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

Similar documents
Automorphism Groups and Invariant Theory on PN

Moduli Spaces for Dynamical Systems Joseph H. Silverman

FIELD THEORY. Contents

Galois theory of quadratic rational functions with a non-trivial automorphism 1

HONDA-TATE THEOREM FOR ELLIPTIC CURVES

Math 201C Homework. Edward Burkard. g 1 (u) v + f 2(u) g 2 (u) v2 + + f n(u) a 2,k u k v a 1,k u k v + k=0. k=0 d

Math 602, Fall 2002, HW 2, due 10/14/2002

Real representations

Abstract Algebra II Groups ( )

Math 3140 Fall 2012 Assignment #3

NUNO FREITAS AND ALAIN KRAUS

Two subgroups and semi-direct products

5 Group theory. 5.1 Binary operations

On Galois cohomology and realizability of 2-groups as Galois groups II. Author copy

The Kummer Pairing. Alexander J. Barrios Purdue University. 12 September 2013

Finite Subgroups of Gl 2 (C) and Universal Deformation Rings

Public-key Cryptography: Theory and Practice

On the modular curve X 0 (23)

Math 121 Homework 3 Solutions

4 Powers of an Element; Cyclic Groups

GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory.

Field Theory Qual Review

Lecture 7 Cyclic groups and subgroups

Cover Page. The handle holds various files of this Leiden University dissertation

The Berkovich Ramification Locus: Structure and Applications

Lecture 6: Etale Fundamental Group

CSIR - Algebra Problems

THE MODULI SPACE OF RATIONAL FUNCTIONS OF DEGREE d

NOTES ON FINITE FIELDS

REPRESENTATION THEORY. WEEK 4

Part II Galois Theory

is an isomorphism, and V = U W. Proof. Let u 1,..., u m be a basis of U, and add linearly independent

Chapter 4. Characters and Gauss sums. 4.1 Characters on finite abelian groups

Definitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations

A Number Theorist s Perspective on Dynamical Systems

MA 162B LECTURE NOTES: THURSDAY, FEBRUARY 26

Riemann surfaces with extra automorphisms and endomorphism rings of their Jacobians

Cover Page. The handle holds various files of this Leiden University dissertation.

1 The Galois Group of a Quadratic

GALOIS GROUPS AS PERMUTATION GROUPS

Algebraic subgroups of GL 2 (C)

Spectra of Semidirect Products of Cyclic Groups

MA 162B LECTURE NOTES: FRIDAY, JANUARY 30

A method for construction of Lie group invariants

Math 429/581 (Advanced) Group Theory. Summary of Definitions, Examples, and Theorems by Stefan Gille

Introduction to Arithmetic Geometry Fall 2013 Lecture #24 12/03/2013

Twists of elliptic curves of rank at least four

ORAL QUALIFYING EXAM QUESTIONS. 1. Algebra

ATTRACTING CYCLES IN p-adic DYNAMICS AND HEIGHT BOUNDS FOR POST-CRITICALLY FINITE MAPS. 1. Introduction

Cover Page. The handle holds various files of this Leiden University dissertation.

Mappings of elliptic curves

Galois theory (Part II)( ) Example Sheet 1

Structure of elliptic curves and addition laws

The Galois group of a polynomial f(x) K[x] is the Galois group of E over K where E is a splitting field for f(x) over K.

Regular embeddings of graphs into surfaces

DIHEDRAL GROUPS II KEITH CONRAD

Mathematics for Cryptography

A CRITERION FOR POTENTIALLY GOOD REDUCTION IN NON-ARCHIMEDEAN DYNAMICS

Geometry and Arithmetic of Dominant Rational Self-Maps of Projective Space Joseph H. Silverman

A Little Beyond: Linear Algebra

ALGEBRA HW 9 CLAY SHONKWILER

Representations. 1 Basic definitions

A GRAPHICAL REPRESENTATION OF RINGS VIA AUTOMORPHISM GROUPS

Chapter 5. Modular arithmetic. 5.1 The modular ring

MAT 535 Problem Set 5 Solutions

A Survey of Quandles with some recent developments

Solutions of Assignment 10 Basic Algebra I

Demushkin s Theorem in Codimension One

On the structure of Picard-Vessiot extensions - Joint work with Arne Ledet - Kolchin Seminar in Differential Algebra May 06, 2006

Cover Page. The handle holds various files of this Leiden University dissertation.

SMALL DYNAMICAL HEIGHTS FOR QUADRATIC POLYNOMIALS AND RATIONAL FUNCTIONS

FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS

1 Notations and Statement of the Main Results

Lecture 1. Toric Varieties: Basics

Contradiction. Theorem 1.9. (Artin) Let G be a finite group of automorphisms of E and F = E G the fixed field of G. Then [E : F ] G.

ON THE ORDERS OF AUTOMORPHISM GROUPS OF FINITE GROUPS

A geometric solution of the Kervaire Invariant One problem

Selected exercises from Abstract Algebra by Dummit and Foote (3rd edition).

Galois Theory TCU Graduate Student Seminar George Gilbert October 2015

The Ring of Monomial Representations

24 Artin reciprocity in the unramified case

Math 121 Homework 5: Notes on Selected Problems

Residual modular Galois representations: images and applications

ABSTRACT ALGEBRA 2 SOLUTIONS TO THE PRACTICE EXAM AND HOMEWORK

From K3 Surfaces to Noncongruence Modular Forms. Explicit Methods for Modularity of K3 Surfaces and Other Higher Weight Motives ICERM October 19, 2015

z, w = z 1 w 1 + z 2 w 2 z, w 2 z 2 w 2. d([z], [w]) = 2 φ : P(C 2 ) \ [1 : 0] C ; [z 1 : z 2 ] z 1 z 2 ψ : P(C 2 ) \ [0 : 1] C ; [z 1 : z 2 ] z 2 z 1

= 1 2x. x 2 a ) 0 (mod p n ), (x 2 + 2a + a2. x a ) 2

AN INTRINSICAL DESCRIPTION OF GROUP CODES

Representations and Linear Actions

Hamburger Beiträge zur Mathematik

Math 752 Week s 1 1

Algebraic Structures Exam File Fall 2013 Exam #1

ON GALOIS GROUPS OF ABELIAN EXTENSIONS OVER MAXIMAL CYCLOTOMIC FIELDS. Mamoru Asada. Introduction

Difference sets and Hadamard matrices

Imaginary Quadratic Fields With Isomorphic Abelian Galois Groups

9. Finite fields. 1. Uniqueness

Rational Homotopy Theory Seminar Week 11: Obstruction theory for rational homotopy equivalences J.D. Quigley

COMPLEX MULTIPLICATION: LECTURE 15

MAT534 Fall 2013 Practice Midterm I The actual midterm will consist of five problems.

(Non)-Uniform Bounds for Rational Preperiodic Points in Arithmetic Dynamics

Transcription:

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

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.

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}. The field of moduli of φ is the fixed field K G φ.

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 fσ.

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 fσ. The field of moduli for φ is contained in every field of definition.

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 fσ. The field of moduli for φ is contained in every field of definition. Equality???

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.

FOD = FOM criterion Idea: If [φ] Md N (K ), then you get a cohomology class f : Gal( K /K ) PGL N+1 σ f σ

FOD = FOM criterion Idea: If [φ] Md N (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 cφ K is FOD for φ c φ trivial X cφ /K When gcd(d, N + 1) = 1, we can find a K -rational zero-cycle on X cφ.

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.

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.

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 φ σ = φ f for f = 1 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) ).

Normal Form for M 2 Lemma (Milnor) Let φ Rat 2 have multipliers λ 1, λ 2, λ 3. 1 If not all three multipliers are 1, φ is conjugate to a map of the form: z 2 + λ 1 z λ 2 z + 1. 2 If all three multipliers are 1, φ is conjugate to: z + 1 z.

Normal Form for M 2 Lemma (Milnor) Let φ Rat 2 have multipliers λ 1, λ 2, λ 3. 1 If not all three multipliers are 1, φ is conjugate to a map of the form: z 2 + λ 1 z λ 2 z + 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.

Arithmetic Normal Form for M 2 Theorem (M., Yasufuku) Let φ Rat 2 (K ) have multipliers λ 1, λ 2, λ 3. 1 If the multipliers are distinct or if exactly two multipliers are 1, then φ(z) is conjugate over K to 2z 2 + (2 σ 1 )z + (2 σ 1 ) z 2 + (2 + σ 1 )z + 2 σ 1 σ 2 K (z), 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.

Arithmetic Normal Form for M 2 Theorem (M., Yasufuku) 2 If λ 1 = λ 2 1 and λ 3 λ 1 or if λ 1 = λ 2 = λ 3 = 1, then 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.

Arithmetic Normal Form for M 2 Theorem (M., Yasufuku) 3 If λ 1 = λ 2 = λ 3 = 2, then φ is conjugate over K to θ d,k (z) = kz2 2dz + dk z 2 2kz + 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

φ Hom 1 d Aut(φ) is conjugate to one of the following: 1 Cyclic group of order n: C n = ζ n z. 2 Dihedral group of order 2n: D n = 3 Tetrahedral group: A 4 = 4 Octahedral group: S 4 = 5 Icosahedral group: A 5 = ζ n z, 1 z z, 1 ( z + 1 z, i z 1 iz, 1 z, i ( z + 1 z 1 ζ 5 z, 1 ( z, ζ5 + ζ 1 5 z ( ζ 5 + ζ 1 5. ). ). ) z + 1 ).

φ Hom 2 d 1 Diagonal Abelian Groups (Cyclic Group of order n): H = ζa n 0 0 0 ζn b 0, gcd(a, n) = 1 or gcd(b, n) = 1. 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 0 ζn a 0. 0 0 1

φ Hom 2 d 2 Subgroups of the form 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.

φ Hom 2 d 3 Subgroups that don t come from embedding PGL 2. (Lots of them.) 0 1 0 1 0 0, 0 0 1 0 0 1 1 0 0 0 1 0, 1 0 0 0 1 0, 1 0 0 0 1 0 0 0 1 0 0 1

Higher Dimensions

Higher Dimensions This slide intentionally left blank.

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 )

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.

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 { S 0 = {2} p odd : p 1 2 } [K : Q] and p d(d 2 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 S 0. Then red v : Aut φ (K ) Aut φ (F v ) is a well-defined injective homomorphism for all places v outside S.

Realizing Maps with a Given Automorphism Group Given a finite subgroup Γ PGL 2, Doyle & McMullen give a way to construct all rational maps φ Rat d with Γ Aut(φ). 2 d n inv. hom. one-form 1 1 inv. hom. rational map Fdx + Gdy 1 1 φ = G F

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).

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.

Two useful gadgets Molien Series: Given a finite group Γ (and character χ), outputs the power series dim ( K [ x] k) Γ t k. k=0 Reynolds Operator: Given a finite group Γ, (character χ), and all homogeneous monomials of a given degree, outputs all (relative) Γ-invariants of that degree.

Example Γ = C 4 = ( ) i 0 0 1

Example Molien Series: Γ = C 4 = ( ) i 0 0 1 1 + t 2 + t 4 + t 6 + 3t 8 + 3t 10 + 3t 12 + 3t 14 + 5t 16 + O(t 18 )

Example Molien Series: Γ = C 4 = ( ) i 0 0 1 1 + t 2 + t 4 + t 6 + 3t 8 + 3t 10 + 3t 12 + 3t 14 + 5t 16 + O(t 18 ) Invariants of degree 8: xy x 2 y 2 x 3 y 3 x 4 y 4 x 8 y 8

Example Molien Series: Γ = C 4 = ( ) i 0 0 1 1 + t 2 + t 4 + t 6 + 3t 8 + 3t 10 + 3t 12 + 3t 14 + 5t 16 + O(t 18 ) Invariants of degree 8: xy x 2 y 2 x 3 y 3 x 4 y 4 x 8 y 8 Some maps with Γ Aut(φ): φ 1 (z) = z4 + 16 z 3 φ 2 (z) = z9 + 9z z 8 1

Example Γ = ( ) 1 1 1 1 (also cyclic of order 4).

Example Γ = ( ) 1 1 1 1 Invariants of degree 8: (also cyclic of order 4). x 2 + y 2 x 8 + 12x 6 y 2 20x 4 y 4 + 12x 2 y 6 + y 8 x 4 + 2x 2 y 2 + y 4 x 8 4x 6 y 2 + 22x 4 y 4 4x 2 y 6 + y 8 x 6 + 3x 4 y 2 + 3x 2 y 4 + y 6 x 7 y 7x 5 y 3 + 7x 3 y 5 xy 7 Some maps with Γ Aut(φ): φ 1 (z) = z7 + 24z 6 + 3z 5 40z 4 + 3z 3 + 72z 2 + z + 8 8z 7 z 6 + 72z 5 3z 4 40z 3 3z 2 + 24z 1 φ 2 (z) = z ( 3z 6 39z 4 + 73z 2 13 ) 13z 6 73z 4 + 39z 2 3

Exact Automorphism Groups? Proposition (Hutz, M.) Let A 4 = z, 1 ( ) z + 1 z, i z 1 and S 4 = iz, 1 ( ) z + 1 z, i. z 1 If φ Q(z) satisfies A 4 Aut(φ), then in fact Aut(φ) = S 4.

Exact Automorphism Groups? Proposition (Hutz, M.) Let A 4 = z, 1 ( ) z + 1 z, i z 1 and S 4 = iz, 1 ( ) z + 1 z, i. z 1 If φ Q(z) satisfies A 4 Aut(φ), then in fact Aut(φ) = S 4. 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?

Automorphisms and Twists K -equivalence classes Twist(φ/K ) = of maps ψ Hom N d (K ). such that ψ is K -equivalent to φ Twists give automorphisms of the map φ: f φf 1 = (f φf 1 ) σ = f σ φ(f 1 ) σ. φ = f 1 f σ φ(f σ ) 1 f f 1 f σ Aut(φ).

Uniform Bounds on Preperiodic Points for Twists Proposition (Levy, M., Thompson) Let φ Hom N d (K ). There is a uniform bound B φ such that for all ψ Twist(φ/K ), # PrePer(ψ, P N K ) B φ. Idea: The degree of the field of definition of the twisting map f is bounded by # Aut(φ). Apply Northcott.

Cohomology and Twists For an object X, twists give automorphisms: g σ : X σ(i 1 ) Y A twist gives a one-cocyle: i X. g : Gal( K /K ) Aut(X) σ i σ(i 1 )

Cohomology and Twists For an object X, twists give automorphisms: g σ : X σ(i 1 ) Y A twist gives a one-cocyle: i X. g : Gal( K /K ) Aut(X) σ i σ(i 1 ) Does every one-cocyle come from a twist?

Cohomology and Twists For an object X, twists give automorphisms: g σ : X σ(i 1 ) Y A twist gives a one-cocyle: i X. 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. { ξ H ( Twist(φ/K ) = 1 Gal( K /K ), Aut(φ) ) } : ξ becomes trivial in H ( 1 Gal( K /K )., PGL N+1

Describing Twists Question Given φ Rat d, can we write an explicit formula for all twists of φ?

Describing Twists Question Given φ Rat d, can we write an explicit formula for all twists of φ? Done for Rat 2 by Arithmetic Normal Form Theorem.

Describing Twists Question Given φ Rat d, can we write an explicit formula for all twists of φ? Done for Rat 2 by Arithmetic Normal Form Theorem. If Aut(φ) = {ζ n z}, then we have an isomorphism K /K n Twist(φ/K ) ( b φ z n ) b n. b

Describing Twists Question Given φ Rat d, can we write an explicit formula for all twists of φ? Done for Rat 2 by Arithmetic Normal Form Theorem. If Aut(φ) = {ζ n z}, then we have an isomorphism K /K n Twist(φ/K ) ( b φ z n ) b n. b More general presentation of C n? Other automorphism groups? Higher dimensions?

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback