Calculus on Riemann Surfaces in Pythoncswiercz.info/assets/files/ncsu-symbolic.pdf · Introduction...

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Introduction Algebraic Components Geometric Components Riemann Matrices and Theta Functions Next Steps

Calculus on Riemann Surfaces in Python

Chris Swierczewski

cswiercz@uw.edu

University of WashingtonDepartment of Applied Mathematics

Symbolic Computation SeminarNorth Carolina State University

March 2013

March 19, 2013

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1 IntroductionAcknowledgements and MotivationRiemann Surfaces and Period Matrices

2 Algebraic ComponentsPuiseux SeriesSingularitiesHolomorphic Differentials

3 Geometric ComponentsAnalytic ContinuationMonodromyHomology

4 Riemann Matrices and Theta FunctionsRiemann MatricesRiemann Theta Functions

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Table of Contents

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Acknowledgements

Collaborators:

Bernard Deconinck: advisor, co-author of algcurves forMaple.

Grady Williams: undergraduate at UW, computing Riemanntheta functions using CUDA.

Primary references:

Mumford, “Tata Lectures on Theta I,II”

Bobenko, Klein, “Computational Approach to RiemannSurfaces”

Chapter 2 by Deconinck and Patterson, “Computing withPlane Algebraic Curves and Riemann Surfaces”

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Motivation

Abelian functions: higher genus versions of elliptic functions.Periodic functions on Cg/Λ).

Any Abelian function can be expressed as a rational functionof the Riemann theta function θ : Cg × hg → C and itsderivatives.

θ(z,Ω) =∑n∈Zg

(12n·Ωn+n·z

Algorithm for computing the Riemann theta function byDeconinck, et. al.

Period matrices corresponding to complex plane algebraiccurves appear in Abelian function theory.

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Motivation

Additional applications of period matrices and Riemann thetafunctions:

Periodic solutions to integrable partial differential equations.(Dubrovin)

Bitangents of complex plane quartics.(Vinzant, Plaumann, Sturmfels)

Linear matrix representations of Helton-Vinnikov curves.(Vinzant, Plaumann, Sturmfels)

Igusa polynomials and Siegel modular forms.(Lauter, Yang)

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Riemann Surfaces

Let f ∈ C[x, y]. The Riemann surface, X, corresponding to f isthe desingularization and compactification of

X = (x, y) ∈ C2 : f(x, y) = 0. (1)

Interpret X as a y-covering of the Riemann sphere C∗x. If

f(x, y) = an(x)yn + · · ·+ a1(x)y + a0(x)

then the covering is n-sheeted. Above all but finitely many x ∈ C∗x(the branch points of f) there are n distinct values of y ∈ Cy.

“Fibre at x” or “y-roots at x”.

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Riemann Surfaces

X is homeomorphic a complex g-torus. The homology group of X

H1(X,Z)

is the space of all closed paths (cycles) on X. Basis consists of 2g cyclesa1, . . . , ag, b1, . . . , bg with the intersection properties

ai aj = 0, bi bj = 0, ai bj = δij .

Dual space of holomorphic differentials on X

Γ(X,Ω1)

has dimension g.8 / 40

Period Matrices and Riemann Matrices

Choose the normalized basis of Γ(X,Ω1) where.∮ai

ωj = δij ,

ωj = Ωij ∈ C.

Period matrix (Ig×g Ω

)∈ Cg×2g

where Ω ∈ Cg×g is a Riemann matrix:

Ω = ΩT ,

Im Ω 0.

The period matrix is used to define the Jacobian of a Riemannsurface.

Jac(X) := Cg/(Ig×gZg + ΩZg)

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Software: abelfunctions

abelfunctions is a Python library for computing with Riemannsurfaces and (eventually) Abelian functions.

A Python + C port of Deconinck, van Hoeij’s algcurves

package in Maple.

Open-source and free.

Uses numpy, scipy, sympy, networkx.

Available on GitHub:https://github.com/cswiercz/abelfunctions

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Software: Path to Riemann Matrices

Puiseux Series

Monodromy

Integral Basis

Singularities

HomologyAnalytic Continuation

Riemann Matrices

Holomorphic Differentials

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Table of Contents

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Puiseux Series

Taylor and Laurent series locally describe single-valued functions ata point. Puiseux series are a multi-valued extension of these.Can locally describe

regular points,

branch points,

singular points.

“point” = a value in the complex x-plane. “place” a location on the

Riemann surface, X.

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Puiseux Series: Definition

A place P on a Riemann surface X lying above some x = α isdetermined by expansions of the form

x = α+ λte,y =

∑∞k=0 βkt

where · · · < nk < nk+1 < · · · with only finitely many negativeexponents. Solve for t = t(x):

y = y(x) =

∞∑k=0

αk(x− α)nk/e.

Local description of y as a function of x on X.

Algorithm by D. Duval using Newton polygons.

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Puiseux series at places on complex plane algebraic curves.

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Singularities

Must resolve the singularities of an algebraic curve to obtaincorresponding Riemann surface. We use Puiseux series to do so.

Puiseux series define local behavior. =⇒ How to go around /pass through singular points.

Use to determine coordinate chart at each singular point,providing the manifold structure.

Definition: Singularities

Let F (X,Y, Z) ∈ C[X,Y, Z] be the homogenization off(x, y) ∈ C[x, y]. The singular points, P ∈ P 2C, occur where

∂XF (P ) = ∂Y F (P ) = ∂ZF (P ) = 0

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Singularity Data

Important pieces of information corresponding to singular pointsP ∈ P 2C.

multiplicity: the sum of the ramification indices of thePuiseux series at P .

delta invariant: the number of double points at P . Appearsin the genus formula: if d is the total degree of f then thegenus of the curves corresp. Riemann surface is

g = (d− 1)(d− 2)/2−∑P∈X

branching number: the number of Puiseux series at P .

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Singularities. Computing the genus of a curve.

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Differentials

Holomorphic differentials on X : f(x, y) = 0 are all of the form

ωk =Pk(x, y)

∂yf(x, y)dx, Pk(x, y) =

∑i+j≤d−3

ckijxiyj

where d is the degree of f and Pk(x, y) are the “adjointpolynomials” of f are chosen such that ωk has no poles on X.

If X is nonsingular then all polynomials Pk(x, y) of degree≤ d− 3 give rise to a holomorphic differential. Consistentwith genus formula:

g = (d− 1)(d− 2)/2

If X has singularities then more conditions are imposed on thePk(x, y)’s. (Pk must, at least, vanish at the singularities.)

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Algorithm

Based on observation by Mnuk: let

OA(X) = C[x, y]/(f) ⊂ C(x, y).

Adj(X) =P (x, y) ∈ C[x, y] ‖ OA(X) · P (x, y) ⊂ C[x, y]

OA(X) is finite dimensional over C[x, y]. Let, β1, . . . , βm be abasis. Then Mnuk’s theorem is equivalent to requiring

βjP (x, y) ∈ C[x, y],∀j = 1, . . . ,m.

Algorithm for computing βj is due to van Hoeij and uses Puiseuxseries.

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Integral bases of algebraic functions fields and bases of the spaceof holomorphic differentials.

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Table of Contents

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Continuing y-Roots Along x-Paths

The y-roots / fibre y = y1, . . . , yn ⊂ Cy of

f(x, y) =

n∑j=0

aj(x)yj = 0

are continuous as a function of x ∈ Cx. Much of the “geometricside” involves continuing a fibre y along a path γ ⊂ Cx.

In particular, we select a “base point” a ∈ Cx and an orderingof the y-roots at that point. (yj is on sheet j of the Riemannsurface.)

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Continuing y-roots Along x-Paths

Two-step process:1 Taylor step.

A proper step size is determined based on the “movement” ofall of the roots.

Let yi = (yi1, . . . , yin) be the fibre at xi ∈ Cx and

dxi = xi+1 − xi. Then

yapproxj = yij − dxif(xi, y

∂yf(xi, yij)

2 Compute roots yi+1j at xi+1.

3 Determine if Taylor approximates yapproxj are “close enough”to the new roots.

If so, match approximates with new roots and iterate.If not, take smaller step and try again.

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Monodromy Group

Let b1, · · · , bN ∈ C∗x be the branch points of f . The monodromygroup of f is the fundamental group

π1(C∗x\b1, . . . , bN, a).

where a ∈ Cx is a fixed point. (Not a branch point.)

Specifically, we compute the fibrey = y1, . . . , yn at x = a andsee how the roots are permutedwhen we analytically continuearound each branch point bi.

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Monodromy: Algorithm

1 Fix a base point a ∈ Cx andan ordering of the fibre yabove it. (Fixed for allfuture comptuations.)

2 Fix an ordering of thebranch points, bi.

3 Compute a minimalspanning tree with thebranch points as the nodes.The root is the branch pointclosest to a. Draw circularpaths around each branchpoint.

For numerical accuracypurposes we stay awayfrom the branch points.

Figure: The collection of all x-pathsover which we analytically continuethe “base fibre”.

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Monodromy: Algorithm

4 Determine appropriatemonodromy group paths,given the above ordering.

The path around branchpoint bi must lie belowthe path around branchpoint bi+1.

5 Analytically continue thebase fibre along each path.

6 When returning to the basepoint, record how the fibreelements / sheets werepermuted.

Figure: The collection of all x-pathsover which we analytically continuethe “base fibre”.

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The monodromy group π1 (C∗x\b0, . . . , bN, a)

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Homology H1(X,Z)

Monodromy says how to get from one sheet to another on X byanalytically continuing along closed paths in Cx. But we wantclosed paths on the Riemann surface, itself.

Algorithm due to Tretkoff and Tretkoff takes monodromyinformation and returns a- and b-cycles.

“Base place” P0 = (a, y1).

Linear combinations of intermediate “c-cycles”.

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Homology basis of X: H1(X,Z)

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Table of Contents

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Riemann Matrices: Putting It Together

Puiseux Series

Monodromy

Integral Basis

Singularities

HomologyAnalytic Continuation

Riemann Matrices

Holomorphic Differentials

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Riemann Matrices

Compute

ωj Bij =

using Gauss-Legendre quadrature. In particular, parameterize thepath γ ∈ H1(X,Z) by t ∈ [0, 1] and compute∫

γω =

0ω (x(t), y(x(t)))x′(t)dt.

Set,Ω = A−1B.

(This Ω is equal to that which is chosen by normalizing our basisof holomorphic differentials.)

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Period matrices and Riemann matrices.

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Riemann Theta Functions

Building block of Abelian functions: θ : Cg × hg → C.

θ(z,Ω) =∑n∈Zg

(12n·Ωn+n·z

Algorithm by Deconinck, et. al.: separates doubly exponentiallygrowing part and “oscillatory part”.

Fast Python + C implementation. (Multiprecise on the way.)

Optional CUDA-enabled implementation by S. and Williams.

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Riemann Theta Functions: Example

We plot real and imaginary parts of the oscillatory part of

θ ([x+ iy, 0, 0],Ω)

for x ∈ [0, 1], y ∈ [0, 3] with N = 65536 z = [x+ iy, 0, 0] valuesand for

−12 + i 1

2 −12 i −

12 −

12 i i 0

−12 −

12 i 0 i

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CUDA evaluation of a slice of a genus g = 3 Riemann thetafunction.

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Thank You

emailcswiercz@uw.edu

abelfunctionshttps://github.com/cswiercz/abelfunctions

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Future Work

Performance improvements.

csympy by Ondrej Certık.Use Cython and GMP for numerical portions.

Abel Map A : X → Jac(X)

A(P ) =

(∫ P

ω1, . . . ,

(Ph.D. thesis work of Matthew Patterson.)

Vector of Riemann Constants: 2K ≡ −A(C) where C is thecanonical class divisor. (The equivalence class of all divisors ofholomorphic differentials.)(Ph.D. thesis work of Matthew Patterson.)

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Future Work

Fay’s Prime Form: Let α, β ∈ [0, 1)g be such that(∂

∂z1θ[α, β](0,Ω), . . . ,

∂zgθ[α, β](0,Ω)

)6= 0.

Define the holomorphic differential

g∑j=1

ωj∂

∂zjθ[α, β](0,Ω).

Then the prime form E : X ×X → C is defined by

E(P1, P2) = θ[α, β](∫ P2

P1ω,Ω

)/√ζ(P1)

√ζ(P2)

where ω = (ω1, . . . , ωg).

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