Variations on a Theme: Fields of Definition, Fields of Moduli, - - PowerPoint PPT Presentation

Fields of Definition / Fields of Moduli Automorphism Groups Twists

Variations on a Theme: Fields of Definition, Fields of Moduli, Automorphisms, and Twists

Michelle Manes (mmanes@math.hawaii.edu) ICERM Workshop Moduli Spaces Associated to Dynamical Systems 17 April, 2012

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Definitions

Definition Let φ ∈ RatN

d . A field K ′/K is a field of definition for φ if

φf ∈ RatN

d (K ′) for some f ∈ PGLN+1.

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Definitions

Definition Let φ ∈ RatN

d . A field K ′/K is a field of definition for φ if

φf ∈ RatN

d (K ′) for some f ∈ PGLN+1.

Definition Let φ ∈ RatN

d , and define

Gφ = {σ ∈ GK | φσ is K equivalent to φ}. The field of moduli of φ is the fixed field K

Gφ.

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Definitions

The field of moduli of φ is the smallest field L with the property that for every σ ∈ Gal(K/L) there is some fσ ∈ PGLN+1 such that φσ = φfσ.

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Definitions

The field of moduli of φ is the smallest field L with the property that for every σ ∈ Gal(K/L) there is some fσ ∈ PGLN+1 such that φσ = φfσ. The field of moduli for φ is contained in every field of definition.

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Definitions

The field of moduli of φ is the smallest field L with the property that for every σ ∈ Gal(K/L) there is some fσ ∈ PGLN+1 such that φσ = φfσ. The field of moduli for φ is contained in every field of definition. Equality???

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

FOD = FOM criterion

Proposition (Hutz, M.) Let ξ ∈ MN

d (K) be a dynamical system with Autφ = {id},

and let D = N

j=0 dN.

If gcd(D, N + 1) = 1, then K is a field of definition of ξ.

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

FOD = FOM criterion

Idea: If [φ] ∈ MN

d (K), then you get a cohomology class

f : Gal( ¯ K/K) → PGLN+1 σ → fσ

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

FOD = FOM criterion

Idea: If [φ] ∈ MN

d (K), then you get a cohomology class

f : Gal( ¯ K/K) → PGLN+1 σ → fσ Twists of PN are in 1-1 correspondence with cocyles: i : PN → X σ → i−1iσ [φ] ∈ MN

d (K) ❀ cocycle cφ ❀ Xcφ

K is FOD for φ ⇐ ⇒ cφ trivial ⇐ ⇒ Xcφ/K When gcd(D, N + 1) = 1, we can find a K-rational zero-cycle on Xcφ.

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

FOD = FOM criterion

If N = 1, then D = d + 1, and the test is on gcd(d + 1, 2). Corollary (Silverman) If d is even, then the field of moduli is a field of definition.

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

FOD = FOM criterion

If N = 1, then D = d + 1, and the test is on gcd(d + 1, 2). Corollary (Silverman) If d is even, then the field of moduli is a field of definition. Result in P1 doesn’t require Aut(φ) = id. Proof requires knowledge of the possible automorphism groups and “cohomology lifting.”

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Example (Silverman) φ(z) = i z−1

z+1

• 3. So Q(i) is a field of definition for φ.

Let σ represent complex conjugation, then φσ = φf for f = −1 z . Hence, Q is the field of moduli for φ. K is a field of definition for φ iff −1 ∈ NK(i)/K (K(i)∗).

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Normal Form for M2

Lemma (Milnor) Let φ ∈ Rat2 have multipliers λ1, λ2, λ3.

1

If not all three multipliers are 1, φ is conjugate to a map of the form: z2 + λ1z λ2z + 1 .

2

If all three multipliers are 1, φ is conjugate to: z + 1 z .

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Normal Form for M2

Lemma (Milnor) Let φ ∈ Rat2 have multipliers λ1, λ2, λ3.

1

If not all three multipliers are 1, φ is conjugate to a map of the form: z2 + λ1z λ2z + 1 .

2

If all three multipliers are 1, φ is conjugate to: z + 1 z . Possible that φ ∈ K(z) but the conjugate map is not.

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Arithmetic Normal Form for M2

Theorem (M., Yasufuku) Let φ ∈ Rat2(K) have multipliers λ1, λ2, λ3.

1

If the multipliers are distinct or if exactly two multipliers are 1, then φ(z) is conjugate over K to 2z2 + (2 − σ1)z + (2 − σ1) −z2 + (2 + σ1)z + 2 − σ1 − σ2 ∈ K(z), where σ1 and σ2 are the first two symmetric functions

• f the multipliers.

Furthermore, no two distinct maps of this form are conjugate to each other over K.

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Arithmetic Normal Form for M2

Theorem (M., Yasufuku)

2

If λ1 = λ2 = 1 and λ3 = λ1 or if λ1 = λ2 = λ3 = 1, then ψ is conjugate over K to a map of the form φk,b(z) = kz + b z with k = λ1+1

2 , and b ∈ K ∗.

Furthermore, two such maps φk,b and φk′,b′ are conjugate over K if and only if k = k ′; they are conjugate over K if in addition b/b′ ∈ (K ∗)2.

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Arithmetic Normal Form for M2

Theorem (M., Yasufuku)

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If λ1 = λ2 = λ3 = −2, then φ is conjugate over K to θd,k(z) = kz2 − 2dz + dk z2 − 2kz + d , with k ∈ K, d ∈ K ∗, and k 2 = d. All such maps are conjugate over K. Furthermore, θd,k(z) and θd′,k′(z) are conjugate over K if and only if ugly, but easily testable condition

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

φ ∈ Hom1

d

Aut(φ) is conjugate to one of the following:

1

Cyclic group of order n: Cn = ζnz .

2

Dihedral group of order 2n: Dn =

• ζnz, 1

z

• .

3

Tetrahedral group: A4 =

• −z, 1

z , i z + 1 z − 1

• .

4

Octahedral group: S4 =

• iz, 1

z , i z + 1 z − 1

• .

5

Icosahedral group: A5 =

• ζ5z, −1

z ,

• ζ5 + ζ−1

5

• z + 1

z −

• ζ5 + ζ−1

5

• .

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

φ ∈ Hom2

d

1

Diagonal Abelian Groups (Cyclic Group of order n): H =   ζa

n

ζb

n

1   , gcd(a, n) = 1 or gcd(b, n) = 1. Proposition Let r be the number of solutions to x2 ≡ 1 mod n. There are n + r/2 − ϕ(n)/2 representations of Cn of the form   ζn ζa

n

1   .

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

φ ∈ Hom2

d

2

Subgroups of the form   ζp ai bi ci di  

• ,

where the lower right 2 × 2 matrices come from embedding the PGL2 automorphism groups.

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

φ ∈ Hom2

d

3

Subgroups that don’t come from embedding PGL2. (Lots of them.)   1 1 1   ,   1 1 1 0,     −1 1 −1   ,   1 −1 −1  

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Higher Dimensions

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Higher Dimensions

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Computing the Absolute Automorphism Group

Algorithm (Faber, M., Viray) Input: a nonconstant rational function φ ∈ K(z), an Autφ( ¯ K)-invariant subset T = {τ1, . . . , τn} ⊂ P1(E) with n ≥ 3. Output: the set Autφ( ¯ K)

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Computing the Absolute Automorphism Group

Algorithm (Faber, M., Viray) create an empty list L. for each triple of distinct integers i, j, k ∈ {1, . . . , n}:

compute s ∈ PGL2( ¯ K) by solving the linear system s(τ1) = τi, s(τ2) = τj, s(τ3) = τk. if s ◦ φ = φ ◦ s: append s to L.

return L.

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Computing the Automorphism Group for a Given Map

Proposition (Faber, M., Viray) Let K be a number field and let φ ∈ K(z) a rational function of degree d ≥ 2. Define S0 to be the set of rational primes given by S0 = {2} ∪

• p odd : p − 1

2

• [K : Q] and p | d(d2 − 1)
• ,

and let S be the (finite) set of places of K of bad reduction for φ along with the places that divide a prime in S0. Then redv : Autφ(K) → Autφ(Fv) is a well-defined injective homomorphism for all places v outside S.

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Realizing Maps with a Given Automorphism Group

Given a finite subgroup Γ ∈ PGL2, Doyle & McMullen give a way to construct all rational maps φ ∈

• 2≤d≤n

Ratd with Γ ⊆ Aut(φ).

• inv. hom. one-form

1−1

← → inv. hom. rational map Fdx + Gdy

1−1

← → φ = −G F

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Realizing Maps with a Given Automorphism Group

It is enough to find all (relative) invariant homogeneous polynomials, i.e. for each γ ∈ Γ there is a character χ: γ∗F = F(γ¯ x) = χ(γ)F(¯ x).

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Realizing Maps with a Given Automorphism Group

It is enough to find all (relative) invariant homogeneous polynomials, i.e. for each γ ∈ Γ there is a character χ: γ∗F = F(γ¯ x) = χ(γ)F(¯ x). λ = (xdy − ydx)/2. Every invariant one-form has the form: Fλ + dG, where deg F + 2 = deg G, F and G invariant with the same character.

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Two useful gadgets

Molien Series: Given a finite group Γ (and character χ),

• utputs the power series

∞

• k=0

dim

• K[¯

x]Γ

k

• tk.

Reynolds Operator: Given a finite group Γ, (character χ), and all homogeneous monomials of a given degree, outputs all (relative) Γ-invariants

• f that degree.

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Example

Γ = C4 =

• i

1

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Example

Γ = C4 =

• i

1

• Molien Series:

1 + t2 + t4 + t6 + 3t8 + 3t10 + 3t12 + 3t14 + 5t16 + O(t18)

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Example

Γ = C4 =

• i

1

• Molien Series:

1 + t2 + t4 + t6 + 3t8 + 3t10 + 3t12 + 3t14 + 5t16 + O(t18) Invariants of degree ≤ 8: xy x2y 2 x3y 3 x4y 4 x8 y 8

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Example

Γ = C4 =

• i

1

• Molien Series:

1 + t2 + t4 + t6 + 3t8 + 3t10 + 3t12 + 3t14 + 5t16 + O(t18) Invariants of degree ≤ 8: xy x2y 2 x3y 3 x4y 4 x8 y 8 Some maps with Γ ⊆ Aut(φ): φ1(z) = z4 + 16 z3 φ2(z) = z9 + 9z z8 − 1

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Example

Γ =

• 1

−1 1 1

• (also cyclic of order 4).

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Example

Γ =

• 1

−1 1 1

• (also cyclic of order 4).

Invariants of degree ≤ 8:

x2 + y2 x8 + 12x6y2 − 20x4y4 + 12x2y6 + y8 x4 + 2x2y2 + y4 x8 − 4x6y2 + 22x4y4 − 4x2y6 + y8 x6 + 3x4y2 + 3x2y4 + y6 x7y − 7x5y3 + 7x3y5 − xy7 Some maps with Γ ⊆ Aut(φ): φ1(z) = − z7 + 24z6 + 3z5 − 40z4 + 3z3 + 72z2 + z + 8 8z7 − z6 + 72z5 − 3z4 − 40z3 − 3z2 + 24z − 1 φ2(z) = −z

• 3z6 − 39z4 + 73z2 − 13
• 13z6 − 73z4 + 39z2 − 3

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Exact Automorphism Groups?

Proposition (Hutz, M.) Let A4 =

• −z, 1

z , i z + 1 z − 1

• and S4 =
• iz, 1

z , i z + 1 z − 1

• .

If φ ∈ Q(z) satisfies A4 ⊆ Aut(φ), then in fact Aut(φ) = S4.

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Exact Automorphism Groups?

Proposition (Hutz, M.) Let A4 =

• −z, 1

z , i z + 1 z − 1

• and S4 =
• iz, 1

z , i z + 1 z − 1

• .

If φ ∈ Q(z) satisfies A4 ⊆ Aut(φ), then in fact Aut(φ) = S4. Question How to construct maps φ ∈ K(z) with Γ = Aut(φ) (or decide there are none)? How to construct maps φ ∈ K(z) with a subgroup of Aut(φ) conjugate to (or equal to) Γ?

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Automorphisms and Twists

Twist(φ/K) =    K-equivalence classes

• f maps ψ ∈ HomN

d (K)

such that ψ is K-equivalent to φ    . Twists give automorphisms of the map φ: fφf −1 = (fφf −1)σ = f σφ(f −1)σ. φ = f −1f σφ(f σ)−1f f −1f σ ∈ Aut(φ).

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Uniform Bounds on Preperiodic Points for Twists

Proposition (Levy, M., Thompson) Let φ ∈ HomN

d (K). There is a uniform bound Bφ such that

for all ψ ∈ Twist(φ/K), # PrePer(ψ, PN

K) ≤ Bφ.

Idea: The degree of the field of definition of the twisting map f is bounded by # Aut(φ). Apply Northcott.

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Cohomology and Twists

For an object X, twists give automorphisms: gσ : X

σ(i−1)

− − − − → Y

i

− → X. A twist gives a one-cocyle: g : Gal( ¯ K/K) → Aut(X) σ → i ◦ σ(i−1)

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Cohomology and Twists

For an object X, twists give automorphisms: gσ : X

σ(i−1)

− − − − → Y

i

− → X. A twist gives a one-cocyle: g : Gal( ¯ K/K) → Aut(X) σ → i ◦ σ(i−1) Does every one-cocyle come from a twist?

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Cohomology and Twists

For an object X, twists give automorphisms: gσ : X

σ(i−1)

− − − − → Y

i

− → X. A twist gives a one-cocyle: g : Gal( ¯ K/K) → Aut(X) σ → i ◦ σ(i−1) Does every one-cocyle come from a twist? For algebraic varieties, yes. For morphisms, sometimes. Twist(φ/K) =

• ξ ∈ H1

Gal( ¯ K/K), Aut(φ)

• :

ξ becomes trivial in H1 Gal( ¯ K/K

• , PGLN+1
• .

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Describing Twists

Question Given φ ∈ Ratd, can we write an explicit formula for all twists of φ?

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Describing Twists

Question Given φ ∈ Ratd, can we write an explicit formula for all twists of φ? Done for Rat2 by Arithmetic Normal Form Theorem.

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Describing Twists

Question Given φ ∈ Ratd, can we write an explicit formula for all twists of φ? Done for Rat2 by Arithmetic Normal Form Theorem. If Aut(φ) = {ζnz}, then we have an isomorphism K ∗/K ∗n → Twist(φ/K) b →   φ

• z

n

√ b

• n

√ b   .

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Fields of Definition / Fields of Moduli Automorphism Groups Twists

Describing Twists

Question Given φ ∈ Ratd, can we write an explicit formula for all twists of φ? Done for Rat2 by Arithmetic Normal Form Theorem. If Aut(φ) = {ζnz}, then we have an isomorphism K ∗/K ∗n → Twist(φ/K) b →   φ

• z

n

√ b

• n

√ b   . More general presentation of Cn? Other automorphism groups? Higher dimensions?

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