Abel-Jacobi map on complex hyperelliptic curves Pascal Molin - - PowerPoint PPT Presentation

Abel-Jacobi map on complex hyperelliptic curves

Pascal Molin

CARAMEL, Loria Nancy

june 2011

Hyperelliptic curve

C : y2 =

2g+1

∏

i=1

(x − ai)

• Riemann surface with two

sheets.

• ai = branch points.

color = argument of y

Abel-Jacobi map

u : Div0(C ) → Cg/ΩZ2g Q − P → Q

P ωj

mod ΩZ2g

• Homology H1(C ) = Zγ1 ⊕ · · · ⊕ Zγ2g
• Holomorphic differentials H 1(C ) = Cω1 ⊕ · · · ⊕ Cωg
• Ω = period matrix =
• γi ωj
• ∈ Mg×2g(C).

Abel-Jacobi map

u : Div0(C ) → Cg/ΩZ2g Q − P → Q

P ωj

mod ΩZ2g

• Homology H1(C ) = ZA1 ⊕ · · · ⊕ ZBg

intersection product (antisymetric) symplectic basis A1, . . . Ag, B1 . . . Bg such that (Ai · Bj = δi,j), Ai · Aj = Bi · Bj = 0.

B A

A · B = +1

• Holomorphic differentials H 1(C ) = Cω1 ⊕ · · · ⊕ Cωg
• Ω = period matrix =
• γi ωj
• ∈ Mg×2g(C).

Abel-Jacobi map

u : Div0(C ) → Cg/ΩZ2g Q − P → Q

P ωj

mod ΩZ2g

• Homology H1(C ) = ZA1 ⊕ · · · ⊕ ZBg
• Holomorphic differentials H 1(C ) = Cω1 ⊕ · · · ⊕ Cωg

take dual basis ωi s.t.

Bj ωi = δi,j

• Ω = period matrix = (τIg), τ =
• Aj ωi
• ∈ Hg.

Computation of the period matrix

• standard branch cuts
• standard symplectic basis

when roots are well ordered

• canonical holomorphic forms

dxi y , i = 1 . . . g

a7 a6 a5 a2 a1 a3 a4 A1 A2 A3 B1 B2 B3

Need to parametrize and integrate efficiently over the loops.

Numerical integration

Theorem (Double-exponential integration)

Assume f :]a, b[→ C admits a holomorphic continuation to a domain Rτ = {tanh(λ sinh R ± it), t < τ}, and that |f | < M on Rτ. Then for any D > 0, there exists n, h > 0 such that n ∼ D log D 2πτ and

• b

a f − n

∑

k=−n

f (zk) dzk

• e−D

with

• zk = b+a

2 + b−a 2 tanh(λ sinh(kh))

• dzk = b−a

2 λh cosh(kh) cosh2(λ sinh(kh)).

Convergence of DE integration

Dependency in the holomorphy domain Rτ :

Rπ/10

a b

n ∼ 0.5D log D

a b

Rπ/5

n ∼ 0.25D log D

a b

Rπ/4

n ∼ 0.2D log D

Van Wamelen

Low variations : integrate far from branch points ai.

a7 a6 a5 a2 a1 a3 a4

Van Wamelen

Low variations : integrate far from branch points ai.

a7 a6 a5 a2 a1 a3 a4 A2 B2

Van Wamelen

Low variations : integrate far from branch points ai.

a7 a6 a5 a2 a1 a3 a4 A2 B2

more than 25 integrals (minimum 5g + 3 = 18)

Between branch points

a7 a6 a5 a2 a1 a3 a4 A1 A2 A3 B1 B2 B3

Between branch points

a7 a6 a5 a2 a1 a3 a4 A1 A2 B1 B2 A3 B3

Between branch points

a7 a6 a5 a2 a1 a3 a4 A1 A2 B1 B2 B3 A3 The double-exponential change of variable desingularizes the branch points.

• B3

dx y = 2

a6

a7

dx y =

• R

λ(a6 − a7) sinh(t) dt cosh(λ sinh(t))

• − ∏k=6,7 ϕ6,7(t) − ak

, ϕi,j(t) = ai + aj 2 + aj − ai 2 tanh(sinh(t))

Between branch points

a7 a6 a5 a2 a1 a3 a4 A1 A2 A3 B1 B2 B3 B′

2

B′

2 =

B2 − B3 B′

3 =

B3 Impossible for B1, B2 → change basis.

Between branch points

a7 a6 a5 a2 a1 a3 a4 A1 A2 A3 B1 B3 B′

2

B′

2 =

B2 − B3 B′

3 =

B3

Between branch points

a7 a6 a5 a2 a1 a3 a4 A1 A2 A3 B3 B′

2

B′

1

B′

1 = B1

−B2 B′

2 =

B2 − B3 B′

3 =

B3

Between branch points

a7 a6 a5 a2 a1 a3 a4 A3 B3 B′

2

B′

1

A1 A2 B′

1 = B1

−B2 B′

2 =

B2 − B3 B′

3 =

B3

Between branch points

a7 a6 a5 a2 a1 a3 a4 B3 A3 A1 A2 B′

2

B′

1

B′

1 = B1

−B2 B′

2 =

B2 − B3 B′

3 =

B3 Only 2g intégrals. But here

B′

2 is slow...

Best homology basis

Chain of points :

a7 a6 a5 a2 a1 a3 a4 B3 A3 A1 A2 B′

2

B′

1

τ = 0.46

a7 a6 a5 a2 a1 a3 a4 B3 A3 A1 A2

τ = 0.52

Best homology basis

Chain of points :

a7 a6 a5 a2 a1 a3 a4 B3 A3 A1 A2 B′

2

B′

1

τ = 0.46

a7 a6 a5 a2 a1 a3 a4 B3 A3 A1 A2

τ = 0.52 Any loop around two branch points → homology class = 0

a7 a6 a5 a2 a1 a3 a4 C1 C2 C3 C4 C5 C6

τ = 0.70 Select 2g cheapest integrals. Homology basis ⇔ spanning tree.

Numerical branch cuts

How to compute y(x) on a loop ? must avoid branch cuts. y =

• ∏(x − ai)

y = ∏ √x − ai custom yai,aj (x) Parallelize locally yai,aj (x) = ∏

k

ai ,aj

√x − ak. The (sign of the) loop Ci is determined by the square root chosen for the computation of

Cj ωi.

Numerical branch cuts

How to compute y(x) on a loop ? must avoid branch cuts. y =

• ∏(x − ai)

y = ∏ √x − ai custom yai,aj (x) Parallelize locally yai,aj (x) = ∏

k

ai ,aj

√x − ak. The (sign of the) loop Ci is determined by the square root chosen for the computation of

Cj ωi.

Retrieve symplecticity

Compute intersection Ci · Cj using local square root at end point.

a7 a6 a5 a2 a1 a3 a4 C1 C2 C3 C4 C5 C6

⊖ a b d

τ

π+τ 2

p

Lemma

Let Ci and Cj such that

Ci dx y = 2 b a dx ya,b(x)

Cj dx y = 2 d b dx yb,d(x).

Then arg

• ∏ak=a,b

a,b

√ b − ak ∏ak=b,d

b,d

√ b − ak

• = (Ci · Cj)π + τ

2

Retrieve symplecticity

Compute intersection Ci · Cj using local square root at end point.

a7 a6 a5 a2 a1 a3 a4 C1 C2 C3 C4 C5 C6

⊖ a b d

τ

π+τ 2

p

symplectic reduction    

−1 1 −1 1 −1 −1 1 −1 −1 1 1 1 1 −1

    →    

1 1 1 −1 −1 −1

   

Algorithm

1 compute integration cost for every edge (ai, aj) 2 find minimum spanning tree (C1, . . . C2g) 3 compute periods ΩC =

• Cj

dxi y

• i,j

4 compute intersection matrix X = (Ci · Cj) 5 perform symplectic reduction tPXP = Jg 6 big period matrix Ω = ΩCP = (Ω0Ω1) 7 small period matrix τ = Ω−1 0 Ω1 ∈ Hg

Complexity : O

• g2D2 log2(D)
• bit operations.

Incomplete integrals

• integrate from P to closest branch point ai
• aj

a1 = 2-torsion

(u(aj − a1)) = Ω        

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

       

Timings

Computation of the period matrix : Genus precision Magma DE factor g=3 100 1.3s 0.2s 6.5 1 000 6mn40 16s 23 2 000 82h36 1mn04 4600 3 000

• 3mn
• 10 000
• 1h05
• g=15

2 000

• 28mn52
• Timings on a single core. Integration highly parallelisable.