Towers of function fields over finite fields and their sequences of - PowerPoint PPT Presentation

Towers of function fields over finite fields and their sequences of zeta functions Alexey Zaytsev I. Kant Baltic Federal University Kalinigrad Russia November 12, 2013 joint work with Alexey Zykin Alexey Zaytsev Towers of function fields over finite fields and their sequences of zeta

Towers Definition A tower of function fields over F q is an infinite sequence F = ( F 1 , F 2 , . . . ) of function fields F i / F q with properties F 1 ⊂ F 2 ⊂ F 3 ⊂ . . . , [ F i : F i − 1 ] > 1 for i > 1 , the genus g ( F j ) > 0 for some j . Alexey Zaytsev Towers of function fields over finite fields and their sequences of zeta

Towers Definition A tower of function fields over F q is an infinite sequence F = ( F 1 , F 2 , . . . ) of function fields F i / F q with properties F 1 ⊂ F 2 ⊂ F 3 ⊂ . . . , [ F i : F i − 1 ] > 1 for i > 1 , the genus g ( F j ) > 0 for some j . remark 1 g ( F i ) → ∞ as i → ∞ , 2 lim N ( F n ) g ( F n ) exits and called λ ( F ) . Alexey Zaytsev Towers of function fields over finite fields and their sequences of zeta

Definition Let F = ( F n ) n ≥ 1 be a tower of function fields over F q . Then F is asymptotically good, if λ ( F ) > 0 , F is asymptotically bad, if λ ( F ) = 0 , F is optimal, if λ ( F ) = A ( q ) . Alexey Zaytsev Towers of function fields over finite fields and their sequences of zeta

Garcia-Stichtenoth optimal tower Let T 1 be a rational function field F 4 ( x 1 ). Then we define the function field T n as following x 3 n − 1 x 2 T n = T n − 1 ( x n ) , where n + x n = . x 2 n − 1 + x n − 1 F ( X , Y ) = ( Y 2 + Y )( X + 1) + X 2 . it is optimal, in other words √ N 1 ( T n ) lim g ( T n ) = 4 − 1 = 1 , n →∞ genus of function field T n is (2 n / 2 − 1) 2 � if i even , g ( T n ) = (2 ( n +1) / 2 − 1)(2 ( n − 1) / 2 − 1) if i odd , Alexey Zaytsev Towers of function fields over finite fields and their sequences of zeta

Geer-Vlugt tower Let F = ( F n ) n ≥ 1 be a tower of function field over F 8 where F 1 = F 8 ( x 1 ) and x 2 F n = F n − 1 ( x n ) , where n + x n = x n − 1 + 1 + 1 / x n − 1 . So the tower F is a recursive tower given by an irreducible polynomial F ( X , Y ) = ( Y 2 + Y ) X − X 2 − X − 1 ∈ F 8 [ X , Y ] . Alexey Zaytsev Towers of function fields over finite fields and their sequences of zeta

The following proposition describes the behavior of the tower and its ramification locus. Let F be a tower over finite field F 8 defined by the polynomial F ( X , Y ). Then the following properties hold: it is a good tower with limit attaining the Ihara bound g ( F n ) = 2( p 2 − 1) N 1 ( F n ) lim = 3 / 2 , p + 2 n →∞ if Q ∈ P ( F n ) is a ramification place of an extension F n / F 1 then Q ∩ F 1 is either a pole of x 1 or a zero x 1 − a , where a ∈ {± 1 , ρ, ρ 2 } , with ρ 2 + ρ + 1 = 0 , genus of F n equals ( n + 10)2 i / 2 − 1 � for i even g ( F n ) = 2 n +2 +1 − ( n + 2[ i / 4] + 15)2 ( i − 3) / 2 for i odd Alexey Zaytsev Towers of function fields over finite fields and their sequences of zeta

tower of Kummer extensions Let K = ( K n ) n ≥ 1 be a tower of function fields over F 9 where F 1 = F 9 ( x 1 ) and x 2 n = ( x 2 K n = K n − 1 ( x n ) , where n − 1 + 1) / (2 x n − 1 ) . So the tower K is a recursive optimal tower given by an absolutely irreducible polynomial F ( X , Y ) = 2 XY 2 − ( X 2 + 1) ∈ F 9 [ X , Y ] . Alexey Zaytsev Towers of function fields over finite fields and their sequences of zeta

Goal Let T be a function field over F q then the zeta function of T is N m ( T ) L T ( x ) x m = � log Z T ( x ) = m (1 − x )(1 − qx ) m ≥ 1 where N m ( T ) is a number of F q m − rational points of T and L T ( x ) = a 0 + a 1 x + · · · + a 2 g ( T ) x 2 g ( T ) Alexey Zaytsev Towers of function fields over finite fields and their sequences of zeta

Goal Let T be a function field over F q then the zeta function of T is N m ( T ) L T ( x ) x m = � log Z T ( x ) = m (1 − x )(1 − qx ) m ≥ 1 where N m ( T ) is a number of F q m − rational points of T and L T ( x ) = a 0 + a 1 x + · · · + a 2 g ( T ) x 2 g ( T ) For each function field in a tower T = ( T n ) n ≥ 1 L T n ( x ) = a (0 , n ) + a (1 , n ) x + · · · + a (2 g ( T n ) , n ) x 2 g ( T n ) Alexey Zaytsev Towers of function fields over finite fields and their sequences of zeta

Goal Let T be a function field over F q then the zeta function of T is N m ( T ) L T ( x ) x m = � log Z T ( x ) = m (1 − x )(1 − qx ) m ≥ 1 where N m ( T ) is a number of F q m − rational points of T and L T ( x ) = a 0 + a 1 x + · · · + a 2 g ( T ) x 2 g ( T ) For each function field in a tower T = ( T n ) n ≥ 1 L T n ( x ) = a (0 , n ) + a (1 , n ) x + · · · + a (2 g ( T n ) , n ) x 2 g ( T n ) Question Can we find explicitly functions a ( i , n ) as functions in i , n for at least one given good tower? Alexey Zaytsev Towers of function fields over finite fields and their sequences of zeta

Asymptotic zeta function Let T = ( T n ) n ≥ 1 be a tower. Then one can define an asymptotic zeta function N n ( T m ) µ n = lim g ( T m ) m →∞ µ n � n x n log Z T ( x ) = n ≥ 1 Alexey Zaytsev Towers of function fields over finite fields and their sequences of zeta

Asymptotic zeta function Let T = ( T n ) n ≥ 1 be a tower. Then one can define an asymptotic zeta function N n ( T m ) µ n = lim g ( T m ) m →∞ µ n � n x n log Z T ( x ) = n ≥ 1 Garcia-Stictenoth tower 1 Z T ( t ) = (1 − t ) tower of Kummer extensions 1 Z T ( t ) = (1 − t ) 2 Alexey Zaytsev Towers of function fields over finite fields and their sequences of zeta

Asymptotic zeta function of the Geer-Vlugt tower Base on Lenstre relation Peter Beelen proved that locus of split completely places is bounded and lies in V ( G 1 F ). Then according to the Perron-Frobenius theorem it follows that number of paths of length m in the graph G i ( F ) is completely determined by a maximum eigenvalue. Therefore µ i ( F ) is a constant. Alexey Zaytsev Towers of function fields over finite fields and their sequences of zeta

Asymptotic zeta function of the Geer-Vlugt tower Base on Lenstre relation Peter Beelen proved that locus of split completely places is bounded and lies in V ( G 1 F ). Then according to the Perron-Frobenius theorem it follows that number of paths of length m in the graph G i ( F ) is completely determined by a maximum eigenvalue. Therefore µ i ( F ) is a constant. Hence Geer-Vlugt tower 1 Z F ( t ) = (1 − t ) 3 / 2 . Alexey Zaytsev Towers of function fields over finite fields and their sequences of zeta

L-polynomials of Garcia-Stichtenoth tower L T 1 1 1 + 3 T + 4 T 2 L T 2 (1 + 3 T + 4 T 2 ) 3 L T 3 (1 − T + 4 T 2 ) 2 (1 + 3 T + 4 T 2 ) 7 L T 4 (1 − T + 4 T 2 ) 4 (1 + 3 T + 4 T 2 ) 11 (1 + T + 4 T 2 ) 2 L T 5 (1 + 2 T + T 2 + 8 T 3 + 16 T 4 ) 2 (1 − T + 4 T 2 ) 4 (1 + T + 4 T 2 ) 10 (1 + 2 T + T 2 + 8 T 3 + 16 T 4 ) 6 L T 6 (1 + 3 T + 4 T 2 ) 17 (1 + T − T 2 + 3 T 3 − 4 T 4 + 16 T 5 + 64 T 6 ) 2 Alexey Zaytsev Towers of function fields over finite fields and their sequences of zeta

Galois Group and Kani-Rosen decomposition Proposition If n ≥ 3, then the extension T n over T n − 2 is Galois and Gal ( T n / T n − 2 ) ∼ = Z / 2 Z × Z / 2 Z . We will always let C n denote a curve with function field T n . The Galois covering C n → C n − 2 implies a decomposition of the Jacobian of the curve C n . If we denote Galois automorphism group by � σ, τ � then we have the following diagram of coverings Alexey Zaytsev Towers of function fields over finite fields and their sequences of zeta

� � � � Galois Group and Kani-Rosen decomposition C n � � ���������������� � � � � 2:1 2:1 � � � 2:1 � � � � � � C n − 1 ∼ = C n / � σ � C n / � στ � C n / � τ � � � � ������������ � � 2:1 2:1 � � � 2:1 � � � � � � � C n − 2 ∼ = C n / � σ, τ � and the following isogeny of Jacobians Jac ( C n ) × Jac ( C n − 2 ) 2 ∼ Jac ( C n − 1 ) × Jac ( C n / � στ � ) × Jac ( C n / � τ � ) , which implies decomposition of L − polynomials L C n ( T ) L C n − 2 ( T ) 2 = L C n − 1 ( T ) L C n / � στ � ( T ) L C n / � τ � ( T ) . Alexey Zaytsev Towers of function fields over finite fields and their sequences of zeta

Recurrence relations and the general zeta function Decomposition of Pic 0 ( T n ) and the L -polynomial of T n . Corollary The L -polynomial of the function field T n has the following factorization L T n = L 2 n − 3 × L 2 n − 6 X 2 , 1 × L 2 n − 8 Y 3 , 1 × · · · × L 2 Y n − 2 , 1 , X 1 or more precisely L T n = ( T 2 + T + 4) 2 n − 8 ( T 2 + 3 T + 4) 12 n − 49 ( T 2 − T + 4) 6 n − 26 ( T 4 + 2 T 3 + T 2 + 8 T + 16) 6 n − 24 ( T 6 + T 5 − T 4 + 3 T 3 − 4 T 2 + 16 T + 64) 2 n − 10 L 2 n − 12 · · · L 2 Y 5 , 1 Y n − 2 , 1 The order of the finite group # Pic 0 ( T n )( F 4 ) = 2 58 n − 243 3 2 n − 8 5 2 n − 10 L 2 n − 12 (1) ... L 2 Y n − 2 , 1 (1) . Y 5 , 1 Alexey Zaytsev Towers of function fields over finite fields and their sequences of zeta