Abelain Function Theory - cs.bath.ac.ukme350/Conferences/PhDSem08.pdf · 1 Elliptic function theory Elliptic functions The Weierstrass elliptic function Connection with elliptic curves

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



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




Elliptic functionsThe Weierstrass elliptic functionConnection with elliptic curves

Outline








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.






They are complex functions with two independent periods.







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.






Key contributors to elliptic function theory

Karl Weierstrass1815-1897

Carl Jacobi1804-1851





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 ℘′.





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)]








σ(u; Λ) = u′∏

m,n

(1− z

Λm,n

)exp

[u

Λm,n+

12

(u

Λm,n

)2]


℘(u) = − d2

du2 ln[σ(u)]








σ(u; Λ) = u′∏

m,n

(1− z

Λm,n

)exp

[u

Λm,n+

12

(u

Λm,n

)2]


℘(u) = − d2

du2 ln[σ(u)]







σ(u; Λ) = u′∏

m,n

(1− z

Λm,n

)exp

[u

Λm,n+

12

(u

Λm,n

)2]


℘(u) = − d2

du2 ln[σ(u)]

Doubly periodic with respect to (ω1, ω2).Only singularities are poles when when u = Λm,n.





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.





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.





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)





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)





What are elliptic curves?

DefinitionAn elliptic curve is anon-singularalgebraic curvewith equation

y2 = x3 + ax + b

for constants a, b





The ℘-function parametrises elliptic curves


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







y2 = x3 + ax + b (∗)

for constants a, b


Consider

[z ′]2 = 4z3 − Az − B (∗∗)

Recall

[℘′(u)]2 = 4℘(u)3−g2℘(u)−g3



z = ℘(u + α)








y2 = x3 + ax + b (∗)

for constants a, b


Consider

[z ′]2 = 4z3 − Az − B (∗∗)

Recall

[℘′(u)]2 = 4℘(u)3−g2℘(u)−g3



z = ℘(u + α)





The higher genus curves and functionsReview of higher genus work

Outline








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





Cyclic (n,s)-curves

DefinitionA cyclic (n, s)-curve is an algebraic curve with equation

yn = xs + λs−1xs−1 + ... + λ1x + λ0



ExampleAn elliptic curve is labelled a(2, 3)-curve and will have genus

g = 12(2− 1)(3− 1) = 1





Cyclic (n,s)-curves

DefinitionAn (n, s)-curve is an algebraic curve with equation

yn = xs + λs−1xs−1 + ... + λ1x + λ0



ExampleA (2, 5)-curve will have genus

g = 12(2− 1)(5− 1) = 2





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.





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.





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 ≡ ℘.







σ = σ(u) = σ(u1, u2, ..., ug)


℘ij = − ∂2

∂ui∂ujln σ(u), i ≤ j ∈ {1, 2, ..., g}









σ = σ(u) = σ(u1, u2, ..., ug)


℘ij = − ∂2

∂ui∂ujln σ(u), i ≤ j ∈ {1, 2, ..., g}







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}





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}





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





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)







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)







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

...







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

...





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.





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.





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.





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.




The cyclic (4,5)-curveThe sigma-function expansionNew results

Outline








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







y4 = x5 + λ4x4 + λ3x3 + λ2x2 + λ1x + λ0

Example: wt(λ3x3)=−8 + 3(−4) = −20









y4 = x5 + λ4x4 + λ3x3 + λ2x2 + λ1x + λ0

Example: wt(λ3x3)=−8 + 3(−4) = −20








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







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





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





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.





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.






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:







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:






Heavy use ofMaple!

Used Distributed Mapleon cluster of machines

Polynomial # TermsC19 50C27 386C35 2193C43 8463C51 28359C59* 81832

∗ # possible terms = 120964





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.





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.





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.





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}.





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}.





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





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





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





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 + ...





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.








℘666666 = 12 ∂∂u6

(℘66℘666

)− 3℘5566 + 4℘4666


Wxxxx = 12 ∂∂x

(WW6

)− 3Wyy + 4Wxt


]x + 3Wyy = 0









℘666666 = 12 ∂∂u6

(℘66℘666

)− 3℘5566 + 4℘4666


Wxxxx = 12 ∂∂x

(WW6

)− 3Wyy + 4Wxt


]x + 3Wyy = 0






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/


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