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OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Theta Functions and the Quintic
Ali Uncu & Frank Patane
www.math.ufl.edu/∼akuncu www.math.ufl.edu/∼frankpatane
October 18, 2011
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
1 A Brief History of the Quintic
2 Transformations of the Quintic EquationOur approachTschirnhaus TransformationInverse Transformation
3 Definition and Properties of Theta Functions
4 The General Scheme
5 The Modular Equation
6 Solving the quintic
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
A Quick Background on the Quintic
Let the general monic quintic be denoted byx5 + a4x
4 + a3x3 + a2x
2 + a1x + a0, ai ∈ Q.
Mathematicians such as Leibniz, Tschirnhausen, Euler,Vandermonde, Lagrange, and Ruffini, all tried to solve the generalquintic in radicals, but were succesful only in special cases.
In 1824 Abel delivered the first correct proof of what is now knownas the Abel-Ruffini Theorem or Abel’s Impossibility Theorem.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
A Quick Background on the Quintic
Let the general monic quintic be denoted byx5 + a4x
4 + a3x3 + a2x
2 + a1x + a0, ai ∈ Q.
Mathematicians such as Leibniz, Tschirnhausen, Euler,Vandermonde, Lagrange, and Ruffini, all tried to solve the generalquintic in radicals, but were succesful only in special cases.
In 1824 Abel delivered the first correct proof of what is now knownas the Abel-Ruffini Theorem or Abel’s Impossibility Theorem.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
A Quick Background on the Quintic
Let the general monic quintic be denoted byx5 + a4x
4 + a3x3 + a2x
2 + a1x + a0, ai ∈ Q.
Mathematicians such as Leibniz, Tschirnhausen, Euler,Vandermonde, Lagrange, and Ruffini, all tried to solve the generalquintic in radicals, but were succesful only in special cases.
In 1824 Abel delivered the first correct proof of what is now knownas the Abel-Ruffini Theorem or Abel’s Impossibility Theorem.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
A Quick Background on the Quintic
Abel’s proof is one of the first achievements of group theory, andlater evolved into what is now known as Galois Theory.
Indeed, we may use the notion of a solvable group to determinewhether a polynomial is solvable in radicals.
Any quadratic, cubic, or quartic is solvable in radicals because theGalois group of such a polynomial must be a subgroup of S2,S3, orS4 which are all solvable groups.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
A Quick Background on the Quintic
Abel’s proof is one of the first achievements of group theory, andlater evolved into what is now known as Galois Theory.
Indeed, we may use the notion of a solvable group to determinewhether a polynomial is solvable in radicals.
Any quadratic, cubic, or quartic is solvable in radicals because theGalois group of such a polynomial must be a subgroup of S2,S3, orS4 which are all solvable groups.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
A Quick Background on the Quintic
Abel’s proof is one of the first achievements of group theory, andlater evolved into what is now known as Galois Theory.
Indeed, we may use the notion of a solvable group to determinewhether a polynomial is solvable in radicals.
Any quadratic, cubic, or quartic is solvable in radicals because theGalois group of such a polynomial must be a subgroup of S2, S3, orS4 which are all solvable groups.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
A Quick Background on the Quintic
The general quintic is not solvable since it may have Galois groupisomorphic to A5 or S5.
Thus one can ask if a given quintic is solvable.
As we will soon see, we are able to transform the general quinticinto the Bring-Jerrard quintic: x5 + 5x − a
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
A Quick Background on the Quintic
The general quintic is not solvable since it may have Galois groupisomorphic to A5 or S5.
Thus one can ask if a given quintic is solvable.
As we will soon see, we are able to transform the general quinticinto the Bring-Jerrard quintic: x5 + 5x − a
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
A Quick Background on the Quintic
The general quintic is not solvable since it may have Galois groupisomorphic to A5 or S5.
Thus one can ask if a given quintic is solvable.
As we will soon see, we are able to transform the general quinticinto the Bring-Jerrard quintic: x5 + 5x − a
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
A Quick Background on the Quintic
Theorem(Spearman Williams 1994) An irreducible quintic P(x) inBring-Jerrard form is solvable by radicals if and only if there existrationals r , s such that P(x) is of the form:
x5 +5r4(4s + 3)
s2 + 1− 4r4(2s − 11)
s2 + 1= 0.
Let us now see that we can transform the general quintic toBring-Jerrard form.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
A Quick Background on the Quintic
Theorem(Spearman Williams 1994) An irreducible quintic P(x) inBring-Jerrard form is solvable by radicals if and only if there existrationals r , s such that P(x) is of the form:
x5 +5r4(4s + 3)
s2 + 1− 4r4(2s − 11)
s2 + 1= 0.
Let us now see that we can transform the general quintic toBring-Jerrard form.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
A Quick Background on the Quintic
Theorem(Spearman Williams 1994) An irreducible quintic P(x) inBring-Jerrard form is solvable by radicals if and only if there existrationals r , s such that P(x) is of the form:
x5 +5r4(4s + 3)
s2 + 1− 4r4(2s − 11)
s2 + 1= 0.
Let us now see that we can transform the general quintic toBring-Jerrard form.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Our approachTschirnhaus TransformationInverse Transformation
General Quintic Equation:
x5 + a4x4 + a3x
3 + a2x2 + a1x + a0 = 0
l
Bring-Jerrard Form:
x5 + c1x + c0 = 0
l
x5 + 5x + a = 0Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Our approachTschirnhaus TransformationInverse Transformation
General Quintic Equation:
f (x) = x5 + a4x4 + a3x
3 + a2x2 + a1x + a0
Assume that f (x) has distinct roots x1, . . . , x5.
Suppose that
y = φ(x) = p4x4 + p3x
3 + p2x2 + p1x + p0
is the Tschirnhaus transformation where yi = y(xi ) beingdistinct roots of another quintic equation. When one is solving forf (x) = 0;
y = p4x4 + p3x
3 + p2x2 + p1x + p0
xy = p4x5 + p3x
4 + p2x3 + p1x
2 + p0xxy = p4(−a4x
4−a3x3−a2x
2−a1x−a0)+p3x4 +p2x
3 +p1x2 +p0x
xy = p′4x4 + p′3x
3 + p′2x2 + p′1x + p′0
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Our approachTschirnhaus TransformationInverse Transformation
General Quintic Equation:
f (x) = x5 + a4x4 + a3x
3 + a2x2 + a1x + a0
Assume that f (x) has distinct roots x1, . . . , x5.
Suppose that
y = φ(x) = p4x4 + p3x
3 + p2x2 + p1x + p0
is the Tschirnhaus transformation where yi = y(xi ) beingdistinct roots of another quintic equation. When one is solving forf (x) = 0;
y = p4x4 + p3x
3 + p2x2 + p1x + p0
xy = p4x5 + p3x
4 + p2x3 + p1x
2 + p0xxy = p4(−a4x
4−a3x3−a2x
2−a1x−a0)+p3x4 +p2x
3 +p1x2 +p0x
xy = p′4x4 + p′3x
3 + p′2x2 + p′1x + p′0
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Our approachTschirnhaus TransformationInverse Transformation
General Quintic Equation:
f (x) = x5 + a4x4 + a3x
3 + a2x2 + a1x + a0
Assume that f (x) has distinct roots x1, . . . , x5.
Suppose that
y = φ(x) = p4x4 + p3x
3 + p2x2 + p1x + p0
is the Tschirnhaus transformation where yi = y(xi ) beingdistinct roots of another quintic equation.
When one is solving forf (x) = 0;
y = p4x4 + p3x
3 + p2x2 + p1x + p0
xy = p4x5 + p3x
4 + p2x3 + p1x
2 + p0xxy = p4(−a4x
4−a3x3−a2x
2−a1x−a0)+p3x4 +p2x
3 +p1x2 +p0x
xy = p′4x4 + p′3x
3 + p′2x2 + p′1x + p′0
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Our approachTschirnhaus TransformationInverse Transformation
General Quintic Equation:
f (x) = x5 + a4x4 + a3x
3 + a2x2 + a1x + a0
Assume that f (x) has distinct roots x1, . . . , x5.
Suppose that
y = φ(x) = p4x4 + p3x
3 + p2x2 + p1x + p0
is the Tschirnhaus transformation where yi = y(xi ) beingdistinct roots of another quintic equation. When one is solving forf (x) = 0;
y = p4x4 + p3x
3 + p2x2 + p1x + p0
xy = p4x5 + p3x
4 + p2x3 + p1x
2 + p0xxy = p4(−a4x
4−a3x3−a2x
2−a1x−a0)+p3x4 +p2x
3 +p1x2 +p0x
xy = p′4x4 + p′3x
3 + p′2x2 + p′1x + p′0
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Our approachTschirnhaus TransformationInverse Transformation
Similarly one can write
x2y = p′′4x4 + p′′3x
3 + p′′2x2 + p′′1x + p′′0
x3y = p′′′4 x4 + p′′′3 x3 + p′′′2 x2 + p′′′1 x + p′′′0
x4y = p′′′′4 x4 + p′′′′3 x3 + p′′′′2 x2 + p′′′′1 x + p′′′′0
So that turning the question of finding coefficients of y into anEigenvalue problem, (A− yI5)x̄ = 0;
p0 − y p1 p2 p3 p4
p′0 p′1 − y p′2 p′3 p′4p′′0 p′′1 p′′2 − y p′′3 p′′4p′′′0 p′′′1 p′′′2 p′′′3 − y p′′′4
p′′′′0 p′′′′1 p′′′′2 p′′′′3 p′′′′4 − y
1xx2
x3
x4
= 0
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Our approachTschirnhaus TransformationInverse Transformation
If the characteristic polynomial
det(yI5 − A) = y5 + b4y4 + b3y
3 + b2y2 + b1y + b0
has no multiple roots, we can solve for b4 = b3 = b2 = 0.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Our approachTschirnhaus TransformationInverse Transformation
Solution due to Bring:
b4(p0, p1, p2, p3, p4) = 0 is a linear equation so solving for p0,we can write every other p0 in the other equations in pi fori = 1, . . . , 4.
b3(p1, p2, p3, p4) = 0 is a quadratic equation and can bewritten as u2
1 − v21 + u2
2 − v22 where WLOG u1(p1, p2, p3, p4),
v1(p2, p3, p4), u2(p3, p4) and v2(p4) are all linear so that onecan solve ui = vi resulting in writing every other variable in p4
and p2.
Choosing a suitable p2 we have all equations in p4.
b2(p4) = 0 is a cubic equation in p4 hence can be solved.
After finding p4 we can solve p3, p1 and p0 in this order.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Our approachTschirnhaus TransformationInverse Transformation
Hence we can transform the general quintic to Bring-JerrardForm.
y5 + b1y + b0 = 0
We can transform the quintic Bring-Jerrard Form to
z5 + 5z + a = 0
by a linear change of variables y 7→ 4√
5/b1z , which we can solveusing theta functions.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Our approachTschirnhaus TransformationInverse Transformation
This has the inverse function z 7→ 4√b1/5y which can be easily
computed once one finds roots for z5 + 5z + a = 0, thus giving thesolutions for y5 + b1y + b0 = 0.
Similarly the inverse map of the Tschirnhaus transformation canbe calculated as a rational function in variable y from the matrix:
A− yI =
p0 − y p1 p2 p3 p4
p′0 p′1 − y p′2 p′3 p′4p′′0 p′′1 p′′2 − y p′′3 p′′4p′′′0 p′′′1 p′′′2 p′′′3 − y p′′′4
p′′′′0 p′′′′1 p′′′′2 p′′′′3 p′′′′4 − y
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Our approachTschirnhaus TransformationInverse Transformation
One can find the inverse transformation by eliminating the matrixyI5 − A upto a certain level:
∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ 0∗ ∗ ∗ 0 0
k(y) h(y) 0 0 00 0 0 0 0
Then since multiplying this matrix with (1, x , x2, x3, x4)t gives zerothe inverse transformation is defined by x = −k(y)/h(y).
Moreover k(y), h(y) has degree both at most 4 in y sincey5 + b1y + b0 = 0.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Our approachTschirnhaus TransformationInverse Transformation
Furthermore, one can replace the inverse map
x = −k(y)/h(y)
with a polynomial of degree at most 4 which shows that there is asymmetry between mapping a Bring-Jerrard form to a generalquintic and vice versa.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Our approachTschirnhaus TransformationInverse Transformation
Example: Let us take the quintic function:
f := x5 + x4 − x3 + x2 + 2x + 1
and the transformation:
y := p4x4 + p3x
3 + p2x2 + p1x + p0
. Recall that we have the freedom of choosing one pi (except p4
being zero). Then the characteristic polynomial in y of matrixyI5 − A becomes:
y5 + b4y4 + b3y
3 + b2y2 + b1y + E0 = 0
where bi s are going to be chosen in order to make
b4 = b3 = b2 = 0
.Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Our approachTschirnhaus TransformationInverse Transformation
Here equations are:
b4 = 7p3 − 3p4 + p1 − 5p0 − 3p2
b3 = 13p4p1 − 5p2p3 − 4p1p0 − 29p24
. . .+ 10p20 + 12bp4b0
b2 = −12p3p0p1 − 42p23p0 + p3
1 + 2p3p22
· · · − 24p3p0p4 − 23p2p1p4
We can solve p0 using b4 in the other variables as:
p0 = (7/5)p3 − (3/5)p4 + (1/5)p1 − (3/5)p2
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Our approachTschirnhaus TransformationInverse Transformation
We can rewrite the second equation as:
b3 = u21 − v2
1 + u22 − v2
2
where
u1 =
√(−7
5
)(p1 −
11
2p4 −
16
7p2 +
4
7p3
)v1 =
1
7
√−47√
7
(p2 +
56
47p4 +
57
94p3
)u2 =
1
188
√−1431
√188
(p3 −
592
1431p4
)v2 =
1
5724
√−8705
√5724p4
and solve ui = vi .Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Our approachTschirnhaus TransformationInverse Transformation
Picking p2 = 1 as our free choice, we get1
p4 = −0.552 . . . , p3 = −0.475 . . . , p1 = −0.36 . . . , p0 = −1.006 . . .
in this order. More importantly the characteristic polynomial turnsinto Bring-Jerrard form:
y5 + By + C = 0
Now we can replace Y by ((1/5)B)(1/4) ∗ Z and normalize to getthe polynomial:
Z 5 + 5Z − 3.14924 · · · = 0
1Exact numbers are all algebraic, yet some exceed maybe 20 pages.Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Intro. to Theta Functions
Before we move further, let us gain some motivation byconsidering one way to solve the cubic.
We may easily depress the general cubic with a linear transformtionto get x3 + px + q = 0.
The key to our argument lies in 4 cos3(θ)− 3 cos(θ)− cos(3θ) = 0
We may exploit the trig identity to write the roots of the cubic interms or cos and cos−1
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Intro. to Theta Functions
Before we move further, let us gain some motivation byconsidering one way to solve the cubic.
We may easily depress the general cubic with a linear transformtionto get x3 + px + q = 0.
The key to our argument lies in 4 cos3(θ)− 3 cos(θ)− cos(3θ) = 0
We may exploit the trig identity to write the roots of the cubic interms or cos and cos−1
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Intro. to Theta Functions
Before we move further, let us gain some motivation byconsidering one way to solve the cubic.
We may easily depress the general cubic with a linear transformtionto get x3 + px + q = 0.
The key to our argument lies in 4 cos3(θ)− 3 cos(θ)− cos(3θ) = 0
We may exploit the trig identity to write the roots of the cubic interms or cos and cos−1
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Intro. to Theta Functions
Before we move further, let us gain some motivation byconsidering one way to solve the cubic.
We may easily depress the general cubic with a linear transformtionto get x3 + px + q = 0.
The key to our argument lies in 4 cos3(θ)− 3 cos(θ)− cos(3θ) = 0
We may exploit the trig identity to write the roots of the cubic interms or cos and cos−1
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Intro. to Theta Functions
The Jacobi theta functions are defined for two complex variablesν, τ with ν any complex number, and τ in the upper half plane H.
We first set q := eπiτ and z := e2πiν and see that Im(τ) > 0implies |q| < 1.
The main Jacobi theta function is defined by the series,
θ3(ν|τ) = θ3(ν) =∞∑
m=−∞qm
2zm
Note that |q| < 1 implies that θ3(ν) is an entire function.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Intro. to Theta Functions
The Jacobi theta functions are defined for two complex variablesν, τ with ν any complex number, and τ in the upper half plane H.
We first set q := eπiτ and z := e2πiν and see that Im(τ) > 0implies |q| < 1.
The main Jacobi theta function is defined by the series,
θ3(ν|τ) = θ3(ν) =∞∑
m=−∞qm
2zm
Note that |q| < 1 implies that θ3(ν) is an entire function.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Intro. to Theta Functions
The Jacobi theta functions are defined for two complex variablesν, τ with ν any complex number, and τ in the upper half plane H.
We first set q := eπiτ and z := e2πiν and see that Im(τ) > 0implies |q| < 1.
The main Jacobi theta function is defined by the series,
θ3(ν|τ) = θ3(ν) =∞∑
m=−∞qm
2zm
Note that |q| < 1 implies that θ3(ν) is an entire function.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Intro. to Theta Functions
The other Jacobi theta functions are a shift of θ3. We have:
θ0(ν) = θ3(ν +1
2) =
∑(−1)mqm
2zm
θ1(ν) = ie−πi(ν−τ4
)θ3(ν +1− τ
2) = ie−πiν
∑(−1)mq(m− 1
2)2zm
θ2(ν) = e−πi(ν−τ4
)θ3(ν − τ
2) = e−πiν
∑q(m− 1
2)2zm
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Intro. to Theta Functions
The other Jacobi theta functions are a shift of θ3. We have:
θ0(ν) = θ3(ν +1
2) =
∑(−1)mqm
2zm
θ1(ν) = ie−πi(ν−τ4
)θ3(ν +1− τ
2) = ie−πiν
∑(−1)mq(m− 1
2)2zm
θ2(ν) = e−πi(ν−τ4
)θ3(ν − τ
2) = e−πiν
∑q(m− 1
2)2zm
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Intro. to Theta Functions
The other Jacobi theta functions are a shift of θ3. We have:
θ0(ν) = θ3(ν +1
2) =
∑(−1)mqm
2zm
θ1(ν) = ie−πi(ν−τ4
)θ3(ν +1− τ
2) = ie−πiν
∑(−1)mq(m− 1
2)2zm
θ2(ν) = e−πi(ν−τ4
)θ3(ν − τ
2) = e−πiν
∑q(m− 1
2)2zm
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Intro. to Theta Functions
The other Jacobi theta functions are a shift of θ3. We have:
θ0(ν) = θ3(ν +1
2) =
∑(−1)mqm
2zm
θ1(ν) = ie−πi(ν−τ4
)θ3(ν +1− τ
2) = ie−πiν
∑(−1)mq(m− 1
2)2zm
θ2(ν) = e−πi(ν−τ4
)θ3(ν − τ
2) = e−πiν
∑q(m− 1
2)2zm
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Intro. to Theta Functions
Much of the time we set ν = 0, and consider the properties ofθ3(0|τ) = θ3.
Similarly we set θi (0|τ) = θi .
The Jacobi Identity states: θ43 = θ4
2 + θ40
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Intro. to Theta Functions
Much of the time we set ν = 0, and consider the properties ofθ3(0|τ) = θ3.
Similarly we set θi (0|τ) = θi .
The Jacobi Identity states: θ43 = θ4
2 + θ40
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Intro. to Theta Functions
Much of the time we set ν = 0, and consider the properties ofθ3(0|τ) = θ3.
Similarly we set θi (0|τ) = θi .
The Jacobi Identity states: θ43 = θ4
2 + θ40
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Product Representation of Theta
Valid for complex q, z with |q| < 1, z 6= 0 the Jacobi TripleProduct Identity gives:
θ3(ν|τ) =∞∑
m=−∞qm
2zm =
∞∏m=1
(1−q2m)(1+zq2m−1)(1+z−1q2m−1).
With ν = 0 we get
θ3 =∞∏
m=1
(1− q2m)(1 + q2m−1)2.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Product Representation of Theta
Valid for complex q, z with |q| < 1, z 6= 0 the Jacobi TripleProduct Identity gives:
θ3(ν|τ) =∞∑
m=−∞qm
2zm =
∞∏m=1
(1−q2m)(1+zq2m−1)(1+z−1q2m−1).
With ν = 0 we get
θ3 =∞∏
m=1
(1− q2m)(1 + q2m−1)2.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Definition of η, f , f1, f2
Using the product representation for θ3 we get productrepresentations for θ0 and θ2:
In particular, θ3 = η(τ)f 2(τ), θ0 = η(τ)f 21 (τ), and
θ2 = η(τ)f 22 (τ) where
η(τ) = q1
12∏
(1− q2k), f (τ) = q−124∏
(1 + q2k−1),
f1(τ) = q−124∏
(1− q2k−1) f2(τ) =√
2q1
12∏
(1 + q2k).
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Definition of η, f , f1, f2
Using the product representation for θ3 we get productrepresentations for θ0 and θ2:
In particular, θ3 = η(τ)f 2(τ), θ0 = η(τ)f 21 (τ), and
θ2 = η(τ)f 22 (τ) where
η(τ) = q1
12∏
(1− q2k), f (τ) = q−124∏
(1 + q2k−1),
f1(τ) = q−124∏
(1− q2k−1) f2(τ) =√
2q1
12∏
(1 + q2k).
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Definition of η, f , f1, f2
Using the product representation for θ3 we get productrepresentations for θ0 and θ2:
In particular, θ3 = η(τ)f 2(τ), θ0 = η(τ)f 21 (τ), and
θ2 = η(τ)f 22 (τ) where
η(τ) = q1
12∏
(1− q2k), f (τ) = q−124∏
(1 + q2k−1),
f1(τ) = q−124∏
(1− q2k−1) f2(τ) =√
2q1
12∏
(1 + q2k).
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Definition of η, f , f1, f2
Using the identity θ43 = θ4
2 + θ40 we derive f 8 = f 8
1 + f 82 .
We will also need the property f1f2f =√
2:
f1f2f =√
2∏
(1−q2k−1)(1+q2k)(1+q2k−1) =√
2∏
(1−q2k−1)(1+qk) =√
2∏ (1− qk)(1 + qk)
(1− q2k)
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Definition of η, f , f1, f2
Using the identity θ43 = θ4
2 + θ40 we derive f 8 = f 8
1 + f 82 .
We will also need the property f1f2f =√
2:
f1f2f =√
2∏
(1−q2k−1)(1+q2k)(1+q2k−1) =√
2∏
(1−q2k−1)(1+qk) =√
2∏ (1− qk)(1 + qk)
(1− q2k)
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Definition of η, f , f1, f2
Using the identity θ43 = θ4
2 + θ40 we derive f 8 = f 8
1 + f 82 .
We will also need the property f1f2f =√
2:
f1f2f =√
2∏
(1−q2k−1)(1+q2k)(1+q2k−1) =√
2∏
(1−q2k−1)(1+qk) =√
2∏ (1− qk)(1 + qk)
(1− q2k)
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
The Modular Equation
Let us set u = f (τ), v∞ = f (5τ) , vc = f ( τ+c5 ) , c ∈ Z.
Since f has period 48, there are only 5 · 48 distinct vc .
We only consider the 5 vc with c ≡ 0 mod 48. Reducing the indexmodulo 5 we may write
v±2 = f (τ ∓ 48
5); v±1 = f (
τ ± 96
5)
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
The Modular Equation
Let us set u = f (τ), v∞ = f (5τ) , vc = f ( τ+c5 ) , c ∈ Z.
Since f has period 48, there are only 5 · 48 distinct vc .
We only consider the 5 vc with c ≡ 0 mod 48. Reducing the indexmodulo 5 we may write
v±2 = f (τ ∓ 48
5); v±1 = f (
τ ± 96
5)
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
The Modular Equation
Let us set u = f (τ), v∞ = f (5τ) , vc = f ( τ+c5 ) , c ∈ Z.
Since f has period 48, there are only 5 · 48 distinct vc .
We only consider the 5 vc with c ≡ 0 mod 48. Reducing the indexmodulo 5 we may write
v±2 = f (τ ∓ 48
5); v±1 = f (
τ ± 96
5)
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
The Modular Equation
Our first goal is to establish the modular equation:
(u
v)3 + (
v
u)3 = (uv)2 − 4
(uv)2
or equivalently : v6 − u5v5 + 4uv + u6 = 0
with u = f (τ) and v = vc or v∞.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
The Modular Equation
Our first goal is to establish the modular equation:
(u
v)3 + (
v
u)3 = (uv)2 − 4
(uv)2
or equivalently : v6 − u5v5 + 4uv + u6 = 0
with u = f (τ) and v = vc or v∞.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
The Modular Equation
Our first goal is to establish the modular equation:
(u
v)3 + (
v
u)3 = (uv)2 − 4
(uv)2
or equivalently : v6 − u5v5 + 4uv + u6 = 0
with u = f (τ) and v = vc or v∞.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
The Modular Equation
Our first goal is to establish the modular equation:
(u
v)3 + (
v
u)3 = (uv)2 − 4
(uv)2
or equivalently : v6 − u5v5 + 4uv + u6 = 0
with u = f (τ) and v = vc or v∞.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Modular to Quintic
Thus we will have shown that the coefficients of a sixth degreepolynomial with roots vc can be expressed in terms of u.
We will then show that the roots of the quinticw(w2 + 5)2 − u12 + 64u−12 are a polynomial in the vc .
After the substitution y =f 81 −f 8
2f 2(w2+5)
,
we will have y5 + 5y =f 81 −f 8
2f 2 .
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Modular to Quintic
Thus we will have shown that the coefficients of a sixth degreepolynomial with roots vc can be expressed in terms of u.
We will then show that the roots of the quinticw(w2 + 5)2 − u12 + 64u−12 are a polynomial in the vc .
After the substitution y =f 81 −f 8
2f 2(w2+5)
,
we will have y5 + 5y =f 81 −f 8
2f 2 .
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Modular to Quintic
Thus we will have shown that the coefficients of a sixth degreepolynomial with roots vc can be expressed in terms of u.
We will then show that the roots of the quinticw(w2 + 5)2 − u12 + 64u−12 are a polynomial in the vc .
After the substitution y =f 81 −f 8
2f 2(w2+5)
,
we will have y5 + 5y =f 81 −f 8
2f 2 .
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Modular to Quintic
Thus we will have shown that the coefficients of a sixth degreepolynomial with roots vc can be expressed in terms of u.
We will then show that the roots of the quinticw(w2 + 5)2 − u12 + 64u−12 are a polynomial in the vc .
After the substitution y =f 81 −f 8
2f 2(w2+5)
,
we will have y5 + 5y =f 81 −f 8
2f 2 .
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Transformations of u and v
We first consider the transformation τ → τ + 2 on u and v .
When seeing f has period 48 we noted η(τ + 1) = eπi12 η(τ) implies
f (τ + 2) = e−πi
12 f (τ)
Thus τ → τ + 2 sends u to e−πi
12 u.
Now τ → τ + 2 takes v∞ = f (5τ) to f (5τ + 10) = e−5πi
12 f (5τ).
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Transformations of u and v
We first consider the transformation τ → τ + 2 on u and v .
When seeing f has period 48 we noted η(τ + 1) = eπi12 η(τ) implies
f (τ + 2) = e−πi
12 f (τ)
Thus τ → τ + 2 sends u to e−πi
12 u.
Now τ → τ + 2 takes v∞ = f (5τ) to f (5τ + 10) = e−5πi
12 f (5τ).
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Transformations of u and v
We first consider the transformation τ → τ + 2 on u and v .
When seeing f has period 48 we noted η(τ + 1) = eπi12 η(τ) implies
f (τ + 2) = e−πi
12 f (τ)
Thus τ → τ + 2 sends u to e−πi
12 u.
Now τ → τ + 2 takes v∞ = f (5τ) to f (5τ + 10) = e−5πi
12 f (5τ).
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Transformations of u and v
We first consider the transformation τ → τ + 2 on u and v .
When seeing f has period 48 we noted η(τ + 1) = eπi12 η(τ) implies
f (τ + 2) = e−πi
12 f (τ)
Thus τ → τ + 2 sends u to e−πi
12 u.
Now τ → τ + 2 takes v∞ = f (5τ) to f (5τ + 10) = e−5πi
12 f (5τ).
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Transformations of u and v
In general the substitution τ → τ + 2 replaces vc with
f (τ + 2 + c
5) = f (
τ + 50− 48 + c
5) = e
−5πi12 f (
τ + c ′
5)
where c ′ = c − 48 ≡ 0mod 48 and c ′ ≡ c + 2 mod 5.
In other words vc → e−5πi
12 vc+2.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Transformations of u and v
In general the substitution τ → τ + 2 replaces vc with
f (τ + 2 + c
5) = f (
τ + 50− 48 + c
5) = e
−5πi12 f (
τ + c ′
5)
where c ′ = c − 48 ≡ 0mod 48 and c ′ ≡ c + 2 mod 5.
In other words vc → e−5πi
12 vc+2.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Transformations of u and v
In general the substitution τ → τ + 2 replaces vc with
f (τ + 2 + c
5) = f (
τ + 50− 48 + c
5) = e
−5πi12 f (
τ + c ′
5)
where c ′ = c − 48 ≡ 0mod 48 and c ′ ≡ c + 2 mod 5.
In other words vc → e−5πi
12 vc+2.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Transformations of u and v
In general the substitution τ → τ + 2 replaces vc with
f (τ + 2 + c
5) = f (
τ + 50− 48 + c
5) = e
−5πi12 f (
τ + c ′
5)
where c ′ = c − 48 ≡ 0mod 48 and c ′ ≡ c + 2 mod 5.
In other words vc → e−5πi
12 vc+2.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Transformations of u and v
Thus we have uv → e−πi
2 uv and uv → e
πi3uv .
where the index of v transforms like τ → τ + 2.
One can also show the τ → −1τ takes uv → uv , u
v →uv , and
τ → τ−1τ+1 takes uv → −2
uv , uv →
−vu with the index of v changing in
the same way as τ .
uv uv
τ → τ + 2 e−πi
2 uv eπi3uv
τ → −1τ uv u
vτ → τ−1
τ+1−2uv
−vu
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Transformations of u and v
Thus we have uv → e−πi
2 uv and uv → e
πi3uv .
where the index of v transforms like τ → τ + 2.
One can also show the τ → −1τ takes uv → uv , u
v →uv , and
τ → τ−1τ+1 takes uv → −2
uv , uv →
−vu with the index of v changing in
the same way as τ .
uv uv
τ → τ + 2 e−πi
2 uv eπi3uv
τ → −1τ uv u
vτ → τ−1
τ+1−2uv
−vu
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Transformations of u and v
Thus we have uv → e−πi
2 uv and uv → e
πi3uv .
where the index of v transforms like τ → τ + 2.
One can also show the τ → −1τ takes uv → uv , u
v →uv , and
τ → τ−1τ+1 takes uv → −2
uv , uv →
−vu with the index of v changing in
the same way as τ .
uv uv
τ → τ + 2 e−πi
2 uv eπi3uv
τ → −1τ uv u
vτ → τ−1
τ+1−2uv
−vu
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Transformations of u and v
Thus we have uv → e−πi
2 uv and uv → e
πi3uv .
where the index of v transforms like τ → τ + 2.
One can also show the τ → −1τ takes uv → uv , u
v →uv , and
τ → τ−1τ+1 takes uv → −2
uv , uv →
−vu with the index of v changing in
the same way as τ .
uv uv
τ → τ + 2 e−πi
2 uv eπi3uv
τ → −1τ uv u
vτ → τ−1
τ+1−2uv
−vu
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Transformations of u and v
Thus we have uv → e−πi
2 uv and uv → e
πi3uv .
where the index of v transforms like τ → τ + 2.
One can also show the τ → −1τ takes uv → uv , u
v →uv , and
τ → τ−1τ+1 takes uv → −2
uv , uv →
−vu with the index of v changing in
the same way as τ .
uv uv
τ → τ + 2 e−πi
2 uv eπi3uv
τ → −1τ uv u
vτ → τ−1
τ+1−2uv
−vu
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Functions invariant under τ + 2, −1τ , τ−1
τ+1
The function f 24(τ) is fixed under the transformations τ → τ + 2and τ → −1
τ .
However τ → τ−1τ+1 is an involution for f (τ). We have
f (τ)→ 212
f 24(τ).
Hence the function F (τ) = f 24(τ) + 212
f 24(τ)is invariant under all 3
transformations.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Functions invariant under τ + 2, −1τ , τ−1
τ+1
The function f 24(τ) is fixed under the transformations τ → τ + 2and τ → −1
τ .
However τ → τ−1τ+1 is an involution for f (τ). We have
f (τ)→ 212
f 24(τ).
Hence the function F (τ) = f 24(τ) + 212
f 24(τ)is invariant under all 3
transformations.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Functions invariant under τ + 2, −1τ , τ−1
τ+1
The function f 24(τ) is fixed under the transformations τ → τ + 2and τ → −1
τ .
However τ → τ−1τ+1 is an involution for f (τ). We have
f (τ)→ 212
f 24(τ).
Hence the function F (τ) = f 24(τ) + 212
f 24(τ)is invariant under all 3
transformations.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Functions invariant under τ + 2, −1τ , τ−1
τ+1
Lemma: Let g(τ) be meromorphic in H, and lacks an essentialsingularity at τ = i∞. If g(τ) is invariant under τ → τ + 2,τ → −1
τ , and τ → τ−1τ+1 , then g(τ) is a rational function of F (τ).
Moreover, the equation F (τ) = c is solvable for any c 6= 0.
Compare the above to modular functions being a rationalexpression in j and j(τ) = c .
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Functions invariant under τ + 2, −1τ , τ−1
τ+1
Lemma: Let g(τ) be meromorphic in H, and lacks an essentialsingularity at τ = i∞. If g(τ) is invariant under τ → τ + 2,τ → −1
τ , and τ → τ−1τ+1 , then g(τ) is a rational function of F (τ).
Moreover, the equation F (τ) = c is solvable for any c 6= 0.
Compare the above to modular functions being a rationalexpression in j and j(τ) = c .
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Functions invariant under τ + 2, −1τ , τ−1
τ+1
Lemma: Let g(τ) be meromorphic in H, and lacks an essentialsingularity at τ = i∞. If g(τ) is invariant under τ → τ + 2,τ → −1
τ , and τ → τ−1τ+1 , then g(τ) is a rational function of F (τ).
Moreover, the equation F (τ) = c is solvable for any c 6= 0.
Compare the above to modular functions being a rationalexpression in j and j(τ) = c .
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Functions invariant under τ + 2, −1τ , τ−1
τ+1
Thus if g(τ) = R(F (τ)) is finite for all τ with F (τ) 6=∞ then R isa polynomial.
If R(F (τ)) is finite for q = 0 (F =∞) as well as being finite for allτ with F (τ) 6=∞, then R is a constant
We are now in a position to deduce the modular equation.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Functions invariant under τ + 2, −1τ , τ−1
τ+1
Thus if g(τ) = R(F (τ)) is finite for all τ with F (τ) 6=∞ then R isa polynomial.
If R(F (τ)) is finite for q = 0 (F =∞) as well as being finite for allτ with F (τ) 6=∞, then R is a constant
We are now in a position to deduce the modular equation.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Functions invariant under τ + 2, −1τ , τ−1
τ+1
Thus if g(τ) = R(F (τ)) is finite for all τ with F (τ) 6=∞ then R isa polynomial.
If R(F (τ)) is finite for q = 0 (F =∞) as well as being finite for allτ with F (τ) 6=∞, then R is a constant
We are now in a position to deduce the modular equation.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deducing the Modular Equation
Recall that we would like to show (uv )3 + ( vu )3 = (uv)2 − 4(uv)2
Let us first set Ac = ( uvc
)3 + ( vcu )3 and Bc = (uvc)2 − 4(uvc )2 .
Using the transformations for u and v we get:
A Bτ → τ + 2 −A −Bτ → −1
τ A Bτ → τ−1
τ+1 −A −B
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deducing the Modular Equation
Recall that we would like to show (uv )3 + ( vu )3 = (uv)2 − 4(uv)2
Let us first set Ac = ( uvc
)3 + ( vcu )3 and Bc = (uvc)2 − 4(uvc )2 .
Using the transformations for u and v we get:
A Bτ → τ + 2 −A −Bτ → −1
τ A Bτ → τ−1
τ+1 −A −B
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deducing the Modular Equation
Recall that we would like to show (uv )3 + ( vu )3 = (uv)2 − 4(uv)2
Let us first set Ac = ( uvc
)3 + ( vcu )3 and Bc = (uvc)2 − 4(uvc )2 .
Using the transformations for u and v we get:
A Bτ → τ + 2 −A −Bτ → −1
τ A Bτ → τ−1
τ+1 −A −B
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deducing the Modular Equation
Recall that we would like to show (uv )3 + ( vu )3 = (uv)2 − 4(uv)2
Let us first set Ac = ( uvc
)3 + ( vcu )3 and Bc = (uvc)2 − 4(uvc )2 .
Using the transformations for u and v we get:
A Bτ → τ + 2 −A −Bτ → −1
τ A Bτ → τ−1
τ+1 −A −B
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deducing the Modular Equation
By the previous table, we see that the function∏
c(Ac − Bc)2 isfixed under the mentioned transformations of τ .
Using the definition of f (τ) we may gather the first few terms ofthe expansions of A∞ and B∞.
A∞ = q−12 (1− 2q + . . .) and B∞ = q
−12 (1− 2q + . . .).
Thus A∞ − B∞ vanishes at q = 0.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deducing the Modular Equation
By the previous table, we see that the function∏
c(Ac − Bc)2 isfixed under the mentioned transformations of τ .
Using the definition of f (τ) we may gather the first few terms ofthe expansions of A∞ and B∞.
A∞ = q−12 (1− 2q + . . .) and B∞ = q
−12 (1− 2q + . . .).
Thus A∞ − B∞ vanishes at q = 0.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deducing the Modular Equation
By the previous table, we see that the function∏
c(Ac − Bc)2 isfixed under the mentioned transformations of τ .
Using the definition of f (τ) we may gather the first few terms ofthe expansions of A∞ and B∞.
A∞ = q−12 (1− 2q + . . .) and B∞ = q
−12 (1− 2q + . . .).
Thus A∞ − B∞ vanishes at q = 0.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deducing the Modular Equation
By the previous table, we see that the function∏
c(Ac − Bc)2 isfixed under the mentioned transformations of τ .
Using the definition of f (τ) we may gather the first few terms ofthe expansions of A∞ and B∞.
A∞ = q−12 (1− 2q + . . .) and B∞ = q
−12 (1− 2q + . . .).
Thus A∞ − B∞ vanishes at q = 0.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deducing the Modular Equation
For c ≡ 0 mod 48 we have, v∞(τ) = f (5τ) = u(5τ − c) andvc(5τ − c) = f ( 5τ−c+c
5 ) = f (τ) = u(τ).
Moreover, A and B are symmetric in u and v , so we haveAc(5τ − c) = A∞(τ) and Bc(5τ − c) = B∞(τ).
Letting τ → i∞ (q → 0) we get that Ac − Bc vanishes at q = 0.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deducing the Modular Equation
For c ≡ 0 mod 48 we have, v∞(τ) = f (5τ) = u(5τ − c) andvc(5τ − c) = f ( 5τ−c+c
5 ) = f (τ) = u(τ).
Moreover, A and B are symmetric in u and v , so we haveAc(5τ − c) = A∞(τ) and Bc(5τ − c) = B∞(τ).
Letting τ → i∞ (q → 0) we get that Ac − Bc vanishes at q = 0.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deducing the Modular Equation
For c ≡ 0 mod 48 we have, v∞(τ) = f (5τ) = u(5τ − c) andvc(5τ − c) = f ( 5τ−c+c
5 ) = f (τ) = u(τ).
Moreover, A and B are symmetric in u and v , so we haveAc(5τ − c) = A∞(τ) and Bc(5τ − c) = B∞(τ).
Letting τ → i∞ (q → 0) we get that Ac − Bc vanishes at q = 0.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deducing the Modular Equation
Hence the function∏
c(Ac − Bc)2 vanishes at q = 0 and by theLemma is a constant.
But then Ac − Bc = 0 for some c .
Hence Ac − Bc = 0 for all c since, we establishedAc(5τ − c) = A∞(τ) and Bc(5τ − c) = B∞(τ).
This establishes the modular equation.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deducing the Modular Equation
Hence the function∏
c(Ac − Bc)2 vanishes at q = 0 and by theLemma is a constant.
But then Ac − Bc = 0 for some c .
Hence Ac − Bc = 0 for all c since, we establishedAc(5τ − c) = A∞(τ) and Bc(5τ − c) = B∞(τ).
This establishes the modular equation.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deducing the Modular Equation
Hence the function∏
c(Ac − Bc)2 vanishes at q = 0 and by theLemma is a constant.
But then Ac − Bc = 0 for some c .
Hence Ac − Bc = 0 for all c since, we establishedAc(5τ − c) = A∞(τ) and Bc(5τ − c) = B∞(τ).
This establishes the modular equation.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deducing the Modular Equation
Hence the function∏
c(Ac − Bc)2 vanishes at q = 0 and by theLemma is a constant.
But then Ac − Bc = 0 for some c .
Hence Ac − Bc = 0 for all c since, we establishedAc(5τ − c) = A∞(τ) and Bc(5τ − c) = B∞(τ).
This establishes the modular equation.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
We have seen that the coefficients of a sixth degree polynomialwith roots vc can be expressed in terms of u.
We now show that the coefficients of the fifth degree polynomialwith roots
wi =(v∞ − vi )(vi+1 − vi−1)(vi+2 − vi−2)√
5u3
can be expressed in terms of u and find the explicit form of thepolynomial.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
We have seen that the coefficients of a sixth degree polynomialwith roots vc can be expressed in terms of u.
We now show that the coefficients of the fifth degree polynomialwith roots
wi =(v∞ − vi )(vi+1 − vi−1)(vi+2 − vi−2)√
5u3
can be expressed in terms of u and find the explicit form of thepolynomial.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
We have seen that the coefficients of a sixth degree polynomialwith roots vc can be expressed in terms of u.
We now show that the coefficients of the fifth degree polynomialwith roots
wi =(v∞ − vi )(vi+1 − vi−1)(vi+2 − vi−2)√
5u3
can be expressed in terms of u and find the explicit form of thepolynomial.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
We have seen that the coefficients of a sixth degree polynomialwith roots vc can be expressed in terms of u.
We now show that the coefficients of the fifth degree polynomialwith roots
wi =(v∞ − vi )(vi+1 − vi−1)(vi+2 − vi−2)√
5u3
can be expressed in terms of u and find the explicit form of thepolynomial.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
Using our established transformations for u and v , we find thetransformations of wi :
w0 w1 w2 w3 w4
τ → τ + 2 −w2 −w3 −w4 −w0 −w1
τ → −1τ w0 w2 w1 w4 w3
τ → τ−1τ+1 −w0 −w3 −w4 −w2 −w1
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
Using our established transformations for u and v , we find thetransformations of wi :
w0 w1 w2 w3 w4
τ → τ + 2 −w2 −w3 −w4 −w0 −w1
τ → −1τ w0 w2 w1 w4 w3
τ → τ−1τ+1 −w0 −w3 −w4 −w2 −w1
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
Consider the polynomial∏(w − wi ) = w5 + A1w
4 + A2w3 + A3w
2 + A4w + A5.
It’s coefficients are finite at u 6= 0,∞.
Also A21,A2,A
23,A4,A
25 are invariant under τ → τ + 2, τ → −1
τ ,and τ → τ−1
τ+1 .
Thus they are polynomials inF (τ) = u24 + 212u−24 = q−1 + 24 + . . .
Such a polynomial is nonconstant only if its power series expansionin q begins with cqr , where r ≤ −1, c 6= 0.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
Consider the polynomial∏(w − wi ) = w5 + A1w
4 + A2w3 + A3w
2 + A4w + A5.
It’s coefficients are finite at u 6= 0,∞.
Also A21,A2,A
23,A4,A
25 are invariant under τ → τ + 2, τ → −1
τ ,and τ → τ−1
τ+1 .
Thus they are polynomials inF (τ) = u24 + 212u−24 = q−1 + 24 + . . .
Such a polynomial is nonconstant only if its power series expansionin q begins with cqr , where r ≤ −1, c 6= 0.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
Consider the polynomial∏(w − wi ) = w5 + A1w
4 + A2w3 + A3w
2 + A4w + A5.
It’s coefficients are finite at u 6= 0,∞.
Also A21,A2,A
23,A4,A
25 are invariant under τ → τ + 2, τ → −1
τ ,and τ → τ−1
τ+1 .
Thus they are polynomials inF (τ) = u24 + 212u−24 = q−1 + 24 + . . .
Such a polynomial is nonconstant only if its power series expansionin q begins with cqr , where r ≤ −1, c 6= 0.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
Consider the polynomial∏(w − wi ) = w5 + A1w
4 + A2w3 + A3w
2 + A4w + A5.
It’s coefficients are finite at u 6= 0,∞.
Also A21,A2,A
23,A4,A
25 are invariant under τ → τ + 2, τ → −1
τ ,and τ → τ−1
τ+1 .
Thus they are polynomials inF (τ) = u24 + 212u−24 = q−1 + 24 + . . .
Such a polynomial is nonconstant only if its power series expansionin q begins with cqr , where r ≤ −1, c 6= 0.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
Consider the polynomial∏(w − wi ) = w5 + A1w
4 + A2w3 + A3w
2 + A4w + A5.
It’s coefficients are finite at u 6= 0,∞.
Also A21,A2,A
23,A4,A
25 are invariant under τ → τ + 2, τ → −1
τ ,and τ → τ−1
τ+1 .
Thus they are polynomials inF (τ) = u24 + 212u−24 = q−1 + 24 + . . .
Such a polynomial is nonconstant only if its power series expansionin q begins with cqr , where r ≤ −1, c 6= 0.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
If we calculate the first term in the expansion of wi , we would find
that the first term of the expansion of Ak is q−k10 .
Therefore A21,A2,A
23,A4 are constant while A2
5 depends linearly onF (τ) = u24 + 212u−24.
Comparing expansions we get A25 = u24 + 212u−24 + C .
To calculate the value of the constants A1,A2,A3,A4,C , wecalculate vc(i).
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
If we calculate the first term in the expansion of wi , we would find
that the first term of the expansion of Ak is q−k10 .
Therefore A21,A2,A
23,A4 are constant while A2
5 depends linearly onF (τ) = u24 + 212u−24.
Comparing expansions we get A25 = u24 + 212u−24 + C .
To calculate the value of the constants A1,A2,A3,A4,C , wecalculate vc(i).
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
If we calculate the first term in the expansion of wi , we would find
that the first term of the expansion of Ak is q−k10 .
Therefore A21,A2,A
23,A4 are constant while A2
5 depends linearly onF (τ) = u24 + 212u−24.
Comparing expansions we get A25 = u24 + 212u−24 + C .
To calculate the value of the constants A1,A2,A3,A4,C , wecalculate vc(i).
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
If we calculate the first term in the expansion of wi , we would find
that the first term of the expansion of Ak is q−k10 .
Therefore A21,A2,A
23,A4 are constant while A2
5 depends linearly onF (τ) = u24 + 212u−24.
Comparing expansions we get A25 = u24 + 212u−24 + C .
To calculate the value of the constants A1,A2,A3,A4,C , wecalculate vc(i).
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
Since f1(τ) = f2(−1τ ) we have f1(i) = f2(i).
Using f1f2f =√
2 and f 8 = f 81 + f 8
2 we get u = f (i) = 214 .
Also f (τ) = f (−1τ ) implies v3 = f ( i+48
5 ) = f ( i−25 + 10) =
e−10πi
24 f ( i−25 ) = e
−10πi24 f (i + 2) = e
−πi2 f (i) = −i2
14 .
Similarly v2 = i214 = iδ, with δ defined as 2
14 .
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
Since f1(τ) = f2(−1τ ) we have f1(i) = f2(i).
Using f1f2f =√
2 and f 8 = f 81 + f 8
2 we get u = f (i) = 214 .
Also f (τ) = f (−1τ ) implies v3 = f ( i+48
5 ) = f ( i−25 + 10) =
e−10πi
24 f ( i−25 ) = e
−10πi24 f (i + 2) = e
−πi2 f (i) = −i2
14 .
Similarly v2 = i214 = iδ, with δ defined as 2
14 .
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
Since f1(τ) = f2(−1τ ) we have f1(i) = f2(i).
Using f1f2f =√
2 and f 8 = f 81 + f 8
2 we get u = f (i) = 214 .
Also f (τ) = f (−1τ ) implies v3 = f ( i+48
5 ) = f ( i−25 + 10) =
e−10πi
24 f ( i−25 ) = e
−10πi24 f (i + 2) = e
−πi2 f (i) = −i2
14 .
Similarly v2 = i214 = iδ, with δ defined as 2
14 .
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
Since f1(τ) = f2(−1τ ) we have f1(i) = f2(i).
Using f1f2f =√
2 and f 8 = f 81 + f 8
2 we get u = f (i) = 214 .
Also f (τ) = f (−1τ ) implies v3 = f ( i+48
5 ) = f ( i−25 + 10) =
e−10πi
24 f ( i−25 ) = e
−10πi24 f (i + 2) = e
−πi2 f (i) = −i2
14 .
Similarly v2 = i214 = iδ, with δ defined as 2
14 .
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
So for τ = i we have already found two roots of the modularequation v6 − δ5v5 + δ9v + δ6 = 0.
Dividing the modular equation by (v − v2)(v − v3) = v2 + δ2 weget:
v4 − δ5v3 + δ2v2 + δ7v + δ4 = (v − α)2(v − β)2,
where α + β = δ and αβ = −δ2.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
So for τ = i we have already found two roots of the modularequation v6 − δ5v5 + δ9v + δ6 = 0.
Dividing the modular equation by (v − v2)(v − v3) = v2 + δ2 weget:
v4 − δ5v3 + δ2v2 + δ7v + δ4 = (v − α)2(v − β)2,
where α + β = δ and αβ = −δ2.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
So for τ = i we have already found two roots of the modularequation v6 − δ5v5 + δ9v + δ6 = 0.
Dividing the modular equation by (v − v2)(v − v3) = v2 + δ2 weget:
v4 − δ5v3 + δ2v2 + δ7v + δ4 = (v − α)2(v − β)2,
where α + β = δ and αβ = −δ2.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
So for τ = i we have already found two roots of the modularequation v6 − δ5v5 + δ9v + δ6 = 0.
Dividing the modular equation by (v − v2)(v − v3) = v2 + δ2 weget:
v4 − δ5v3 + δ2v2 + δ7v + δ4 = (v − α)2(v − β)2,
where α + β = δ and αβ = −δ2.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
Thus we can solve for α, β and we get α = δ 1+√
52 and β = δ 1−
√5
2 .
Lastly we see v∞ = f (5i) = f (−15i ) = f ( i
5 ) = v0 and v∞ > 0implies
v0 = v∞ = δ 1+√
52 and v1 = v4 = δ 1−
√5
2 .
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
Thus we can solve for α, β and we get α = δ 1+√
52 and β = δ 1−
√5
2 .
Lastly we see v∞ = f (5i) = f (−15i ) = f ( i
5 ) = v0 and v∞ > 0implies
v0 = v∞ = δ 1+√
52 and v1 = v4 = δ 1−
√5
2 .
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
Thus we can solve for α, β and we get α = δ 1+√
52 and β = δ 1−
√5
2 .
Lastly we see v∞ = f (5i) = f (−15i ) = f ( i
5 ) = v0 and v∞ > 0implies
v0 = v∞ = δ 1+√
52 and v1 = v4 = δ 1−
√5
2 .
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
Plugging in we find w0 = 0,w1 = w2 = i√
5,w3 = w4 = −i√
5.
Thus our quintic∏(w − wi ) = w(w − i
√5)2(w + i
√5)2 = w(w2 + 5)2.
So our constant term A5(i) = 0 and we calculate C = −27.
We deduce A5 = −u12 + 64u−12.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
Plugging in we find w0 = 0,w1 = w2 = i√
5,w3 = w4 = −i√
5.
Thus our quintic∏(w − wi ) = w(w − i
√5)2(w + i
√5)2 = w(w2 + 5)2.
So our constant term A5(i) = 0 and we calculate C = −27.
We deduce A5 = −u12 + 64u−12.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
Plugging in we find w0 = 0,w1 = w2 = i√
5,w3 = w4 = −i√
5.
Thus our quintic∏(w − wi ) = w(w − i
√5)2(w + i
√5)2 = w(w2 + 5)2.
So our constant term A5(i) = 0 and we calculate C = −27.
We deduce A5 = −u12 + 64u−12.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
Plugging in we find w0 = 0,w1 = w2 = i√
5,w3 = w4 = −i√
5.
Thus our quintic∏(w − wi ) = w(w − i
√5)2(w + i
√5)2 = w(w2 + 5)2.
So our constant term A5(i) = 0 and we calculate C = −27.
We deduce A5 = −u12 + 64u−12.
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
Thus we have w(w2 + 5)2 = u12 − 64u−12.
We use f 8 = f 81 + f 8
2 and f1f2f =√
2 to get
u12 − 64
u12=
f 24 − 64
f 12= (
f 81 − f 8
2
f 2)2.
Thus we have√w = ± f 8
1 −f 82
f 2(w2+5).
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
Thus we have w(w2 + 5)2 = u12 − 64u−12.
We use f 8 = f 81 + f 8
2 and f1f2f =√
2 to get
u12 − 64
u12=
f 24 − 64
f 12= (
f 81 − f 8
2
f 2)2.
Thus we have√w = ± f 8
1 −f 82
f 2(w2+5).
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
Thus we have w(w2 + 5)2 = u12 − 64u−12.
We use f 8 = f 81 + f 8
2 and f1f2f =√
2 to get
u12 − 64
u12=
f 24 − 64
f 12= (
f 81 − f 8
2
f 2)2.
Thus we have√w = ± f 8
1 −f 82
f 2(w2+5).
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
Thus we have w(w2 + 5)2 = u12 − 64u−12.
We use f 8 = f 81 + f 8
2 and f1f2f =√
2 to get
u12 − 64
u12=
f 24 − 64
f 12= (
f 81 − f 8
2
f 2)2.
Thus we have√w = ± f 8
1 −f 82
f 2(w2+5).
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
So we let y =f 81 −f 8
2f 2(w2+5)
.
Thenf 81 −f 8
2f 2 = y(w2 + 5) = y(y4 + 5) = y5 + 5y .
Hence to find the roots of the Bring-Jerrard quintic
y5 + 5y − a = 0 we only need to find a τ such that a =f 81 −f 8
2f 2 .
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
So we let y =f 81 −f 8
2f 2(w2+5)
.
Thenf 81 −f 8
2f 2 = y(w2 + 5) = y(y4 + 5) = y5 + 5y .
Hence to find the roots of the Bring-Jerrard quintic
y5 + 5y − a = 0 we only need to find a τ such that a =f 81 −f 8
2f 2 .
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Deriving the Quintic
So we let y =f 81 −f 8
2f 2(w2+5)
.
Thenf 81 −f 8
2f 2 = y(w2 + 5) = y(y4 + 5) = y5 + 5y .
Hence to find the roots of the Bring-Jerrard quintic
y5 + 5y − a = 0 we only need to find a τ such that a =f 81 −f 8
2f 2 .
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Solving for τ
Squaring the equation f 81 (τ)− f 8
2 (τ) = af 2(τ), and usingf 81 + f 8
2 = f 8, f1f2f =√
2 we get
f 24 − a2f 12 − 64 = 0 and can solve for f 12(τ).
Let j be Klein’s j-invariant. Then j = f 24−16f 24 .
The ability to solve j(τ) = c for any complex c shows that we cansolve for a τ such that f 8
1 (τ)− f 82 (τ) = af 2(τ).
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Solving for τ
Squaring the equation f 81 (τ)− f 8
2 (τ) = af 2(τ), and usingf 81 + f 8
2 = f 8, f1f2f =√
2 we get
f 24 − a2f 12 − 64 = 0 and can solve for f 12(τ).
Let j be Klein’s j-invariant. Then j = f 24−16f 24 .
The ability to solve j(τ) = c for any complex c shows that we cansolve for a τ such that f 8
1 (τ)− f 82 (τ) = af 2(τ).
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Solving for τ
Squaring the equation f 81 (τ)− f 8
2 (τ) = af 2(τ), and usingf 81 + f 8
2 = f 8, f1f2f =√
2 we get
f 24 − a2f 12 − 64 = 0 and can solve for f 12(τ).
Let j be Klein’s j-invariant. Then j = f 24−16f 24 .
The ability to solve j(τ) = c for any complex c shows that we cansolve for a τ such that f 8
1 (τ)− f 82 (τ) = af 2(τ).
Ali Uncu & Frank Patane Theta Functions and the Quintic
OutlineA Brief History of the Quintic
Transformations of the Quintic EquationDefinition and Properties of Theta Functions
The General SchemeThe Modular Equation
Solving the quintic
Solving for τ
Squaring the equation f 81 (τ)− f 8
2 (τ) = af 2(τ), and usingf 81 + f 8
2 = f 8, f1f2f =√
2 we get
f 24 − a2f 12 − 64 = 0 and can solve for f 12(τ).
Let j be Klein’s j-invariant. Then j = f 24−16f 24 .
The ability to solve j(τ) = c for any complex c shows that we cansolve for a τ such that f 8
1 (τ)− f 82 (τ) = af 2(τ).
Ali Uncu & Frank Patane Theta Functions and the Quintic