Decomposing Jacobian Varieties Jen Paulhus - - PowerPoint PPT Presentation

Decomposing Jacobian Varieties

Jen Paulhus

www.math.grinnell.edu/~paulhusj

Plan of Attack

• Why decompose Jacobian Varieties?
• How to use group actions to decompose them.
• Some results.
• An intermediate interlude.
• Some work in progress.

Motivating Questions

dim(JX) = g

• I. For a fixed genus what is the largest positive

integer such that there is some genus curve

• ver , and some elliptic curve with Jacobian

variety ? g g

t

ℚ E JX ∼ Et × A so the largest can be is . g

t

Motivating Questions

• II. In genus 2, Jacobians of curves with nontrivial

automorphism group decompose into the product

• f two elliptic curves. Often those elliptic curves

have interesting arithmetic properties.

• III. Ekedahl and Serre [1993] find curves in various

genera up to 1297 with a curve X of that genus with JX ∼ E1 × E2 × ⋯ × Eg

Motivating Questions

• IV. Once we have a decomposition, what are the

factors? Do they have complex multiplication? Are there fixed factors across families of curves?

• V. Equations of curves are generally hard to come by,

so can we answer these questions without access to an equation?

Given a compact Riemann surface of genus , let (or a possible subgroup) and let be the quotient, of genus

Group Actions

X/G = XG The following technique will work over any field, as long as you know the automorphism group of the corresponding curve over that field. X g ≥ 2 G = Aut(X) h .

Riemann’s Existence Theorem

A finite group G acts on a compact Riemann surface X of genus g >1 if and only if there are elements of the group and so that satisfy the Riemann Hurwitz formula

g = 1 + |G|(h − 1) + |G| 2

r

∑

j=1 (1 − 1

mj) .

a1, b1, …, ah, bh, c1, …, cr

which generate the group, satisfy the following equation, mj = |cj|

h

∏

i=1

[ai, bi]

r

∏

j=1

cj = 1G

Signature: Generating vector:

(h; m1, …, mr) (a1, b1, …, ah, bh, c1, …, cr)

η : ℚ[G] → Endℚ(JX)

An idempotent in produces a factor of the Jacobian: . f Endℚ(JX) f(JX) But the endomorphism ring is complicated, so we start with and translate using the natural map ℚ[G]

From a theorem of Wedderburn we know that ℚ[G] ≅ Mn1(Δ1) × ⋯Mns(Δs) where the are division rings. Δi 1ℚ[G] = ∑

i,j

πi,j . is the idempotent of with the zero matrix in every component except the ith component where it has a 1 in the position only. Then πi,j ℚ[G] j, j

This translates by a result of Kani and Rosen [1989] to an isogeny We call this the group algebra decomposition.

JX ∼ Bn1

1 × ⋯Bns s

1ℚ[G] = ∑

i,j

πi,j . JX ∼ η(π1,1)JX × η(π1,2)JX × ⋯ × η(π1,n1)JX × ⋯ × η(πs,ns)JX

For a special -representation V called the Hurwitz representation with character

ℚ χV

where the are the irreducible -characters of G.

dim Bi = 1 2⟨χi, χV⟩

χi ℚ

H1(X, ℤ) ⊗ ℚ . is the representation of on V

G

Definition

χV = 2χ0 + 2(h − 1)ρ⟨1G⟩ +

r

∑

i=1

(ρ⟨1G⟩ − ρ⟨ci⟩) Given a branched cover with monodromy X → XG c1, …, cr, where is the character of induced from the trivial character of the subgroup , and is the trivial character of ⟨ci⟩ . χ0 G . G ρ⟨ci⟩

• Automorphism group of X
• Monodromy of the cover
• Irreducible -characters

JX ∼ Bn1

1 × ⋯ × Bns s

X → XG To compute the dimensions of the factors of JX, we need

ℚ

where dim Bi = 1

2 ⟨χi, χV⟩

Some Results

• I. Hyperelliptic Curves

Their automorphism groups are well known [2013].

• II. Hurwitz Curves

Curves with the largest possible automorphism groups for a fixed genus, with signature [2016] and current work with students.

[0; 2,3,7], e.g. PSL(2,q), An .

• III. Completely Decomposable Jacobians

with Anita Rojas [2017]

Ekedahl and Serre [1993] demonstrate various curves up to genus 1297 with completely decomposable Jacobian

• varieties. However, there are numerous “gaps” in their data.

Yamauchi [2007] gives lists of integers N so that the Jacobian

• f the modular curve X0(N) is completely decomposable. His

work adds genus 113, 161, and 205 to the list.

JX ∼ E1 × E2 × ⋯ × Eg

where the are (possibly isogenous) elliptic curves.

Given a genus curve , its Jacobian variety is called completely decomposable if g X JX Ei

• All curves up to genus 48 [Breuer, 2000]
• Curves with automorphism group larger than 4(g-1) for

genus up to 101 [Conder, 2010]

• Even higher genus using LowIndexNormalSubgroup( , n)

We applied the technique above to many curves up to genus 101, and a few strategically chosen curves up to genus 500, and found 7 new examples for g = {36, 46, 81, 85, 91, 193, 244} .

Γ

The rest of the talk is joint work with Anita Rojas from Universidad de Chile.

An Intermediate Interlude

For we find

Example

Y : y2 = (x2 − 4)(x3 − 3x + a) .

JX ∼ E2

Group actions will not tell the whole story.

X : y2 = x(x6 + ax3 + 1)

Aut(Y) ≅ C2

• JY ∼ E1 × E2 .

Consider the curve

so can’t decompose by previous technique. We can also compute and so

JX ∼ JY × E1

An Example There is a genus 101 curve with automorphism group (800, 980) whose Jacobian decomposes as: JX ∼ E × A × E2 × E8 × ⋯ × E8

12

where A is an abelian variety of dimension 2 . = C2

10 ⋊ C8

where is the subspace of fixed by . if is a subgroup of then the group algebra decomposition of is given as Given a Galois cover and Carocca and Rodriguez [2006] X → XG JX ∼ B

1

× ⋯ × B

s

, G

JXH

JXH ∼ B

1

× ⋯ × B

s

,

VH

H H

Vi

an irreducible -representation from the ith

Vi ℂ

irreducible -character, and the Schur index. mi

ℚ

JX ∼ B

1

× ⋯ × B

s

JXH ∼ B

1

× ⋯ × B

s

We look for large genus decompositions that need not have all elliptic curves in their decomposition … We can compute as

dim VH

i

⟨Vi, ρH⟩ .

We can compute as

JX ∼ B

1

× ⋯ × B

s

JXH ∼ B

1

× ⋯ × B

s

We look for large genus decompositions that need not have all elliptic curves in their decomposition … … then we compute the dimensions of the exponents hoping they are 0 for any i which is not an elliptic curve in the decomposition of JX.

dim VH

i

⟨Vi, ρH⟩ .

An Example (again)

There is a genus 101 curve with automorphism group (800, 980) whose Jacobian decomposes as:

JX ∼ E × A × E2 × E8 × ⋯ × E8

where A is an abelian variety of dimension 2 . The group has 3 subgroups which produce quotients

• f genus 51. Using the previous slide, we get:

JXH ∼ E × E2 × E4 × ⋯ × E4

• P. and Rojas (2017)

For every integer g in the following list, there is a curve

• f genus g with completely decomposable Jacobian

variety found using a group acting on a curve.

1–29, 30, 31, 32, 33, 34–36, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51–52, 53, 54, 55, 57, 58, 61, 62, 63, 64, 65, 67, 69, 71, 72–73, 79–81, 82, 85, 89, 91, 93, 95, 97, 103, 105–107, 109, 118, 121, 125, 129, 142, 145, 154, 161, 163, 193, 199, 211, 213, 217, 244, 257, 325, 433 The numbers in pink are new genera, the others are different examples from those Ekedahl and Serre found.

The rest of the talk is work in early stages.

Algorithmifying

Use prior work of Auffarth, Behn, Lange, Rodríguez, Rojas

a complex torus where is a complex vector space of dimension and is a lattice in . VA g LA VA A polarization on is a non-degenerate real alternating form satisfying

E(iu, iv) = E(u, v)

E(LA × LA) ⊆ ℤ .

A E A polarization is of type if there exists a basis for such that the matrix for with respect to that basis has the form LA E E = ( D −D 0) . (d1, d2, …, dg) and A = VA/LA where is the diagonal matrix formed from the type: D

(d1, d2, …, dg) .

where is a complex symmetric matrix such that is positive definite. The period matrix for is the matrix ΠA = (D Z) A g × g ℑ(Z) Z Such a basis is called symplectic. γA = {α1, …, αg, β1, …, βg} g × 2g

Goal: Given a principally polarized abelian variety , its period matrix, and an idempotent , compute the period matrix of the corresponding factor .

• If the factor is an elliptic curve: the period matrix

will be of the form

• If the factor is higher dimensional: Auffarth,

Lange and Rojas [2017] give criterion on Neron- Severi group to determine additional factorization. (1 τ) .

A f f(A)

Let be an idempotent defining a subvariety

• f dimension .

B = f(A) = VB/LB f ∈ Endℚ(A) h Lift uniquely to a map and then restrict this map to the lattice: ρr : LA → LA . But because we have a group action, this induces a map: ρr : G → GL(LA ⊗ ℚ) . VA → VA

is sometimes called the rational representation of the action of on . I prefer symplectic representation. ρr G A

Since the group action respects the polarization, for all , Similarly, we can compute the induced polarization. g ∈ G ρr(g) ∈ Sp2g(ℤ) . This representation is the key to determining the induced polarization and lattice of the factor . B E.g., the lattice of is given by the intersection of the basis formed by the columns of matrix intersected with . B LA ℤ ρr(f )

• a symplectic basis of
• Since then the form a basis of
• Determine a symplectic basis of , say

.

• The and can be written with respect to

γA = {α1, …, αg, β1, …, βg} LA . ΠA = (Ig Z) αi LB γB = {u1, …, uh, v1, …, vh} . ui vi

γA .

VA .

• Use to replace any with .
• We get a matrix whose columns are written

with respect to a -basis of . Call it

• and will then each form a -basis of written

with respect to -basis of

• But then is too! We want

basis.

ΠA

βj αi g × 2h γB ℂ VA MB = (C1 C2) . C1 C2 ℂ VB ℂ VA . C1D−1 { u1 d1 , …, uh dh}

• Find change of basis matrix from to in Siegel

upper half-space. Call it

• Then columns of are coordinates of with

respect to

• Means we can form period matrix:

C1D−1 C2 W . W {v1, …, vh} { u1 d1 , …, uh dh} . ΠB = (D W) .

Results

acts on a family of non-hyperelliptic curve of genus 3 with signature [0; 2,2,4,4] in two topologically inequivalent ways. A generating vector for one action is We can factor the Jacobian as C2 × C4 = ⟨a⟩ × ⟨b⟩ (a, ab2, b, b) .

JX ∼ E1 × E2 × E3 .

ΠA = 1 0 0 i −1 − 1

1 0 −1 i − 1

0 1 − 1

x

MB1 = i −1 1 i

1 2 − 1 2i 1 2 + 1 2 i

MB2 = 1 i i −1 − 1

2 + 1 2 i − 1 2 − 1 2i

MB3 = ( 1 2x − i)

• E1 ∼ ℂ/⟨1,i⟩

E2 ∼ ℂ/⟨1,i⟩ E3 ∼ ℂ/⟨1,2x − i⟩

acts on a hyperelliptic curve of genus 11 with signature [0; 2,4,24]. (96,28) = (C2 × C24) ⋊ C2 The group algebra decomposition gives JX ∼ E1 × E2

2 × E2 3 × E2 4 × A2

where is a surface. A Using our method we get that and by the Neron Severi method, that also factors more. E1 ∼ E3 ∼ ℂ/⟨1 i⟩ A

Issues (so far!)

• What is the period matrix of ambient variety?
• Which parameter value(s) correspond to

Jacobian varieties?

• Higher dimension gets hard!

The End