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```ELLIPSES
AND
ELLIPTIC CURVES
M. Ram Murty
Queen’s University
Planetary orbits are elliptical
x2
a2
What is an ellipse?
y2
+
b2
= 1
An ellipse has two foci
From:
gomath.com/geometry/ellipse.php
Metric mishap causes loss of Mars
orbiter (Sept. 30, 1999)
How to calculate arc length
Let f : [0; 1] ! R2 be given by t 7
! (x(t); y(t)).
p
By Pyt hagoras, ¢ s = (¢ x) 2 + (¢ y) 2
r
ds =
Arc lengt h =
¡
R
1
0
dx
dt
r
³
¢2
+
¡
¢2
dx
dt
dy
dt
´2
³
+
dy
dt
dt
´2
dt
Circumference of an ellipse
We can parametrize the points of an ellipse in the ¯rst quadrant by
f : [0; ¼=2] ! (asint; bcost):
Circumference
=
p
R
4 ¼=2 a2 cos2 t + b2 sin2 tdt:
0
Observe t hat if a = b, we get 2¼a.
We can simplify t his as follows.
Let cos2 µ = 1 ¡ sin2 µ.
T he int egral
becomes
p
R
4a ¼=2 1 ¡ ¸ sin2 µdµ
0
Where ¸ = 1¡ b2 =a2 .
We can expand t he square root using
t heRbinomial
t ³heorem:
´
P
1=2 (¡ 1) n ¸ n sin2n µdµ:
4a ¼=2 1
0
n= 0
n
We
R can use t he fact t hat
2 ¼=2 sin2n µdµ = ¼( 1=2) ( 1=2) + 1) ¢¢¢( ( 1=2) + ( n ¡
0
n!
1) )
The final answer
T he circumference is given by
2¼aF (1=2; ¡ 1=2; 1; ¸ )
where
P
1
( a) n ( b) n zn
F (a; b; c; z) =
n = 0 n !( c) n
is t he hypergeomet ric series and
(a) n = a(a + 1)(a + 2) ¢¢¢(a + n ¡ 1):
We now use Landens t ransformat ion:
F (a; b; 2b; 4x ) = (1 + x) 2a F (a; a ¡ b+ 1=2; b+ 1=2; x 2 ).
(1+ x ) 2
We put a = ¡ 1=2; b = 1=2 and x = (a ¡ b)=(a + b).
T he answer can now be writ t en as ¼(a + b)F (¡ 1=2; ¡ 1=2; 1; x 2 ).
Approximations
• In 1609, Kepler used the
approximation (a+b). The
above formula shows the
perimeter is always greater than
this amount.
• In 1773, Euler gave the
approximation 2√(a²+b²)/2.
• In 1914, Ramanujan gave the
approximation (3(a+b) √(a+3b)(3a+b)).
What kind of number is this?
For example, if a and b are rat ional,
is t he circumeference irrat ional?
In case a = b we have a circle
and t he circumference 2¼a
is irrat ional.
In fact , ¼is t ranscendent al.
What does this mean?
T his means t hat ¼does not sat isfy
an equat ion of t he type
x n + an ¡ 1 x n ¡ 1 + ¢¢¢+ a1 x + a0 = 0
wit h ai rat ional numbers.
In 1882, Ferdinand von Lindemann
proved t hat ¼is t ranscendent al.
1852-1939
This means that you can’t square
the circle!
• There is an ancient problem of
constructing a square with straightedge
and compass whose area equals .
• Theorem: if you can construct a line
segment of length α then α is an algebraic
number.
• Since  is not algebraic, neither is √.
Some other interesting numbers
• The number e is transcendental.
• This was first proved by Charles Hermite
(1822-1901) in 1873.
Is ¼+ e t ranscendent al?
Answer: unknown.
Is ¼e t ranscendent al?
Answer: unknown.
Not both can be algebraic!
• Here’s a proof.
• (x
If ¡both
are
algebraic,
then
2
e)(x¡ ¼) = x ¡ (e + ¼)x + ¼e
is a quadrat ic polynomial wit h algebraic coe± cient s.
T his implies bot h e and ¼ are algebraic,
a cont radict ion t o t he t heorems of
Hermit e and Lindemann.
Conject ure:
¼ and e are algebraically independent .
But what about the case of the
ellipse?
Is t hep int egral
R
¼=2
a2 cos2 t + b2 sin2 tdt
0
t ranscendent al
if a and b are rat ional?
Yes.
T his is a t heorem of
T heodor Schneider (1911-1988)
Let s look at t he int egral again.
Put t ing u = sin t t he int egral becomes
R q
1
0
a 2 ¡ ( a 2 ¡ b2 ) u 2
1¡ u 2
du:
Set k 2 = 1 ¡ b2 =a2 :
T he qint egral becomes
R
a 1 1¡ k 2 u 2 du:
1¡ u 2
0
Put t = 1 ¡ k 2 u2 :
1
2
R
1
1¡ k 2
p
t dt
t ( t ¡ 1) ( t ¡ ( 1¡ k 2 ) )
Elliptic Integrals
Int
R egrals of t he form
p
dx
x 3 + a2 x 2 + a3 x + a4
are called ellipt ic int egrals of t he ¯rst kind.
Int
R egrals of t he form
p
x dx
x 3 + a2 x 2 + a3 x + a4
are called ellipt ic int egrals of t he second kind.
Our int egral is of t he second kind.
Elliptic Curves
y 2 = x(x ¡ 1)(x ¡ (1 ¡ k 2 ))
The equation
is an example of an elliptic curve.
0
1 ¡ k2
1
One can writ e t he equat ion of such a curve as
y2 = 4x 3 ¡ ax ¡ b.
C
Elliptic Curves over
Let L be a lat t ice of rank 2 over R.
T his means t hat L = Z!
©
Z!
1
2
We at t ach t he Weierst rass } -funct ion t o L :
} (z) =
1
z2
+
³
P
06
= ! 2L
´
1
( z¡ ! ) 2
¡
1
!
2
!
2
!
1
T he } funct ion is doubly periodic:
T his means
} (z) = } (z + ! 1 ) = } (z + ! 2 ).
e2¼i = 1
T he exponent ial funct ion is periodic since
ez = ez+ 2¼i :
T he periods of t he exponent ial funct ion
consist of mult iples of 2¼i .
T hat is, t he period \ lat t ice; ;
is of t he form Z(2¼i ).
The Weierstrass function
T he } -funct ion sat is¯es
t he following di®erent ial equat ion:
(} ; (z)) 2 = 4} (z) 3 ¡ g2 } (z)¡ g3 ;
where P
g2 = 60
!¡ 4
06
= ! 2L
and
P
g3 = 140
! ¡ 6:
06
= ! 2L
T his means t hat (} (z); } ; (z))
is a point on t he curve
y 2 = 4x 3 ¡ g2 x¡ g3 :
The Uniformization Theorem
Conversely, every complex point on t he curve
y2 = 4x 3 ¡ g2 x ¡ g3
is of t he form (} (z); } ; (z))
for some z 2 C.
Given any g2 ; g3 2 C,
t here is a } funct ion such t hat
(} ; (z)) 2 = 4} (z) 3 ¡ g2 } (z)¡ g3 :
Even and Odd Functions
Recall t hat a funct ion f is even if
f (z) = f (¡ z).
For example, z2 and
cosz are even funct ions.
P
1
} (z) is even since } (z) =
+
06
= ! 2L
z2
} ; (z) is odd since } ; (z) = ¡
2
z3
¡ 2
Not e t hat } ; (¡ ! =2) = ¡ } ; (! =2)
T his means } ; (! =2) = 0:
A funct ion is called odd if
f (z) = ¡ f (¡ z).
For example, z and
sin z are odd funct ions.
³
´
1
¡ 1
( z¡ ! ) 2
!
2
³
P
06
= ! 2L
´
1
( z¡ ! ) 3
.
In part icular
} ; (! 1 =2) = 0
} ; (! 2 =2) = 0 and
} ; ((! 1 + ! 2 )=2) = 0.
T his means t he numbers
} (! 1 =2); } (! 2 =2); } ((! 1 + ! 2 )=2)
are root s of t he cubic
4x 3 ¡ g2 x¡ g3 = 0:
One can show t hese root s are dist inct .
In part icular, if g2 ; g3 are algebraic,
t he root s are algebraic.
Schneider’s Theorem
If g2 ; g3 are algebraic
and } is t he associat ed
Weierst rass } -funct ion, t hen
for ® algebraic,
} (®) is t ranscendent al.
Since } (! 1 =2),
} (! 2 =2) are algebraic
It follows t hat t he periods
! 1 ; ! 2 must
be t ranscendent al when
g2 ; g3 are algebraic.
T his is t he ellipt ic analog of t he
Hermit e-Lindemann t heorem
t hat says if ® is a non-zero algebraic number,
t hen e® is t ranscendent al.
Not e t hat we get ¼ t ranscendent al by set t ing ® = 2¼i
Why should this interest us?
Let y = sin x.
T hen
( dy ) 2 + y 2 = 1.
dx
Let y = } (x).
T
³ hen
´2
dy
= 4y3 ¡ g2 y ¡ g3 .
dx
T hus p
dy
1¡ y 2
= dx.
T hus p
= dx:
dy
4y 3 ¡ g2 y ¡ g3
Int egrat ing bot h sides, we get Int egrat ing bot h sides, we get
R
R
si n b p dy
} ( ( ! 1 + ! 2 ) =2) p
= b.
dy
= !
0
1¡ y 2
} ( ! 1 =2)
4y 3 ¡ g2 y ¡ g3
2
2
R
Let’s look at our formula for the
circumference of an ellipse again.
1
1¡ k 2
p
where
k 2 = 1¡
t dt
t ( t ¡ 1) ( t ¡ ( 1¡ k 2 ) )
b2
a2
:
T he cubic in t he int egrand is not in Weierst rass form.
It can be put in t his form.
p
But let us look at t he case k = 1= 2.
Put t ing t = s + 1=2, t he int egral becomes
R
1=2 2s+ 1 ds:
p
0
4s 3 ¡ s
T he int egral
R
1=2
p ds
0
4s 3 ¡ s
is a period of t he ellipt ic curve
y2 = 4x 3 ¡ x:
But
what
about
R
1=2
p sds ?
0
4s 3 ¡ s
Let us look at t he Weierst
rass ³ -funct
ion:
³
´
P
1
1 + 1 + z .
³ (z) = +
z
06
= ! 2L
z¡ !
!
!
2
Observe t hat ³ ; (z) = ¡ } (z).
 is not periodic!
• It is “quasi-periodic’’.
• What does this mean?
³ (z + ! ) = ³ (z) + ´ (! ):
Put z = ¡ ! =2 t o get
³ (! =2) = ³ (¡ ! =2) + ´ (! ).
Since ³ is an odd funct ion, we get
´ (! ) = 2³ (! =2).
T his is called a quasi-period.
What are these quasi-periods?
Observe t hat
³ (! 1 =2) = ³ (¡ ! 1 =2 + ! 1 ) = ³ (¡ ! 1 =2) + ´ (! 1 )
But ³ is odd, so ³ (¡ ! 1 =2) = ¡ ³ (! 1 =2) so t hat
2³ (! 1 =2) = ´ (! 1 ):
Similarly, 2³ (! 2 =2) = ´ (! 2 )
Since ´ is a linear funct ion on t he period lat t ice,
we get
2³ ((! 1 + ! 2 )=2) = ´ (! 1 ) + ´ (! 2 ).
T hus
d³ = ¡ } (z)dz.
Recallpt hat
d} =
4} (z) 3 ¡ g2 } (z) ¡ g3 dz:
Hence,
d³ = ¡ p
We can int egrat e bot h sides
from z = ! 1 =2
to z = R
(! 1 + ! 2 )=2 t o get
´ = 2 e2 p x dx
2
e1
x 3 ¡ g2 x ¡ g3
where
e1 = } (! 1 ) and
e2 = } ((! 1 + ! 2 )=2):
Hence, our original int egral is a sum of a
period and a quasi-period.
} ( z) d}
4} ( z) 3 ¡ g2 } ( z ) ¡ g3
Are there triply periodic functions?
• In 1835, Jacobi proved that
such functions of a single
variable do not exist.
• Abel and Jacobi constructed a
function of two variables with
four periods giving the first
example of an abelian variety
of dimension 2.
1804-1851
1802 - 1829
What exactly is a period?
• These are the values of absolutely
convergent integrals of algebraic functions
with algebraic coefficients defined by
domains in Rn given by polynomial
inequalities with algebraic coefficients.
• For example
 is a period.
RR
¼=
dxdy
x2 + y2 · 1
Some unanswered questions
•
•
•
•
•
Is e a period?
Probably not.
Is 1/ a period?
Probably not.
The set of periods P is countable but no
one has yet given an explicit example of a
number not in P.
NµZµQµAµP
```