abelain function theory - cs.bath.ac.ukme350/conferences/phdsem08.pdf · 1 elliptic function theory...
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Elliptic function theoryGeneralising to higher genus
What I have been doing
Abelain Function TheoryNew Results for Abelian functions associated with a cyclic
tetragonal curve of genus six
Matthew England ggggggg Chris Eilbeck
Department of Mathematics, MACSHeriot Watt University, Edinburgh
Heriot Watt Mathematics Postgraduate Seminar17th October 2008
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
Outline
1 Elliptic function theoryElliptic functionsThe Weierstrass elliptic functionConnection with elliptic curves
2 Generalising to higher genusThe higher genus curves and functionsReview of higher genus work
3 What I have been doingThe cyclic (4,5)-curveThe sigma-function expansionNew results
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
Elliptic functionsThe Weierstrass elliptic functionConnection with elliptic curves
Outline
1 Elliptic function theoryElliptic functionsThe Weierstrass elliptic functionConnection with elliptic curves
2 Generalising to higher genusThe higher genus curves and functionsReview of higher genus work
3 What I have been doingThe cyclic (4,5)-curveThe sigma-function expansionNew results
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
Elliptic functionsThe Weierstrass elliptic functionConnection with elliptic curves
What are elliptic functions?
They are complex functions with two independent periods.
We usually label the two periods ω1, ω2.The period lattice Λ is the set of points Λm,n = mω1 + nω2.
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
Elliptic functionsThe Weierstrass elliptic functionConnection with elliptic curves
What are elliptic functions?
They are complex functions with two independent periods.
We usually label the two periods ω1, ω2.The period lattice Λ is the set of points Λm,n = mω1 + nω2.
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
Elliptic functionsThe Weierstrass elliptic functionConnection with elliptic curves
What are elliptic functions?
DefinitionAn elliptic function is ameromorphic function f definedon C for which there existnon-zero complex numbersω1, ω2,
ω1ω2
/∈ R such that
f (u + ω1) = f (u + ω2) = f (u)
for all u ∈ C.
We usually label the two periods ω1, ω2.The period lattice Λ is the set of points Λm,n = mω1 + nω2.
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
Elliptic functionsThe Weierstrass elliptic functionConnection with elliptic curves
Key contributors to elliptic function theory
Karl Weierstrass1815-1897
Carl Jacobi1804-1851
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
Elliptic functionsThe Weierstrass elliptic functionConnection with elliptic curves
Key contributors to elliptic function theory
Karl Weierstrass1815-1897
Most modern authorsfollow the work andnotation of KarlWeierstrass.The field of ellipticfunctions with respect togiven periods is generatedby a Weierstrass℘-function and itsderivative ℘′.
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
Elliptic functionsThe Weierstrass elliptic functionConnection with elliptic curves
The Weierstrass ℘-function
Defined using a complex variable u and the periods (ω1, ω2).
DefinitionDefine the Weierstrass σ-function associated to the lattice Λ
σ(u; Λ) = u′∏
m,n
(1− z
Λm,n
)exp
[u
Λm,n+
12
(u
Λm,n
)2]
Then the Weierstrass ℘-function can be defined as
℘(u) = − d2
du2 ln[σ(u)]
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
Elliptic functionsThe Weierstrass elliptic functionConnection with elliptic curves
The Weierstrass ℘-function
Defined using a complex variable u and the periods (ω1, ω2).
DefinitionDefine the Weierstrass σ-function associated to the lattice Λ
σ(u; Λ) = u′∏
m,n
(1− z
Λm,n
)exp
[u
Λm,n+
12
(u
Λm,n
)2]
Then the Weierstrass ℘-function can be defined as
℘(u) = − d2
du2 ln[σ(u)]
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
Elliptic functionsThe Weierstrass elliptic functionConnection with elliptic curves
The Weierstrass ℘-function
Defined using a complex variable u and the periods (ω1, ω2).
DefinitionDefine the Weierstrass σ-function associated to the lattice Λ
σ(u; Λ) = u′∏
m,n
(1− z
Λm,n
)exp
[u
Λm,n+
12
(u
Λm,n
)2]
Then the Weierstrass ℘-function can be defined as
℘(u) = − d2
du2 ln[σ(u)]
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
Elliptic functionsThe Weierstrass elliptic functionConnection with elliptic curves
The Weierstrass ℘-function
DefinitionDefine the Weierstrass σ-function associated to the lattice Λ
σ(u; Λ) = u′∏
m,n
(1− z
Λm,n
)exp
[u
Λm,n+
12
(u
Λm,n
)2]
Then the Weierstrass ℘-function can be defined as
℘(u) = − d2
du2 ln[σ(u)]
Doubly periodic with respect to (ω1, ω2).Only singularities are poles when when u = Λm,n.
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
Elliptic functionsThe Weierstrass elliptic functionConnection with elliptic curves
Differential equations satisfied by the ℘-function
The Differential EquationThe Weierstrass ℘-functions satisfies
[℘′(u)]2 = 4℘(u)3 − g2℘(u)− g3
where g2 and g3 are the elliptic invariants defined as
g2 = 60′∑
m,n
Λ−4m,n g3 = 140
′∑m,n
Λ−6m,n
Differentiating gives℘′′(u) = 6℘(u)2 − 1
2g2
We can use (℘, ℘′) to express all higher order derivatives.
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
Elliptic functionsThe Weierstrass elliptic functionConnection with elliptic curves
Differential equations satisfied by the ℘-function
The Differential EquationThe Weierstrass ℘-functions satisfies
[℘′(u)]2 = 4℘(u)3 − g2℘(u)− g3
where g2 and g3 are the elliptic invariants defined as
g2 = 60′∑
m,n
Λ−4m,n g3 = 140
′∑m,n
Λ−6m,n
Differentiating gives℘′′(u) = 6℘(u)2 − 1
2g2
We can use (℘, ℘′) to express all higher order derivatives.
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
Elliptic functionsThe Weierstrass elliptic functionConnection with elliptic curves
Other important results
The σ-function has a power series expansion
σ(u) = u − 1240
g2u5 − 1840
g3u7 − 1161280
g22u9 − ...
Both ℘(u) and σ(u) satisfy addition formula.
℘(u + v) =14
[℘′(u)− ℘′(v)
℘(u)− ℘(v)
]2
− ℘(u)− ℘(v)
−σ(u + v)σ(u − v)
σ(u)2σ(v)2 = ℘(u)− ℘(v)
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
Elliptic functionsThe Weierstrass elliptic functionConnection with elliptic curves
Other important results
The σ-function has a power series expansion
σ(u) = u − 1240
g2u5 − 1840
g3u7 − 1161280
g22u9 − ...
Both ℘(u) and σ(u) satisfy addition formula.
℘(u + v) =14
[℘′(u)− ℘′(v)
℘(u)− ℘(v)
]2
− ℘(u)− ℘(v)
−σ(u + v)σ(u − v)
σ(u)2σ(v)2 = ℘(u)− ℘(v)
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
Elliptic functionsThe Weierstrass elliptic functionConnection with elliptic curves
What are elliptic curves?
DefinitionAn elliptic curve is anon-singularalgebraic curvewith equation
y2 = x3 + ax + b
for constants a, b
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
Elliptic functionsThe Weierstrass elliptic functionConnection with elliptic curves
The ℘-function parametrises elliptic curves
DefinitionAn elliptic curve is anon-singularalgebraic curvewith equation
y2 = x3 + ax + b (∗)
for constants a, b
Rescale (∗)y2 = 4x3−Ax−B
Consider
[z ′]2 = 4z3 − Az − B (∗∗)
Recall
[℘′(u)]2 = 4℘(u)3−g2℘(u)−g3
providing there existsω1, ω2 such thatg2 = A, g3 = B
=⇒ a solution to (∗∗) is
z = ℘(u + α)
So we say (℘, ℘′) parametrises an elliptic curve
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
Elliptic functionsThe Weierstrass elliptic functionConnection with elliptic curves
The ℘-function parametrises elliptic curves
DefinitionAn elliptic curve is anon-singularalgebraic curvewith equation
y2 = x3 + ax + b (∗)
for constants a, b
Rescale (∗)y2 = 4x3−Ax−B
Consider
[z ′]2 = 4z3 − Az − B (∗∗)
Recall
[℘′(u)]2 = 4℘(u)3−g2℘(u)−g3
providing there existsω1, ω2 such thatg2 = A, g3 = B
=⇒ a solution to (∗∗) is
z = ℘(u + α)
So we say (℘, ℘′) parametrises an elliptic curve
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
Elliptic functionsThe Weierstrass elliptic functionConnection with elliptic curves
The ℘-function parametrises elliptic curves
DefinitionAn elliptic curve is anon-singularalgebraic curvewith equation
y2 = x3 + ax + b (∗)
for constants a, b
Rescale (∗)y2 = 4x3−Ax−B
Consider
[z ′]2 = 4z3 − Az − B (∗∗)
Recall
[℘′(u)]2 = 4℘(u)3−g2℘(u)−g3
providing there existsω1, ω2 such thatg2 = A, g3 = B
=⇒ a solution to (∗∗) is
z = ℘(u + α)
So we say (℘, ℘′) parametrises an elliptic curve
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The higher genus curves and functionsReview of higher genus work
Outline
1 Elliptic function theoryElliptic functionsThe Weierstrass elliptic functionConnection with elliptic curves
2 Generalising to higher genusThe higher genus curves and functionsReview of higher genus work
3 What I have been doingThe cyclic (4,5)-curveThe sigma-function expansionNew results
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The higher genus curves and functionsReview of higher genus work
Cyclic (n,s)-curves
DefinitionA cyclic (n, s)-curve is an algebraic curve with equation
yn = xs + λs−1xs−1 + ... + λ1x + λ0
for (n, s) coprime with n < s.
This will map to a surface with genus g = 12(n − 1)(s − 1)
ExampleAn elliptic curve is labelled a(2, 3)-curve and will have genus
g = 12(2− 1)(3− 1) = 1
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The higher genus curves and functionsReview of higher genus work
Cyclic (n,s)-curves
DefinitionA cyclic (n, s)-curve is an algebraic curve with equation
yn = xs + λs−1xs−1 + ... + λ1x + λ0
for (n, s) coprime with n < s.
This will map to a surface with genus g = 12(n − 1)(s − 1)
ExampleAn elliptic curve is labelled a(2, 3)-curve and will have genus
g = 12(2− 1)(3− 1) = 1
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The higher genus curves and functionsReview of higher genus work
Cyclic (n,s)-curves
DefinitionAn (n, s)-curve is an algebraic curve with equation
yn = xs + λs−1xs−1 + ... + λ1x + λ0
for (n, s) coprime with n < s.
This will map to a surface with genus g = 12(n − 1)(s − 1)
ExampleA (2, 5)-curve will have genus
g = 12(2− 1)(5− 1) = 2
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The higher genus curves and functionsReview of higher genus work
Abelian functions associated to curves
We construct:du — a basis of holomorphic differentials upon C.{αi , βj}1≤i,j≤g — a basis of cycles upon C.The period matrices
ω1 =(∮
αkdu`
)k ,`=1,...,g
ω2 =(∮
βkdu`
)k ,`=1,...,g
Let M(u) be a meromorphic function of u ∈ Cg . Then M(u) isan Abelian function associated with C if
M(u + ω1nT + ω2mT ) = M(u),
for all integer vectors n, m ∈ Z where M(u) is defined.
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The higher genus curves and functionsReview of higher genus work
Abelian functions associated to curves
We construct:du — a basis of holomorphic differentials upon C.{αi , βj}1≤i,j≤g — a basis of cycles upon C.The period matrices
ω1 =(∮
αkdu`
)k ,`=1,...,g
ω2 =(∮
βkdu`
)k ,`=1,...,g
Let M(u) be a meromorphic function of u ∈ Cg . Then M(u) isan Abelian function associated with C if
M(u + ω1nT + ω2mT ) = M(u),
for all integer vectors n, m ∈ Z where M(u) is defined.
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The higher genus curves and functionsReview of higher genus work
Kleinian ℘-functions I
For a given (n, s) curve we can define the multivariateσ-function associated to it, (in analogy to the elliptic case).
σ = σ(u) = σ(u1, u2, ..., ug)
Kleinian ℘-functionsDefine the Kleinian ℘-functions as the second log derivatives.
℘ij = − ∂2
∂ui∂ujln σ(u), i ≤ j ∈ {1, 2, ..., g}
They are Abelian functions.
Imposing this notation on the elliptic case gives ℘11 ≡ ℘.
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The higher genus curves and functionsReview of higher genus work
Kleinian ℘-functions I
For a given (n, s) curve we can define the multivariateσ-function associated to it, (in analogy to the elliptic case).
σ = σ(u) = σ(u1, u2, ..., ug)
Kleinian ℘-functionsDefine the Kleinian ℘-functions as the second log derivatives.
℘ij = − ∂2
∂ui∂ujln σ(u), i ≤ j ∈ {1, 2, ..., g}
They are Abelian functions.
Imposing this notation on the elliptic case gives ℘11 ≡ ℘.
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The higher genus curves and functionsReview of higher genus work
Kleinian ℘-functions I
For a given (n, s) curve we can define the multivariateσ-function associated to it, (in analogy to the elliptic case).
σ = σ(u) = σ(u1, u2, ..., ug)
Kleinian ℘-functionsDefine the Kleinian ℘-functions as the second log derivatives.
℘ij = − ∂2
∂ui∂ujln σ(u), i ≤ j ∈ {1, 2, ..., g}
They are Abelian functions.
Imposing this notation on the elliptic case gives ℘11 ≡ ℘.
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The higher genus curves and functionsReview of higher genus work
Kleinian ℘-functions II
We can extend this notation to higher order derivatives
℘ijk = − ∂3
∂ui∂uj∂ukln σ(u) i ≤ j ≤ k ∈ {1, 2, ..., g}
℘ijkl = − ∂4
∂ui∂uj∂uk∂ulln σ(u) i ≤ j ≤ k ≤ l ∈ {1, 2, ..., g}
etc.Imposing this notation on the elliptic case would show
℘′ ≡ ℘111 ℘′′ ≡ ℘1111
A curve with g = 3 has 6 ℘ij and 10 ℘ijk :
{℘11, ℘12, ℘13, ℘22, ℘23, ℘33}{℘111, ℘112, ℘113, ℘122, ℘123, ℘133, ℘222, ℘223, ℘233, ℘333}
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The higher genus curves and functionsReview of higher genus work
Kleinian ℘-functions II
We can extend this notation to higher order derivatives
℘ijk = − ∂3
∂ui∂uj∂ukln σ(u) i ≤ j ≤ k ∈ {1, 2, ..., g}
℘ijkl = − ∂4
∂ui∂uj∂uk∂ulln σ(u) i ≤ j ≤ k ≤ l ∈ {1, 2, ..., g}
etc.Imposing this notation on the elliptic case would show
℘′ ≡ ℘111 ℘′′ ≡ ℘1111
A curve with g = 3 has 6 ℘ij and 10 ℘ijk :
{℘11, ℘12, ℘13, ℘22, ℘23, ℘33}{℘111, ℘112, ℘113, ℘122, ℘123, ℘133, ℘222, ℘223, ℘233, ℘333}
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The higher genus curves and functionsReview of higher genus work
Hyperelliptic generalisation
Felix Klein1849-1925
DefinitionA hyperelliptic curveis an algebraic curve
y2 = f (x)
where f is of degreen > 4 H. F. Baker
1866-1956
The simplest is the (2,5)-curve which has genus g = 2.
y2 = x5 + λ4x4 + λ3x3 + λ2x2 + λ1x + λ0
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The higher genus curves and functionsReview of higher genus work
Some results for the (2,5)-curve
Baker derived many results for the Kleinianfunctions associated to a (2,5)-curve, thatgeneralised the elliptic results:
Addition formula for the σ-function:℘qr
Elliptic case: − σ(u + v)σ(u − v)
σ(u)2σ(v)2 = ℘(u)− ℘(v)
(2,5)-case:
σ(u + v)σ(u− v)
σ(u)2σ(v)2 =℘22(u)℘21(v)− ℘11(u)−℘21(u)℘22(v) + ℘11(v)
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The higher genus curves and functionsReview of higher genus work
Some results for the (2,5)-curve
Baker derived many results for the Kleinianfunctions associated to a (2,5)-curve, thatgeneralised the elliptic results:
Addition formula for the σ-function:℘qr
Elliptic case: − σ(u + v)σ(u − v)
σ(u)2σ(v)2 = ℘(u)− ℘(v)
(2,5)-case:
σ(u + v)σ(u− v)
σ(u)2σ(v)2 =℘22(u)℘21(v)− ℘11(u)−℘21(u)℘22(v) + ℘11(v)
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The higher genus curves and functionsReview of higher genus work
Some results for the (2,5)-curve
Baker derived many results for the Kleinianfunctions associated to a (2,5)-curve, thatgeneralised the elliptic results:
PDEs for the 10 possible ℘ijk ·℘lmn using℘qr of order ≤ 3:
Elliptic case: [℘′(u)]2 = 4℘(u)3 − g2℘(u)− g3
(2,5)-case:
℘2222 = 4℘3
22 + 4℘12℘22 + 4℘11 + λ4℘222 + λ2
...
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The higher genus curves and functionsReview of higher genus work
Some results for the (2,5)-curve
Baker derived many results for the Kleinianfunctions associated to a (2,5)-curve, thatgeneralised the elliptic results:
PDEs for the 5 possible ℘ijkl using℘qr of order ≤ 2.
Elliptic case: ℘′′(u) = 6℘(u)2 − 12g2
(2,5)-case:
℘2222 = 6℘222 + 1
2λ3 + λ4℘22 + 4℘12
...
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The higher genus curves and functionsReview of higher genus work
Brief summary of other higher genus work
A theory for hyperelliptic curves of arbitrary genus hasbeen developed by Buchstaber, Enolski and Leykin (1997).
DefinitionA Trigonal curve is an algebraic curve with equation
y3 = f (x)where f is of degree n ≥ 4.
Considerable work has been completed for the (3,4) and(3,5)-curves by Baldwin, Eilbeck, Enolski, Gibbons,Matsutani, Onishi and Previato.
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The higher genus curves and functionsReview of higher genus work
Brief summary of other higher genus work
A theory for hyperelliptic curves of arbitrary genus hasbeen developed by Buchstaber, Enolski and Leykin (1997).
DefinitionA Trigonal curve is an algebraic curve with equation
y3 = f (x)where f is of degree n ≥ 4.
Considerable work has been completed for the (3,4) and(3,5)-curves by Baldwin, Eilbeck, Enolski, Gibbons,Matsutani, Onishi and Previato.
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The higher genus curves and functionsReview of higher genus work
Applications in non-linear wave theory
In the elliptic case it is well know that ℘(u) could be used in thesolution of a number of nonlinear equations.
For example, the function
u(x , t) = A℘(x − ct) + B
gives a travelling wave solution for the KdV equation
ut + 12uux + uxxx = 0
Higher genus Kleinian functions can also be shown to solvenonlinear equations.
For example, rescaling the function ℘33, associated to the(3,4)-curve, gives a solution to the Boussinesq equation.
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The higher genus curves and functionsReview of higher genus work
Applications in non-linear wave theory
In the elliptic case it is well know that ℘(u) could be used in thesolution of a number of nonlinear equations.
For example, the function
u(x , t) = A℘(x − ct) + B
gives a travelling wave solution for the KdV equation
ut + 12uux + uxxx = 0
Higher genus Kleinian functions can also be shown to solvenonlinear equations.
For example, rescaling the function ℘33, associated to the(3,4)-curve, gives a solution to the Boussinesq equation.
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The cyclic (4,5)-curveThe sigma-function expansionNew results
Outline
1 Elliptic function theoryElliptic functionsThe Weierstrass elliptic functionConnection with elliptic curves
2 Generalising to higher genusThe higher genus curves and functionsReview of higher genus work
3 What I have been doingThe cyclic (4,5)-curveThe sigma-function expansionNew results
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The cyclic (4,5)-curveThe sigma-function expansionNew results
The cyclic (4,5)-curve
I have been working on the first tetragonal case, using thecyclic (4,5)-curve, which has genus g = 6.
y4 = x5 + λ4x4 + λ3x3 + λ2x2 + λ1x + λ0
Example: wt(λ3x3)=−8 + 3(−4) = −20
For every (n, s)-curve we can define a set of Sato weights thatrender all equations in the theory homogeneous.
For the (4, 5) curve they are given byx y u1 u2 u3 u4 u5 u6 λ4 λ3 λ2 λ1 λ04 5 -11 -7 -6 -3 -2 -1 4 8 12 16 20
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The cyclic (4,5)-curveThe sigma-function expansionNew results
The cyclic (4,5)-curve
I have been working on the first tetragonal case, using thecyclic (4,5)-curve, which has genus g = 6.
y4 = x5 + λ4x4 + λ3x3 + λ2x2 + λ1x + λ0
Example: wt(λ3x3)=−8 + 3(−4) = −20
For every (n, s)-curve we can define a set of Sato weights thatrender all equations in the theory homogeneous.
For the (4, 5) curve they are given byx y u1 u2 u3 u4 u5 u6 λ4 λ3 λ2 λ1 λ04 5 -11 -7 -6 -3 -2 -1 4 8 12 16 20
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The cyclic (4,5)-curveThe sigma-function expansionNew results
The cyclic (4,5)-curve
I have been working on the first tetragonal case, using thecyclic (4,5)-curve, which has genus g = 6.
y4 = x5 + λ4x4 + λ3x3 + λ2x2 + λ1x + λ0
Example: wt(λ3x3)=−8 + 3(−4) = −20
For every (n, s)-curve we can define a set of Sato weights thatrender all equations in the theory homogeneous.
For the (4, 5) curve they are given byx y u1 u2 u3 u4 u5 u6 λ4 λ3 λ2 λ1 λ04 5 -11 -7 -6 -3 -2 -1 4 8 12 16 20
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The cyclic (4,5)-curveThe sigma-function expansionNew results
The cyclic (4,5)-curve
I have been working on the purely tetragonal case, based uponthe (4,5)-curve, which has genus g = 6.
y4 = x5 + λ4x4 + λ3x3 + λ2x2 + λ1x + λ0
Example: wt(λ3x3)=8 + 3(4) = 20
For every (n, s)-curve we can define a set of Sato weights thatrender all equations in the theory homogeneous.
For the (4, 5) curve they are given byx y u1 u2 u3 u4 u5 u6 λ4 λ3 λ2 λ1 λ04 5 -11 -7 -6 -3 -2 -1 4 8 12 16 20
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The cyclic (4,5)-curveThe sigma-function expansionNew results
The sigma-function expansion
We want to construct a power series expansion for σ(u).The expansion will depend on u = (u1, u2, u3, u4, u5, u6)and the coefficients of the curve, {λ4, λ3, λ2, λ1, λ0}.
In 1999 Buchstaber, Enolski and Leykin showed that thecanonical limit of the sigma function associated to an(n, s)-curve, was equal to the Schur-Weierstrasspolynomial generated by (n,s).
σ(u;λ)∣∣∣λ=0
= σ(u; 0) = SWn,s
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The cyclic (4,5)-curveThe sigma-function expansionNew results
The sigma-function expansion
We want to construct a power series expansion for σ(u).The expansion will depend on u = (u1, u2, u3, u4, u5, u6)and the coefficients of the curve, {λ4, λ3, λ2, λ1, λ0}.In 1999 Buchstaber, Enolski and Leykin showed that thecanonical limit of the sigma function associated to an(n, s)-curve, was equal to the Schur-Weierstrasspolynomial generated by (n,s).
σ(u;λ)∣∣∣λ=0
= σ(u; 0) = SWn,s
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The cyclic (4,5)-curveThe sigma-function expansionNew results
The sigma-function expansion II
For the (4,5)-curve we calculate
SW4,5 = 18382528 u15
6 + 1336 u8
6u25u4 − 1
12 u46u1 − 1
126 u76u3u5 − 1
6 u4u3u5u46
− 172 u3
4u66 −
133264 u11
6 u25 + 1
27 u65u3
6 + 23 u4u3
5u3 − 2u24u6u3u5 − u2
2u6
− 29 u3
5u3u36 − u4u2
3 + 112 u4
4u36 −
13024 u9
6u24 −
1756 u7
6u45 + 1
1008 u86u2
+ 13 u4
5u2 + 13 u3
6u23 −
19 u4u6
5 + 1399168 u12
6 u4 + u4u6u25u2 + 1
4 u54
+ 2 u5u3u2 + 16 u5
2u64u2 + 1
12 u65u2u4 − 1
2 u42u6
2u2 + 12 u4
3u62u5
2
− 13 u4
2u6u54 − 1
36 u54u4u6
4 + u4u6u1 − u52u1
Each term has weight −15.Hence, for the (4,5)-curve, σ(u) has weight −15.
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The cyclic (4,5)-curveThe sigma-function expansionNew results
The sigma-function expansion III
The σ-expansion has weight -15, and contains ui and λj .
Writethe expansion as
σ(u) = C15 + C19 + C23 + ... + C15+4n + ...
where each Ck has weight −k in the ui and (k − 15) in the λj .We have C15 ≡ SW . Find the other Ck in turn by:
1 Identifying the possible terms — those with correct weight.2 Form the sigma function with unidentified coefficients.3 Determine coefficients by satisfying known properties.
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The cyclic (4,5)-curveThe sigma-function expansionNew results
The sigma-function expansion III
The σ-expansion has weight -15, and contains ui and λj . Writethe expansion as
σ(u) = C15 + C19 + C23 + ... + C15+4n + ...
where each Ck has weight −k in the ui and (k − 15) in the λj .We have C15 ≡ SW .
Find the other Ck in turn by:
1 Identifying the possible terms — those with correct weight.2 Form the sigma function with unidentified coefficients.3 Determine coefficients by satisfying known properties.
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The cyclic (4,5)-curveThe sigma-function expansionNew results
The sigma-function expansion III
The σ-expansion has weight -15, and contains ui and λj . Writethe expansion as
σ(u) = C15 + C19 + C23 + ... + C15+4n + ...
where each Ck has weight −k in the ui and (k − 15) in the λj .We have C15 ≡ SW . Find the other Ck in turn by:
1 Identifying the possible terms — those with correct weight.2 Form the sigma function with unidentified coefficients.3 Determine coefficients by satisfying known properties.
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The cyclic (4,5)-curveThe sigma-function expansionNew results
Heavy use ofMaple!
Used Distributed Mapleon cluster of machines
Polynomial # TermsC19 50C27 386C35 2193C43 8463C51 28359C59* 81832
∗ # possible terms = 120964
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The cyclic (4,5)-curveThe sigma-function expansionNew results
The Q-functions I
The σ-function expansion was constructed in tandem witha set of PDEs satisfied by Abelian functions.
These were derived systematically through theconstruction of a basis for such functions.But this required an additional class of Abelian functions —the Q-functions.
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The cyclic (4,5)-curveThe sigma-function expansionNew results
The Q-functions I
The σ-function expansion was constructed in tandem witha set of PDEs satisfied by Abelian functions.These were derived systematically through theconstruction of a basis for such functions.
But this required an additional class of Abelian functions —the Q-functions.
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The cyclic (4,5)-curveThe sigma-function expansionNew results
The Q-functions I
The σ-function expansion was constructed in tandem witha set of PDEs satisfied by Abelian functions.These were derived systematically through theconstruction of a basis for such functions.But this required an additional class of Abelian functions —the Q-functions.
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The cyclic (4,5)-curveThe sigma-function expansionNew results
The Q-functions II
Definition
Hirota’s bilinear operator is defined as ∆i =∂
∂ui− ∂
∂viIt is then simple to check that
℘ij(u) = − 12σ(u)2 ∆i∆jσ(u)σ(v)
∣∣∣v=u
i ≤ j ∈ {1, . . . , 6}.
We extend this to define the n-index Q-functions (for n even).
Qi1,i2,...,in(u) =(−1)
2σ(u)2 ∆i1∆i2 ...∆inσ(u)σ(v)∣∣∣v=u
i1 ≤ ... ≤ in ∈ {1, . . . , 6}.
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The cyclic (4,5)-curveThe sigma-function expansionNew results
The Q-functions II
Definition
Hirota’s bilinear operator is defined as ∆i =∂
∂ui− ∂
∂viIt is then simple to check that
℘ij(u) = − 12σ(u)2 ∆i∆jσ(u)σ(v)
∣∣∣v=u
i ≤ j ∈ {1, . . . , 6}.
We extend this to define the n-index Q-functions (for n even).
Qi1,i2,...,in(u) =(−1)
2σ(u)2 ∆i1∆i2 ...∆inσ(u)σ(v)∣∣∣v=u
i1 ≤ ... ≤ in ∈ {1, . . . , 6}.
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The cyclic (4,5)-curveThe sigma-function expansionNew results
New results for the (4,5)-curve I
New results we have derived for the (4,5)-curve include:A basis for Abelian functions associated to the (4,5)-curve,with poles of order at most 2.
A set of equations that express other such functions usinga linear combinations of the basis entries. For example,
Q5556 = 4℘36 + 4℘56λ4
Q566666 = 20℘36 − 4℘56λ4
...
Q1355 = 12Q2236 + Q1346 + 1
4Q4555λ2 − 12Q2335
... − ℘45λ4λ2 + ℘15λ3
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The cyclic (4,5)-curveThe sigma-function expansionNew results
New results for the (4,5)-curve I
New results we have derived for the (4,5)-curve include:A basis for Abelian functions associated to the (4,5)-curve,with poles of order at most 2.A set of equations that express other such functions usinga linear combinations of the basis entries. For example,
Q5556 = 4℘36 + 4℘56λ4
Q566666 = 20℘36 − 4℘56λ4
...
Q1355 = 12Q2236 + Q1346 + 1
4Q4555λ2 − 12Q2335
... − ℘45λ4λ2 + ℘15λ3
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The cyclic (4,5)-curveThe sigma-function expansionNew results
New results for the (4,5)-curve II
A complete set of PDEs that express the 4-index℘-functions, using Abelian functions of order at most 2.
(4) ℘6666 = 6℘266 − 3℘55 + 4℘46
(5) ℘5666 = 6℘56℘66 − 2℘45
...(20) ℘2336 = 4℘23℘36 + 2℘26℘33 + 8℘16λ3 − 2℘55λ1
+ 2℘35λ2 + 8℘16℘26 − 2℘1356 + 4℘13℘56
+ 4℘15℘36 + 4℘16℘35 − 2℘1266 + 4℘12℘66
...
These are generalisations of the elliptic PDE:
℘′′(u) = 6℘(u)2 − 12g2
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The cyclic (4,5)-curveThe sigma-function expansionNew results
New results for the (4,5)-curve III
A two-term addition formula similar to that found in lowergenus cases.
σ(u + v)σ(u− v)
σ(u)2σ(v)2 = f (u, v)− f (v , u)
where f (u, v) is a (quite large) polynomial constructed ofAbelian functions.
f (u, v) = 14Q114466(u)− 1
6Q4455(v)Q1444(u)
− 6℘55(v)℘66(u)λ24λ1 + 4℘44(v)℘46(u)λ4λ1 + ...
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The cyclic (4,5)-curveThe sigma-function expansionNew results
Applications in Non-Linear Wave Theory
In the (4,5)-case we proved that ℘6666 = 6℘266 − 3℘55 + 4℘46.
Differentiate twice with respect to u6 to give:
℘666666 = 12 ∂∂u6
(℘66℘666
)− 3℘5566 + 4℘4666
Let u6 = x , u5 = y , u4 = t and W (x , y , t) = ℘66(u). Then
Wxxxx = 12 ∂∂x
(WW6
)− 3Wyy + 4Wxt
Rearranging gives[Wxxx − 12WWx − 4Wt
]x + 3Wyy = 0
which is a parametrised form of the KP-equation.
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The cyclic (4,5)-curveThe sigma-function expansionNew results
Applications in Non-Linear Wave Theory
In the (4,5)-case we proved that ℘6666 = 6℘266 − 3℘55 + 4℘46.
Differentiate twice with respect to u6 to give:
℘666666 = 12 ∂∂u6
(℘66℘666
)− 3℘5566 + 4℘4666
Let u6 = x , u5 = y , u4 = t and W (x , y , t) = ℘66(u). Then
Wxxxx = 12 ∂∂x
(WW6
)− 3Wyy + 4Wxt
Rearranging gives[Wxxx − 12WWx − 4Wt
]x + 3Wyy = 0
which is a parametrised form of the KP-equation.
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The cyclic (4,5)-curveThe sigma-function expansionNew results
Applications in Non-Linear Wave Theory
In the (4,5)-case we proved that ℘6666 = 6℘266 − 3℘55 + 4℘46.
Differentiate twice with respect to u6 to give:
℘666666 = 12 ∂∂u6
(℘66℘666
)− 3℘5566 + 4℘4666
Let u6 = x , u5 = y , u4 = t and W (x , y , t) = ℘66(u). Then
Wxxxx = 12 ∂∂x
(WW6
)− 3Wyy + 4Wxt
Rearranging gives[Wxxx − 12WWx − 4Wt
]x + 3Wyy = 0
which is a parametrised form of the KP-equation.
Matthew England Abelian Function Theory
Elliptic function theoryGeneralising to higher genus
What I have been doing
The cyclic (4,5)-curveThe sigma-function expansionNew results
Further Reading
E.T. Whittaker and G.N. WatsonA Course Of Modern Analysis.Cambridge, 1947.
V.M. Buchstaber, V.Z. Enolski, D.V. LeykinKleinian functions, hyperelliptic Jacobians & applications.Reviews in Math. and Math. Physics, 1997, 10:1-125
J.C. Eilbeck, V.Z. Enolski, S. Matsutani, Y. Onishi andE. PreviatoAbelian Functions For Trigonal Curves Of Genus Three.Intl. Math. Research Notices, 2007, Art.ID: rnm140
http://www.risc.uni-linz.ac.at/software/distmaple/http://www.ma.hw.ac.uk/∼matte/
Matthew England Abelian Function Theory