On the Zeta Function of Curves over Finite Fields Nurdag ul Anbar - PowerPoint PPT Presentation

On the Zeta Function of Curves over Finite Fields Nurdag¨ ul Anbar (joint work with Henning Stichtenoth) Sabancı University RICAM, Workshop 2: Algebraic Curves over Finite Fields 11-15 November 2013

Introduction Curves with a prescribed L -polynomial up to some degree Curves with prescribed number of points L -polynomial of a curve X : a nice curve over F q of genus g . The Zeta function of X , L X ( t ) Z X ( t ) = (1 − t )(1 − qt ) , where L X ( t ) ∈ Z [ t ] of degree 2 g . L X ( t ) = a 0 + a 1 t + . . . + a 2 g t 2 g ( L -polynomial of X ) • a 0 = 1 • a 1 = N − ( q + 1), where N is the number of rational points of X • a 2 g − i = q g − i a i for i = 0 , . . . , g

Introduction Curves with a prescribed L -polynomial up to some degree Curves with prescribed number of points L -polynomial of a curve X : a nice curve over F q of genus g . The Zeta function of X , L X ( t ) Z X ( t ) = (1 − t )(1 − qt ) , where L X ( t ) ∈ Z [ t ] of degree 2 g . L X ( t ) = a 0 + a 1 t + . . . + a 2 g t 2 g ( L -polynomial of X ) • a 0 = 1 • a 1 = N − ( q + 1), where N is the number of rational points of X • a 2 g − i = q g − i a i for i = 0 , . . . , g

Introduction Curves with a prescribed L -polynomial up to some degree Curves with prescribed number of points L -polynomial of a curve X : a nice curve over F q of genus g . The Zeta function of X , L X ( t ) Z X ( t ) = (1 − t )(1 − qt ) , where L X ( t ) ∈ Z [ t ] of degree 2 g . L X ( t ) = a 0 + a 1 t + . . . + a 2 g t 2 g ( L -polynomial of X ) • a 0 = 1 • a 1 = N − ( q + 1), where N is the number of rational points of X • a 2 g − i = q g − i a i for i = 0 , . . . , g

Introduction Curves with a prescribed L -polynomial up to some degree Curves with prescribed number of points Some notation Remember: X is defined over F q F d := F q d X d : the curve X over F d N d : the number of rational points of X d S d := N d − ( q d + 1) B r : the number of degree r points of X L ( t ) = L X ( t ) = 1 + a 1 t + . . . + a 2 g t 2 g d − 1 � S d = da d − S d − j a j with S 1 = N 1 − ( q + 1) = a 1 j =1 � � r � ( q d + 1 + S d ) for all r ≥ 1 , rB r = µ d d | r for all r ≥ 1, where µ ( . ) is the M¨ obius function.

Introduction Curves with a prescribed L -polynomial up to some degree Curves with prescribed number of points Some notation Remember: X is defined over F q F d := F q d X d : the curve X over F d N d : the number of rational points of X d S d := N d − ( q d + 1) B r : the number of degree r points of X L ( t ) = L X ( t ) = 1 + a 1 t + . . . + a 2 g t 2 g d − 1 � S d = da d − S d − j a j with S 1 = N 1 − ( q + 1) = a 1 j =1 � � r � ( q d + 1 + S d ) for all r ≥ 1 , rB r = µ d d | r for all r ≥ 1, where µ ( . ) is the M¨ obius function.

Introduction Curves with a prescribed L -polynomial up to some degree Curves with prescribed number of points Some recursively defined functions over Z : σ 0 := 0 and for all r ≥ 1, r − 1 � σ r ( T 1 , . . . , T r ) := rT r − σ r − j ( T 1 , . . . , T r − j ) · T j j =1 � � � r � � r � ( q d + 1) β r ( T 1 , . . . , T r ) := µ σ d ( T 1 , . . . , T d ) + µ d d d | r d | r ϕ r ( T 1 , . . . , T r − 1 ) := rT r − β r ( T 1 , . . . , T r ) σ r ( a 1 , . . . , a r ) = S r = N r − ( q r +1) and β r ( a 1 , . . . , a r ) = rB r = ⇒ ra r = ϕ r ( a 1 , . . . , a r − 1 ) + rB r

Introduction Curves with a prescribed L -polynomial up to some degree Curves with prescribed number of points Some recursively defined functions over Z : σ 0 := 0 and for all r ≥ 1, r − 1 � σ r ( T 1 , . . . , T r ) := rT r − σ r − j ( T 1 , . . . , T r − j ) · T j j =1 � � � r � � r � ( q d + 1) β r ( T 1 , . . . , T r ) := µ σ d ( T 1 , . . . , T d ) + µ d d d | r d | r ϕ r ( T 1 , . . . , T r − 1 ) := rT r − β r ( T 1 , . . . , T r ) σ r ( a 1 , . . . , a r ) = S r = N r − ( q r +1) and β r ( a 1 , . . . , a r ) = rB r = ⇒ ra r = ϕ r ( a 1 , . . . , a r − 1 ) + rB r

Introduction Curves with a prescribed L -polynomial up to some degree Curves with prescribed number of points Some recursively defined functions over Z : σ 0 := 0 and for all r ≥ 1, r − 1 � σ r ( T 1 , . . . , T r ) := rT r − σ r − j ( T 1 , . . . , T r − j ) · T j j =1 � � � r � � r � ( q d + 1) β r ( T 1 , . . . , T r ) := µ σ d ( T 1 , . . . , T d ) + µ d d d | r d | r ϕ r ( T 1 , . . . , T r − 1 ) := rT r − β r ( T 1 , . . . , T r ) σ r ( a 1 , . . . , a r ) = S r = N r − ( q r +1) and β r ( a 1 , . . . , a r ) = rB r = ⇒ ra r = ϕ r ( a 1 , . . . , a r − 1 ) + rB r

Introduction Curves with a prescribed L -polynomial up to some degree Curves with prescribed number of points Necessary conditions on the coefficients of L -polynomial Theorem Let X be a non-singular, absolutely irreducible, projective curve defined over F q and let L X ( t ) = 1 + a 1 t + . . . + a 2 g t 2 g be its L-polynomial. Then the inequalities ra r ≥ ϕ r ( a 1 , . . . , a r − 1 ) hold for r = 1 , . . . , g. Example a 1 ≥ − ( q + 1) 1 + a 1 − ( q 2 − q ) 2 a 2 ≥ a 2 1 + a 1 + 3 a 1 a 2 − ( q 3 − q ) 3 a 3 ≥ − a 3 1 a 2 + 4 a 1 a 3 + 2 a 2 − ( q 4 − q 2 ) 4 a 4 ≥ − a 4 1 − a 2 1 − 4 a 2

Introduction Curves with a prescribed L -polynomial up to some degree Curves with prescribed number of points Necessary conditions on the coefficients of L -polynomial Theorem Let X be a non-singular, absolutely irreducible, projective curve defined over F q and let L X ( t ) = 1 + a 1 t + . . . + a 2 g t 2 g be its L-polynomial. Then the inequalities ra r ≥ ϕ r ( a 1 , . . . , a r − 1 ) hold for r = 1 , . . . , g. Example a 1 ≥ − ( q + 1) 1 + a 1 − ( q 2 − q ) 2 a 2 ≥ a 2 1 + a 1 + 3 a 1 a 2 − ( q 3 − q ) 3 a 3 ≥ − a 3 1 a 2 + 4 a 1 a 3 + 2 a 2 − ( q 4 − q 2 ) 4 a 4 ≥ − a 4 1 − a 2 1 − 4 a 2

Introduction Curves with a prescribed L -polynomial up to some degree Curves with prescribed number of points The converse of the Theorem Problem: Let ( a 1 , a 2 , . . . , a m ) ∈ Z m satisfying ra r ≥ ϕ r ( a 1 , . . . , a r − 1 ) for all r = 1 , . . . , m . Is there a curve X of genus g over F q whose L -polynomial has the form L ( t ) = 1 + a 1 t + a 2 t 2 + . . . + a m t m + . . . ? Not in general! Hasse-Weil Theorem: L ( t ) = � 2 g k =1 (1 − w k t ) with | w k | = √ q � 2 g � · √ q r = ⇒| a r |≤ for r = 1 , . . . , g . r

Introduction Curves with a prescribed L -polynomial up to some degree Curves with prescribed number of points The converse of the Theorem Problem: Let ( a 1 , a 2 , . . . , a m ) ∈ Z m satisfying ra r ≥ ϕ r ( a 1 , . . . , a r − 1 ) for all r = 1 , . . . , m . Is there a curve X of genus g over F q whose L -polynomial has the form L ( t ) = 1 + a 1 t + a 2 t 2 + . . . + a m t m + . . . ? Not in general! Hasse-Weil Theorem: L ( t ) = � 2 g k =1 (1 − w k t ) with | w k | = √ q � 2 g � · √ q r = ⇒| a r |≤ for r = 1 , . . . , g . r

Introduction Curves with a prescribed L -polynomial up to some degree Curves with prescribed number of points The converse of the Theorem Problem: Let ( a 1 , a 2 , . . . , a m ) ∈ Z m satisfying ra r ≥ ϕ r ( a 1 , . . . , a r − 1 ) for all r = 1 , . . . , m . Is there a curve X of genus g over F q whose L -polynomial has the form L ( t ) = 1 + a 1 t + a 2 t 2 + . . . + a m t m + . . . ? Not in general! Hasse-Weil Theorem: L ( t ) = � 2 g k =1 (1 − w k t ) with | w k | = √ q � 2 g � · √ q r = ⇒| a r |≤ for r = 1 , . . . , g . r

Introduction Curves with a prescribed L -polynomial up to some degree Curves with prescribed number of points Theorem (A., Stichtenoth) Let a 1 , . . . , a m be integers such that ra r ≥ ϕ r ( a 1 , . . . , a r − 1 ) for r = 1 , . . . , m. Then there is an integer g 0 ≥ m such that for all g ≥ g 0 , there exists a curve over F q of genus g whose L-polynomial has the form L ( t ) ≡ 1 + a 1 t + . . . + a m t m mod t m +1

Introduction Curves with a prescribed L -polynomial up to some degree Curves with prescribed number of points Sketch of the proof Remember: ra r = ϕ r ( a 1 , . . . , a r − 1 ) + rB r for r ≥ 1. Step 1: For all m ≥ 1 and all ( a 1 , . . . , a m − 1 ) ∈ Z m − 1 , ϕ m ( a 1 , . . . , a m − 1 ) ≡ 0 mod m . Step 2: Define b r := r − 1 ( ra r − ϕ r ( a 1 , . . . , a r − 1 )) for r = 1 , . . . , m . Equivalent statement: Let b 1 , . . . , b m be non-negative integers. Then there is a constant g 0 ≥ m such that for all integers g ≥ g 0 there exists a curve X over F q of genus g such that X has exactly b r points of degree r , for r = 1 , . . . , m .

Introduction Curves with a prescribed L -polynomial up to some degree Curves with prescribed number of points Sketch of the proof Remember: ra r = ϕ r ( a 1 , . . . , a r − 1 ) + rB r for r ≥ 1. Step 1: For all m ≥ 1 and all ( a 1 , . . . , a m − 1 ) ∈ Z m − 1 , ϕ m ( a 1 , . . . , a m − 1 ) ≡ 0 mod m . Step 2: Define b r := r − 1 ( ra r − ϕ r ( a 1 , . . . , a r − 1 )) for r = 1 , . . . , m . Equivalent statement: Let b 1 , . . . , b m be non-negative integers. Then there is a constant g 0 ≥ m such that for all integers g ≥ g 0 there exists a curve X over F q of genus g such that X has exactly b r points of degree r , for r = 1 , . . . , m .