et dt and
sin t dt. Over the complex numbers, any rational differential can be written as a sum of
elementary differentials.
Exercise 1. Show that every rational function f ∈ C(t) can be written as unique C-linear combination
of monomials
R tk with k ∈ Z≥0 and fractions (t − α)−k with α ∈ C and k ∈ Z≥1 . Use this representation
to write f (t)dt as a sum of elementary functions. [Hint: partial fraction expansion.]
Even if one restricts to polynomial or rational functions f , already the problem of com-
puting the length of the graph of f , an old problem
p known as the ‘rectification’ of plane
curves, leads to the non-elementary differential 1 + f 0 (t)2 dt. If R ∈ C(x, y) ispa rational
function and f ∈ C[t] a polynomial that is not a square, the differential R(t, f (t))dt is
called hyperelliptic. We can and will always suppose that p f is separable, i.e., it has no mul-
tiple roots.pIf f is of degree 1, one can transform R(t, f (t))dt into a rational differential
by taking f (t) as a new variable. If f is quadratic, one can apply a linear transformation
t 7→ at + b to reduce to the case f (t) = 1 − t2 . We will see in a moment that the resulting
integrals are closely related to the problem of computing lengths of circular arcs or, what
amounts to the same thing, inverting
p trigonometric functions. If f is of degree 3 or 4 and
squarefree, the differential R(t, f (t))dt is said to be elliptic.
Exercise 2. Show that, for c 6= 0, the length of the ellipse with equation y 2 = c2 (1 − x2 ) in R2 equals
Z r Z
1
1 + (c2 − 1)t2
π/2 p
2 dt = 2 1 + (c2 − 1)sin2 φ dφ,
−1
1 − t2 −π/2
q
1+(c2 −1)t2
and that the differential 1−t2
dt is elliptic for c2 6= 1.
Elliptic differentials lead naturally to the study of elliptic functions and elliptic curves.
In a similar way, the case of f of higher degree gives rise to hyperelliptic curves. More
generally, it has gradually become clear during the 19th century that an algebraic differen-
tial R(t, u)dt, with R a rational function and t and u satisfying some polynomial relation
P (t, u) = 0, should be studied as an object living on the plane algebraic curve defined
by the equation P (x, y) = 0. For hyperelliptic differentials, this is the hyperelliptic curve
given by the equation y 2 = f (x).
4
P. Stevenhagen
Universiteit Leiden
2008
,Date of this online version: March 11, 2008
Mail address of the author:
P. Stevenhagen
Mathematisch Instituut
Universiteit Leiden
Postbus 9512
2300 RA Leiden
Netherlands
,CONTENTS
1. Elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2. Elliptic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3. Complex elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . 22
, Elliptic curves - §1 version March 11, 2008, 22:25
1. Elliptic integrals
The subject of elliptic curves has its roots in the differential and integral calculus, which
was developed in the 17th and 18th century and became the main subject of what is nowa-
days a ‘basic mathematical education’. In calculus, one tries to integrate the differentials
f (t)dt associated with, say, a real-valued function f on the real line. As is well known,
such integrals are related to the area of certain surfaces bounded by the graph of f . Ex-
plicit integration of the differential f (t)dt, which amounts to finding an anti-derivative F
satisfying dF/dt = f , can only be performed for a very limited number of ‘standard inte-
grals’. These include the integrals of polynomial differentials tk dt with k ∈ Z≥0 , rational
differentials as (t − α)−k with k ∈ Z>0 and a few ‘exponential differentials’ as et dt and
sin t dt. Over the complex numbers, any rational differential can be written as a sum of
elementary differentials.
Exercise 1. Show that every rational function f ∈ C(t) can be written as unique C-linear combination
of monomials
R tk with k ∈ Z≥0 and fractions (t − α)−k with α ∈ C and k ∈ Z≥1 . Use this representation
to write f (t)dt as a sum of elementary functions. [Hint: partial fraction expansion.]
Even if one restricts to polynomial or rational functions f , already the problem of com-
puting the length of the graph of f , an old problem
p known as the ‘rectification’ of plane
curves, leads to the non-elementary differential 1 + f 0 (t)2 dt. If R ∈ C(x, y) ispa rational
function and f ∈ C[t] a polynomial that is not a square, the differential R(t, f (t))dt is
called hyperelliptic. We can and will always suppose that p f is separable, i.e., it has no mul-
tiple roots.pIf f is of degree 1, one can transform R(t, f (t))dt into a rational differential
by taking f (t) as a new variable. If f is quadratic, one can apply a linear transformation
t 7→ at + b to reduce to the case f (t) = 1 − t2 . We will see in a moment that the resulting
integrals are closely related to the problem of computing lengths of circular arcs or, what
amounts to the same thing, inverting
p trigonometric functions. If f is of degree 3 or 4 and
squarefree, the differential R(t, f (t))dt is said to be elliptic.
Exercise 2. Show that, for c 6= 0, the length of the ellipse with equation y 2 = c2 (1 − x2 ) in R2 equals
Z r Z
1
1 + (c2 − 1)t2
π/2 p
2 dt = 2 1 + (c2 − 1)sin2 φ dφ,
−1
1 − t2 −π/2
q
1+(c2 −1)t2
and that the differential 1−t2
dt is elliptic for c2 6= 1.
Elliptic differentials lead naturally to the study of elliptic functions and elliptic curves.
In a similar way, the case of f of higher degree gives rise to hyperelliptic curves. More
generally, it has gradually become clear during the 19th century that an algebraic differen-
tial R(t, u)dt, with R a rational function and t and u satisfying some polynomial relation
P (t, u) = 0, should be studied as an object living on the plane algebraic curve defined
by the equation P (x, y) = 0. For hyperelliptic differentials, this is the hyperelliptic curve
given by the equation y 2 = f (x).
4
