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 1

Fields of Definition / Fields of Moduli Automorphism Groups Twists Definitions Definition Let φ ∈ Rat N d . A field K ′ / K is a field of definition for φ if φ f ∈ Rat N d ( K ′ ) for some f ∈ PGL N + 1 . 2

Fields of Definition / Fields of Moduli Automorphism Groups Twists Definitions Definition Let φ ∈ Rat N d . A field K ′ / K is a field of definition for φ if φ f ∈ Rat N d ( K ′ ) for some f ∈ PGL N + 1 . Definition Let φ ∈ Rat N d , and define G φ = { σ ∈ G K | φ σ is K equivalent to φ } . G φ . The field of moduli of φ is the fixed field K 3

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 f σ ∈ PGL N + 1 such that φ σ = φ f σ . 4

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 f σ ∈ PGL N + 1 such that φ σ = φ f σ . The field of moduli for φ is contained in every field of definition. 5

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 f σ ∈ PGL N + 1 such that φ σ = φ f σ . The field of moduli for φ is contained in every field of definition. Equality??? 6

Fields of Definition / Fields of Moduli Automorphism Groups Twists FOD = FOM criterion Proposition (Hutz, M.) Let ξ ∈ M N d ( K ) be a dynamical system with Aut φ = { id } , and let D = � N j = 0 d N . 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 ] ∈ M N d ( K ) , then you get a cohomology class f : Gal ( ¯ K / K ) → PGL N + 1 σ �→ f σ 8

Fields of Definition / Fields of Moduli Automorphism Groups Twists FOD = FOM criterion Idea: If [ φ ] ∈ M N d ( K ) , then you get a cohomology class f : Gal ( ¯ K / K ) → PGL N + 1 σ �→ f σ Twists of P N are in 1-1 correspondence with cocyles: i : P N → X σ �→ i − 1 i σ [ φ ] ∈ M N d ( K ) ❀ cocycle c φ ❀ X c φ K is FOD for φ ⇐ ⇒ c φ trivial ⇐ ⇒ X c φ / K When gcd ( D , N + 1 ) = 1, we can find a K -rational zero-cycle on X c φ . 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. 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 P 1 doesn’t require Aut ( φ ) = id. Proof requires knowledge of the possible automorphism groups and “cohomology lifting.” 11

Fields of Definition / Fields of Moduli Automorphism Groups Twists Example (Silverman) � z − 1 � 3 . So Q ( i ) is a field of definition for φ . φ ( z ) = i z + 1 Let σ represent complex conjugation, then f = − 1 φ σ = φ f for z . Hence, Q is the field of moduli for φ . K is a field of definition for φ iff − 1 ∈ N K ( i ) / K ( K ( i ) ∗ ) . 12

Fields of Definition / Fields of Moduli Automorphism Groups Twists Normal Form for M 2 Lemma (Milnor) Let φ ∈ Rat 2 have multipliers λ 1 , λ 2 , λ 3 . If not all three multipliers are 1 , φ is conjugate to a 1 map of the form: z 2 + λ 1 z λ 2 z + 1 . If all three multipliers are 1 , φ is conjugate to: 2 z + 1 z . 13

Fields of Definition / Fields of Moduli Automorphism Groups Twists Normal Form for M 2 Lemma (Milnor) Let φ ∈ Rat 2 have multipliers λ 1 , λ 2 , λ 3 . If not all three multipliers are 1 , φ is conjugate to a 1 map of the form: z 2 + λ 1 z λ 2 z + 1 . If all three multipliers are 1 , φ is conjugate to: 2 z + 1 z . 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 M 2 Theorem (M., Yasufuku) Let φ ∈ Rat 2 ( K ) have multipliers λ 1 , λ 2 , λ 3 . If the multipliers are distinct or if exactly two 1 multipliers are 1 , then φ ( z ) is conjugate over K to 2 z 2 + ( 2 − σ 1 ) z + ( 2 − σ 1 ) ∈ K ( z ) , − z 2 + ( 2 + σ 1 ) z + 2 − σ 1 − σ 2 where σ 1 and σ 2 are the first two symmetric functions of the multipliers. Furthermore, no two distinct maps of this form are conjugate to each other over K. 15

Fields of Definition / Fields of Moduli Automorphism Groups Twists Arithmetic Normal Form for M 2 Theorem (M., Yasufuku) If λ 1 = λ 2 � = 1 and λ 3 � = λ 1 or if λ 1 = λ 2 = λ 3 = 1 , then 2 ψ is conjugate over K to a map of the form φ k , b ( z ) = kz + b z with k = λ 1 + 1 2 , and b ∈ K ∗ . Furthermore, two such maps φ k , b and φ k ′ , b ′ are conjugate over K if and only if k = k ′ ; they are conjugate over K if in addition b / b ′ ∈ ( K ∗ ) 2 . 16

Fields of Definition / Fields of Moduli Automorphism Groups Twists Arithmetic Normal Form for M 2 Theorem (M., Yasufuku) If λ 1 = λ 2 = λ 3 = − 2 , then φ is conjugate over K to 3 θ d , k ( z ) = kz 2 − 2 dz + dk z 2 − 2 kz + d , with k ∈ K , d ∈ K ∗ , and k 2 � = d . All such maps are conjugate over K. Furthermore, θ d , k ( z ) and θ d ′ , 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 ∈ Hom 1 d Aut ( φ ) is conjugate to one of the following: Cyclic group of order n : C n = � ζ n z � . 1 � ζ n z , 1 � Dihedral group of order 2 n : D n = . 2 z � − z , 1 � z + 1 �� Tetrahedral group: A 4 = z , i . 3 z − 1 � � z + 1 �� iz , 1 Octahedral group: S 4 = z , i . 4 z − 1 Icosahedral group: 5 � � � ζ 5 + ζ − 1 � z + 1 ζ 5 z , − 1 5 A 5 = z , . ζ 5 + ζ − 1 � � z − 5 18

Fields of Definition / Fields of Moduli Automorphism Groups Twists φ ∈ Hom 2 d Diagonal Abelian Groups (Cyclic Group of order n ): 1  ζ a  0 0 n  , ζ b H = 0 0 gcd ( a , n ) = 1 or gcd ( b , n ) = 1 .  n 0 0 1 Proposition Let r be the number of solutions to x 2 ≡ 1 mod n . There are n + r / 2 − ϕ ( n ) / 2 representations of C n of the form   ζ n 0 0  . ζ a 0 0  n 0 0 1 19

Fields of Definition / Fields of Moduli Automorphism Groups Twists φ ∈ Hom 2 d Subgroups of the form 2 �  ζ p 0 0 � 0 a i b i ,   0 c i d i where the lower right 2 × 2 matrices come from embedding the PGL 2 automorphism groups. 20

Fields of Definition / Fields of Moduli Automorphism Groups Twists φ ∈ Hom 2 d Subgroups that don’t come from embedding PGL 2 . 3 (Lots of them.) �    0 1 0 0 0 1  , 1 0 0 1 0 0    0 0 1 0 1 0 ,     − 1 0 0 1 0 0 �  , 0 1 0 0 − 1 0    0 0 − 1 0 0 − 1 21

Fields of Definition / Fields of Moduli Automorphism Groups Twists Higher Dimensions 22

Fields of Definition / Fields of Moduli Automorphism Groups Twists Higher Dimensions This slide intentionally left blank. 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 Aut φ ( ¯ K ) -invariant subset T = { τ 1 , . . . , τ n } ⊂ P 1 ( E ) with n ≥ 3. Output: the set Aut φ ( ¯ K ) 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 ∈ PGL 2 ( ¯ K ) by solving the linear system s ( τ 1 ) = τ i , s ( τ 2 ) = τ j , s ( τ 3 ) = τ k . 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 rational function of degree d ≥ 2 . Define S 0 to be the set of rational primes given by � � p odd : p − 1 � � [ K : Q ] and p | d ( d 2 − 1 ) S 0 = { 2 } ∪ , � 2 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 S 0 . Then red v : Aut φ ( K ) → Aut φ ( F v ) is a well-defined injective homomorphism for all places v outside S. 26