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Fields of Definition / Fields of Moduli Automorphism Groups Twists
Variations on a Theme:Fields of Definition, Fields of Moduli,
Automorphisms, and Twists
Michelle Manes ([email protected])
ICERM WorkshopModuli Spaces Associated to Dynamical Systems
17 April, 2012
1
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Definitions
Definition
Let φ ∈ RatNd . A field K ′/K is a field of definition for φ ifφf ∈ RatNd (K ′) for some f ∈ PGLN+1.
Definition
Let φ ∈ RatNd , and define
Gφ = {σ ∈ GK | φσ is K equivalent to φ}.
The field of moduli of φ is the fixed field KGφ.
2
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Definitions
Definition
Let φ ∈ RatNd . A field K ′/K is a field of definition for φ ifφf ∈ RatNd (K ′) for some f ∈ PGLN+1.
Definition
Let φ ∈ RatNd , and define
Gφ = {σ ∈ GK | φσ is K equivalent to φ}.
The field of moduli of φ is the fixed field KGφ.
3
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Definitions
The field of moduli of φ is the smallest field L with theproperty that for every σ ∈ Gal(K/L) there is somefσ ∈ PGLN+1 such that φσ = φfσ .
The field of moduli for φ is contained in every field ofdefinition.
Equality???
4
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Definitions
The field of moduli of φ is the smallest field L with theproperty that for every σ ∈ Gal(K/L) there is somefσ ∈ PGLN+1 such that φσ = φfσ .
The field of moduli for φ is contained in every field ofdefinition.
Equality???
5
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Definitions
The field of moduli of φ is the smallest field L with theproperty that for every σ ∈ Gal(K/L) there is somefσ ∈ PGLN+1 such that φσ = φfσ .
The field of moduli for φ is contained in every field ofdefinition.
Equality???
6
Fields of Definition / Fields of Moduli Automorphism Groups Twists
FOD = FOM criterion
Proposition (Hutz, M.)
Let ξ ∈ MNd (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.
7
Fields of Definition / Fields of Moduli Automorphism Groups Twists
FOD = FOM criterion
Idea: If [φ] ∈ MNd (K ), then you get a cohomology class
f : Gal(K̄/K )→ PGLN+1
σ 7→ fσ
Twists of PN are in 1-1 correspondence with cocyles:
i : PN → X
σ 7→ i−1iσ
[φ] ∈ MNd (K ) ; cocycle cφ ; Xcφ
K is FOD for φ⇐⇒ cφ trivial ⇐⇒ Xcφ/K
When gcd(D,N + 1) = 1, we can find a K -rationalzero-cycle on Xcφ
.
8
Fields of Definition / Fields of Moduli Automorphism Groups Twists
FOD = FOM criterion
Idea: If [φ] ∈ MNd (K ), then you get a cohomology class
f : Gal(K̄/K )→ PGLN+1
σ 7→ fσ
Twists of PN are in 1-1 correspondence with cocyles:
i : PN → X
σ 7→ i−1iσ
[φ] ∈ MNd (K ) ; cocycle cφ ; Xcφ
K is FOD for φ⇐⇒ cφ trivial ⇐⇒ Xcφ/K
When gcd(D,N + 1) = 1, we can find a K -rationalzero-cycle on Xcφ
.9
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 automorphismgroups and “cohomology lifting.”
10
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 automorphismgroups and “cohomology lifting.”
11
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Example (Silverman)
φ(z) = i(
z−1z+1
)3. So Q(i) is a field of definition for φ.
Let σ represent complex conjugation, then
φσ = φf for f = −1z.
Hence, Q is the field of moduli for φ.
K is a field of definition for φ iff −1 ∈ NK (i)/K (K (i)∗).
12
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 amap of the form:
z2 + λ1zλ2z + 1
.
2 If all three multipliers are 1, φ is conjugate to:
z +1z.
Possible that φ ∈ K (z) but the conjugate map is not.
13
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 amap of the form:
z2 + λ1zλ2z + 1
.
2 If all three multipliers are 1, φ is conjugate to:
z +1z.
Possible that φ ∈ K (z) but the conjugate map is not.
14
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 twomultipliers 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 functionsof the multipliers.
Furthermore, no two distinct maps of this form areconjugate to each other over K .
15
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Arithmetic Normal Form for M2
Theorem (M., Yasufuku)
2 If λ1 = λ2 6= 1 and λ3 6= λ1 or if λ1 = λ2 = λ3 = 1, thenψ is conjugate over K to a map of the form
φk ,b(z) = kz +bz
with k = λ1+12 , and b ∈ K ∗.
Furthermore, two such maps φk ,b and φk ′,b′ areconjugate over K if and only if k = k ′; they areconjugate over K if in addition b/b′ ∈ (K ∗)2.
16
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Arithmetic Normal Form for M2
Theorem (M., Yasufuku)
3 If λ1 = λ2 = λ3 = −2, then φ is conjugate over K to
θd ,k (z) =kz2 − 2dz + dkz2 − 2kz + d
,
with k ∈ K ,d ∈ K ∗, and k2 6= 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
17
Fields of Definition / Fields of Moduli Automorphism Groups Twists
φ ∈ Hom1d
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,
1z
⟩.
3 Tetrahedral group: A4 =
⟨−z,
1z, i(
z + 1z − 1
)⟩.
4 Octahedral group: S4 =
⟨iz,
1z, i(
z + 1z − 1
)⟩.
5 Icosahedral group:
A5 =
⟨ζ5z,−1
z,
(ζ5 + ζ−1
5
)z + 1
z −(ζ5 + ζ−1
5
) ⟩ .18
Fields of Definition / Fields of Moduli Automorphism Groups Twists
φ ∈ Hom2d
1 Diagonal Abelian Groups (Cyclic Group of order n):
H =
ζan 0 00 ζb
n 00 0 1
, gcd(a,n) = 1 or gcd(b,n) = 1.
Proposition
Let r be the number of solutions to x2 ≡ 1 mod n. Thereare n + r/2− ϕ(n)/2 representations of Cn of the formζn 0 0
0 ζan 0
0 0 1
.
19
Fields of Definition / Fields of Moduli Automorphism Groups Twists
φ ∈ Hom2d
2 Subgroups of the form⟨ζp 0 00 ai bi0 ci di
⟩ ,where the lower right 2× 2 matrices come fromembedding the PGL2 automorphism groups.
20
Fields of Definition / Fields of Moduli Automorphism Groups Twists
φ ∈ Hom2d
3 Subgroups that don’t come from embedding PGL2.(Lots of them.)
⟨0 1 01 0 00 0 1
,
0 0 11 0 00 1 0,
−1 0 0
0 1 00 0 −1
,
1 0 00 −1 00 0 −1
⟩
21
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Higher Dimensions
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22
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Higher Dimensions
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23
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(K̄ )-invariant subset T = {τ1, . . . , τn} ⊂ P1(E)with n ≥ 3.
Output: the set Autφ(K̄ )
24
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(K̄ ) 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.
25
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 rationalfunction of degree d ≥ 2. Define S0 to be the set ofrational 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 reductionfor φ along with the places that divide a prime in S0. Thenredv : Autφ(K )→ Autφ(Fv ) is a well-defined injectivehomomorphism for all places v outside S.
26
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Realizing Maps with a Given Automorphism Group
Given a finite subgroup Γ ∈ PGL2, Doyle & McMullen givea 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 = −GF
27
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Realizing Maps with a Given Automorphism Group
It is enough to find all (relative) invariant homogeneouspolynomials, i.e. for each γ ∈ Γ there is a character χ:
γ∗F = F (γx̄) = χ(γ)F (x̄).
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.
28
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Realizing Maps with a Given Automorphism Group
It is enough to find all (relative) invariant homogeneouspolynomials, i.e. for each γ ∈ Γ there is a character χ:
γ∗F = F (γx̄) = χ(γ)F (x̄).
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.
29
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Two useful gadgets
Molien Series: Given a finite group Γ (and character χ),outputs the power series
∞∑k=0
dim(K [x̄ ]Γ
k
)tk .
Reynolds Operator: Given a finite group Γ, (characterχ), and all homogeneous monomials of agiven degree, outputs all (relative) Γ-invariantsof that degree.
30
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Example
Γ = C4 =
⟨(i 00 1
)⟩
Molien Series:
1 + t2 + t4 + t6 + 3t8 + 3t10 + 3t12 + 3t14 + 5t16 + O(t18)
Invariants of degree ≤ 8:
xy x2y2 x3y3
x4y4 x8 y8
Some maps with Γ ⊆ Aut(φ):
φ1(z) =z4 + 16
z3 φ2(z) =z9 + 9zz8 − 1
31
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Example
Γ = C4 =
⟨(i 00 1
)⟩Molien Series:
1 + t2 + t4 + t6 + 3t8 + 3t10 + 3t12 + 3t14 + 5t16 + O(t18)
Invariants of degree ≤ 8:
xy x2y2 x3y3
x4y4 x8 y8
Some maps with Γ ⊆ Aut(φ):
φ1(z) =z4 + 16
z3 φ2(z) =z9 + 9zz8 − 1
32
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Example
Γ = C4 =
⟨(i 00 1
)⟩Molien Series:
1 + t2 + t4 + t6 + 3t8 + 3t10 + 3t12 + 3t14 + 5t16 + O(t18)
Invariants of degree ≤ 8:
xy x2y2 x3y3
x4y4 x8 y8
Some maps with Γ ⊆ Aut(φ):
φ1(z) =z4 + 16
z3 φ2(z) =z9 + 9zz8 − 1
33
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Example
Γ = C4 =
⟨(i 00 1
)⟩Molien Series:
1 + t2 + t4 + t6 + 3t8 + 3t10 + 3t12 + 3t14 + 5t16 + O(t18)
Invariants of degree ≤ 8:
xy x2y2 x3y3
x4y4 x8 y8
Some maps with Γ ⊆ Aut(φ):
φ1(z) =z4 + 16
z3 φ2(z) =z9 + 9zz8 − 1
34
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Example
Γ =
⟨(1 −11 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 + 88z7 − z6 + 72z5 − 3z4 − 40z3 − 3z2 + 24z − 1
φ2(z) = −z(3z6 − 39z4 + 73z2 − 13
)13z6 − 73z4 + 39z2 − 3
35
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Example
Γ =
⟨(1 −11 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 + 88z7 − z6 + 72z5 − 3z4 − 40z3 − 3z2 + 24z − 1
φ2(z) = −z(3z6 − 39z4 + 73z2 − 13
)13z6 − 73z4 + 39z2 − 3
36
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Exact Automorphism Groups?
Proposition (Hutz, M.)Let
A4 =
⟨−z,
1z, i(
z + 1z − 1
)⟩and S4 =
⟨iz,
1z, i(
z + 1z − 1
)⟩.
If φ ∈ Q(z) satisfies A4 ⊆ Aut(φ), then in fact Aut(φ) = S4.
QuestionHow to construct maps φ ∈ K (z) with Γ = Aut(φ) (ordecide there are none)?
How to construct maps φ ∈ K (z) with a subgroup ofAut(φ) conjugate to (or equal to) Γ?
37
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Exact Automorphism Groups?
Proposition (Hutz, M.)Let
A4 =
⟨−z,
1z, i(
z + 1z − 1
)⟩and S4 =
⟨iz,
1z, i(
z + 1z − 1
)⟩.
If φ ∈ Q(z) satisfies A4 ⊆ Aut(φ), then in fact Aut(φ) = S4.
QuestionHow to construct maps φ ∈ K (z) with Γ = Aut(φ) (ordecide there are none)?
How to construct maps φ ∈ K (z) with a subgroup ofAut(φ) conjugate to (or equal to) Γ?
38
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Automorphisms and Twists
Twist(φ/K ) =
K -equivalence classesof 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).
39
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Uniform Bounds on Preperiodic Points for Twists
Proposition (Levy, M., Thompson)
Let φ ∈ HomNd (K ). There is a uniform bound Bφ such that
for all ψ ∈ Twist(φ/K ),
# PrePer(ψ,PNK ) ≤ Bφ.
Idea: The degree of the field of definition of the twistingmap f is bounded by # Aut(φ). Apply Northcott.
40
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Cohomology and Twists
For an object X , twists give automorphisms:
gσ : Xσ(i−1)−−−−→ Y i−→ X .
A twist gives a one-cocyle:
g : Gal(K̄/K )→ Aut(X )
σ 7→ 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(K̄/K ),Aut(φ)
):
ξ becomes trivial in H1(Gal(K̄/K
),PGLN+1
}.
41
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Cohomology and Twists
For an object X , twists give automorphisms:
gσ : Xσ(i−1)−−−−→ Y i−→ X .
A twist gives a one-cocyle:
g : Gal(K̄/K )→ Aut(X )
σ 7→ 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(K̄/K ),Aut(φ)
):
ξ becomes trivial in H1(Gal(K̄/K
),PGLN+1
}.
42
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Cohomology and Twists
For an object X , twists give automorphisms:
gσ : Xσ(i−1)−−−−→ Y i−→ X .
A twist gives a one-cocyle:
g : Gal(K̄/K )→ Aut(X )
σ 7→ 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(K̄/K ),Aut(φ)
):
ξ becomes trivial in H1(Gal(K̄/K
),PGLN+1
}.
43
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Describing Twists
QuestionGiven φ ∈ Ratd , can we write an explicit formula for alltwists 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 7→
φ(
z n√
b)
n√
b
.More general presentation of Cn? Otherautomorphism groups? Higher dimensions?
44
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Describing Twists
QuestionGiven φ ∈ Ratd , can we write an explicit formula for alltwists 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 7→
φ(
z n√
b)
n√
b
.More general presentation of Cn? Otherautomorphism groups? Higher dimensions?
45
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Describing Twists
QuestionGiven φ ∈ Ratd , can we write an explicit formula for alltwists 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 7→
φ(
z n√
b)
n√
b
.
More general presentation of Cn? Otherautomorphism groups? Higher dimensions?
46
Fields of Definition / Fields of Moduli Automorphism Groups Twists
Describing Twists
QuestionGiven φ ∈ Ratd , can we write an explicit formula for alltwists 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 7→
φ(
z n√
b)
n√
b
.More general presentation of Cn? Otherautomorphism groups? Higher dimensions?
47