Pseudorandomness of a Markoff Automorphism over F p Alois Cerbu - PowerPoint PPT Presentation

Pseudorandomness of a Markoff Automorphism over F p Alois Cerbu Elijah Gunther Luke Peilen Yale University 4 August, 2016 1 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

The Markoff Equation 2 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

The Markoff Equation Classical Markoff Equation: x 2 + y 2 + z 2 = 3 xyz 2 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

The Markoff Equation Classical Markoff Equation: x 2 + y 2 + z 2 = 3 xyz Variant: x 2 + y 2 + z 2 = xyz 2 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

Solutions over a finite field 3 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

Solutions over a finite field Definition Define the variety over F p V ( F p ) = { ( x, y, z ) ∈ ( F p ) 3 | x 2 + y 2 + z 2 = xyz } . 3 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

Solutions over a finite field Definition Define the variety over F p V ( F p ) = { ( x, y, z ) ∈ ( F p ) 3 | x 2 + y 2 + z 2 = xyz } . Remark The size of this set is | V ( F p ) | ≃ p 2 . 3 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

Solutions over a finite field Definition Define the variety over F p V ( F p ) = { ( x, y, z ) ∈ ( F p ) 3 | x 2 + y 2 + z 2 = xyz } . Remark The size of this set is | V ( F p ) | ≃ p 2 . Note: We discard the “trivial” solution (0 , 0 , 0) . 3 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

Solutions over a finite field Definition Define the variety over F p V ( F p ) = { ( x, y, z ) ∈ ( F p ) 3 | x 2 + y 2 + z 2 = xyz } . Remark The size of this set is | V ( F p ) | ≃ p 2 . Note: We discard the “trivial” solution (0 , 0 , 0) . We will examine polynomial automorphisms of V ( F p ) . 3 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

Vieta Involutions 4 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

Vieta Involutions ( x − α )( x − β ) = x 2 − ( α + β ) x + αβ = 0 4 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

Vieta Involutions ( x − α )( x − β ) = x 2 − ( α + β ) x + αβ = 0 x 2 − ( yz ) x + ( y 2 + z 2 ) = 0 4 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

Vieta Involutions ( x − α )( x − β ) = x 2 − ( α + β ) x + αβ = 0 x 2 − ( yz ) x + ( y 2 + z 2 ) = 0     x yz − x  �−  , m 1 : y → y   z z 4 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

Vieta Involutions ( x − α )( x − β ) = x 2 − ( α + β ) x + αβ = 0 x 2 − ( yz ) x + ( y 2 + z 2 ) = 0     x yz − x  �−  , m 1 : y → y   z z         x x x x  �−  ,  �− m 2 : y → xz − y m 3 : y → y      z z z xy − z 4 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

Even Sign Changes & S 3 5 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

Even Sign Changes & S 3 x 2 + y 2 + z 2 = xyz     x x  �− n 1 : y → − y    z − z 5 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

Even Sign Changes & S 3 x 2 + y 2 + z 2 = xyz     x x  �− n 1 : y → − y    z − z     x − x  �− n 2 : y → y    z − z  x   − x   �− n 3 : y → − y    z z 5 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

Even Sign Changes & S 3 x 2 + y 2 + z 2 = xyz     x x x 1 := x, x 2 := y, x 3 := z ;  �− n 1 : y → − y    z − z σ ∈ S 3 acts by x i �→ x σ ( i ) .     x − x  �− n 2 : y → y    z − z  x   − x   �− n 3 : y → − y    z z 5 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

Even Sign Changes & S 3 x 2 + y 2 + z 2 = xyz     x x x 1 := x, x 2 := y, x 3 := z ;  �− n 1 : y → − y    z − z σ ∈ S 3 acts by x i �→ x σ ( i ) .     x − x Example  �− n 2 : y → y     x 1   x 2  z − z  �− (1 3 2) : x 2 → x 3    x 3 x 1  x   − x   �− n 3 : y → − y    z z 5 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

The Automorphism Group, Γ 6 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

The Automorphism Group, Γ Theorem (Horowitz, 1975) Vieta involutions, even sign changes, and permutations of the coordinates generate the full group Γ of polynomial automorphisms of the variety. 6 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

The Automorphism Group, Γ Theorem (Horowitz, 1975) Vieta involutions, even sign changes, and permutations of the coordinates generate the full group Γ of polynomial automorphisms of the variety. Conjecture (McCullough, Wanderley, 2013) Strong Approximation : The action of Γ on V ( F p ) \ { (0 , 0 , 0) } is transitive for all primes. 6 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

The Automorphism Group, Γ Theorem (Horowitz, 1975) Vieta involutions, even sign changes, and permutations of the coordinates generate the full group Γ of polynomial automorphisms of the variety. Conjecture (McCullough, Wanderley, 2013) Strong Approximation : The action of Γ on V ( F p ) \ { (0 , 0 , 0) } is transitive for all primes. Theorem (Bourgain, Gamburd, Sarnak, 2016) The action of Γ on V ( F p ) \ { (0 , 0 , 0) } is transitive for almost all primes (all but a small and slowly-growing exceptional set). 6 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

Reduced Variety Remark N = � n 1 , n 2 , n 3 � � Γ 7 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

Reduced Variety Remark N = � n 1 , n 2 , n 3 � � Γ Remark Γ acts on W ( F p ) , the set of N -blocks. 7 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

Reduced Variety Remark N = � n 1 , n 2 , n 3 � � Γ Remark Γ acts on W ( F p ) , the set of N -blocks. Definition We denote by H ( p ) the permutation representation of this action. 7 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

Reduced Variety Remark N = � n 1 , n 2 , n 3 � � Γ Remark Γ acts on W ( F p ) , the set of N -blocks. Definition We denote by H ( p ) the permutation representation of this action. The remainder of the talk concerns this series { H ( p ) } of finite permutation groups. 7 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

Permutation Group, H ( p ) 8 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

Permutation Group, H ( p ) Let | W ( F p ) | = n . Lemma (CGP 2016) H ( p ) ≤ A n if and only p ≡ 3 (mod 16) . 8 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

Permutation Group, H ( p ) Let | W ( F p ) | = n . Lemma (CGP 2016) H ( p ) ≤ A n if and only p ≡ 3 (mod 16) . Conjecture (CGP 2016) H ( p ) ∼ = S n for p �≡ 3 (mod 16) H ( p ) ∼ = A n for p ≡ 3 (mod 16) We have checked this for primes up to 31. 8 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

Pseudorandom Behavior 9 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

Pseudorandom Behavior Question: Does a fixed automorphism behave pseudorandomly, modulo p ? 9 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

Pseudorandom Behavior Question: Does a fixed automorphism behave pseudorandomly, modulo p ? γ ∈ Γ γ 11 ∈ H (11) . . . γ p ∈ H ( p ) . . . γ 5 ∈ H (5) γ 7 ∈ H (7) (Recall H ( p ) is the permutation group generated by the action of Γ on W ( F p ) ) 9 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p

Nonexamples 10 / 17 Alois Cerbu, Elijah Gunther, Luke Peilen Pseudorandomness of a Markoff Automorphism over F p