Bounds for the number of Rational points on curves over finite - PowerPoint PPT Presentation

Preliminaries and the St¨ ohr-Voloch Theory Variation of the St¨ ohr-Voloch approach Results Exemplos Exam Bounds for the number of Rational points on curves over finite fields Herivelto Borges Universidade de S˜ ao Paulo-Brasill Joint work with Nazar Arakelian Workshop on Algebraic curves -Linz-Austria-2013 Herivelto Borges ICMC-USP-S˜ ao Carlos Bounds for the number of Rational points on curves over finite fields

Preliminaries and the St¨ ohr-Voloch Theory Variation of the St¨ ohr-Voloch approach Results Exemplos Exam Classical Bounds Let X be a projective, irreducible, non-singular curve of genus g , defined over F q . If N is the number of F q -rational points of X then Hasse-Weil-Serre: | N − ( q + 1) | ≤ g ⌊ 2 q 1 / 2 ⌋ . ”Zeta”: N 2 ≤ q 2 + 1 + 2 gq − ( N 1 − q − 1) 2 g where N r is the number of F q r -rational points of X . St¨ ohr-Voloch (baby version): If X has a plane model of degree d , and a finite number of inflection points, then N ≤ g − 1 + d ( q + 2) / 2 . Herivelto Borges ICMC-USP-S˜ ao Carlos Bounds for the number of Rational points on curves over finite fields

Preliminaries and the St¨ ohr-Voloch Theory Variation of the St¨ ohr-Voloch approach Results Exemplos Exam Classical Bounds Let X be a projective, irreducible, non-singular curve of genus g , defined over F q . If N is the number of F q -rational points of X then Hasse-Weil-Serre: | N − ( q + 1) | ≤ g ⌊ 2 q 1 / 2 ⌋ . ”Zeta”: N 2 ≤ q 2 + 1 + 2 gq − ( N 1 − q − 1) 2 g where N r is the number of F q r -rational points of X . St¨ ohr-Voloch (baby version): If X has a plane model of degree d , and a finite number of inflection points, then N ≤ g − 1 + d ( q + 2) / 2 . Herivelto Borges ICMC-USP-S˜ ao Carlos Bounds for the number of Rational points on curves over finite fields

Preliminaries and the St¨ ohr-Voloch Theory Variation of the St¨ ohr-Voloch approach Results Exemplos Exam Classical Bounds Let X be a projective, irreducible, non-singular curve of genus g , defined over F q . If N is the number of F q -rational points of X then Hasse-Weil-Serre: | N − ( q + 1) | ≤ g ⌊ 2 q 1 / 2 ⌋ . ”Zeta”: N 2 ≤ q 2 + 1 + 2 gq − ( N 1 − q − 1) 2 g where N r is the number of F q r -rational points of X . St¨ ohr-Voloch (baby version): If X has a plane model of degree d , and a finite number of inflection points, then N ≤ g − 1 + d ( q + 2) / 2 . Herivelto Borges ICMC-USP-S˜ ao Carlos Bounds for the number of Rational points on curves over finite fields

Preliminaries and the St¨ ohr-Voloch Theory Variation of the St¨ ohr-Voloch approach Results Exemplos Exam Morphisms vs. Linear Series Let X be a proj. irred. smooth curve of genus g defined over F q .Associted to a non-degenerated morphism → P n ( K ), there exists a base-point-free φ = ( f 0 : ... : f n ) : X − linear series, of dimension n and degree d , given by � n � � � � D = div a i f i + E | a 0 , ..., a n ∈ K , i =0 where � E := e P P, with e P = − min { v P ( f 0 ) , ..., v P ( f n ) } P ∈X and d = deg E Herivelto Borges ICMC-USP-S˜ ao Carlos Bounds for the number of Rational points on curves over finite fields

Preliminaries and the St¨ ohr-Voloch Theory Variation of the St¨ ohr-Voloch approach Results Exemplos Exam Morphisms vs. Linear Series Let X be a proj. irred. smooth curve of genus g defined over F q .Associted to a non-degenerated morphism → P n ( K ), there exists a base-point-free φ = ( f 0 : ... : f n ) : X − linear series, of dimension n and degree d , given by � n � � � � D = div a i f i + E | a 0 , ..., a n ∈ K , i =0 where � E := e P P, with e P = − min { v P ( f 0 ) , ..., v P ( f n ) } P ∈X and d = deg E Herivelto Borges ICMC-USP-S˜ ao Carlos Bounds for the number of Rational points on curves over finite fields

Preliminaries and the St¨ ohr-Voloch Theory Variation of the St¨ ohr-Voloch approach Results Exemplos Exam Morphisms vs. Linear Series Let X be a proj. irred. smooth curve of genus g defined over F q .Associted to a non-degenerated morphism → P n ( K ), there exists a base-point-free φ = ( f 0 : ... : f n ) : X − linear series, of dimension n and degree d , given by � n � � � � D = div a i f i + E | a 0 , ..., a n ∈ K , i =0 where � E := e P P, with e P = − min { v P ( f 0 ) , ..., v P ( f n ) } P ∈X and d = deg E Herivelto Borges ICMC-USP-S˜ ao Carlos Bounds for the number of Rational points on curves over finite fields

Preliminaries and the St¨ ohr-Voloch Theory Variation of the St¨ ohr-Voloch approach Results Exemplos Exam Morphisms vs. Linear Series Let X be a proj. irred. smooth curve of genus g defined over F q .Associted to a non-degenerated morphism → P n ( K ), there exists a base-point-free φ = ( f 0 : ... : f n ) : X − linear series, of dimension n and degree d , given by � n � � � � D = div a i f i + E | a 0 , ..., a n ∈ K , i =0 where � E := e P P, with e P = − min { v P ( f 0 ) , ..., v P ( f n ) } P ∈X and d = deg E Herivelto Borges ICMC-USP-S˜ ao Carlos Bounds for the number of Rational points on curves over finite fields

Preliminaries and the St¨ ohr-Voloch Theory Variation of the St¨ ohr-Voloch approach Results Exemplos Exam Order sequence For each point P ∈ X , we have φ ( P ) = (( t e P f 0 )( P ) : ... : ( t e P f n )( P )) , where t ∈ K ( X ) is a local parameter at P . For each point P ∈ X , we define a sequence of non-negative integers ( j 0 ( P ) , ..., j n ( P )) where j 0 ( P ) < ... < j n ( P ) , are called ( D , P ) orders.This can be obtained from { j 0 ( P ) , · · · , j n ( P ) } := { v P ( D ) : D ∈ D} . Herivelto Borges ICMC-USP-S˜ ao Carlos Bounds for the number of Rational points on curves over finite fields

Preliminaries and the St¨ ohr-Voloch Theory Variation of the St¨ ohr-Voloch approach Results Exemplos Exam Order sequence For each point P ∈ X , we have φ ( P ) = (( t e P f 0 )( P ) : ... : ( t e P f n )( P )) , where t ∈ K ( X ) is a local parameter at P . For each point P ∈ X , we define a sequence of non-negative integers ( j 0 ( P ) , ..., j n ( P )) where j 0 ( P ) < ... < j n ( P ) , are called ( D , P ) orders.This can be obtained from { j 0 ( P ) , · · · , j n ( P ) } := { v P ( D ) : D ∈ D} . Herivelto Borges ICMC-USP-S˜ ao Carlos Bounds for the number of Rational points on curves over finite fields

Preliminaries and the St¨ ohr-Voloch Theory Variation of the St¨ ohr-Voloch approach Results Exemplos Exam Order sequence For each point P ∈ X , we have φ ( P ) = (( t e P f 0 )( P ) : ... : ( t e P f n )( P )) , where t ∈ K ( X ) is a local parameter at P . For each point P ∈ X , we define a sequence of non-negative integers ( j 0 ( P ) , ..., j n ( P )) where j 0 ( P ) < ... < j n ( P ) , are called ( D , P ) orders.This can be obtained from { j 0 ( P ) , · · · , j n ( P ) } := { v P ( D ) : D ∈ D} . Herivelto Borges ICMC-USP-S˜ ao Carlos Bounds for the number of Rational points on curves over finite fields

Preliminaries and the St¨ ohr-Voloch Theory Variation of the St¨ ohr-Voloch approach Results Exemplos Exam Order sequence For each point P ∈ X , we have φ ( P ) = (( t e P f 0 )( P ) : ... : ( t e P f n )( P )) , where t ∈ K ( X ) is a local parameter at P . For each point P ∈ X , we define a sequence of non-negative integers ( j 0 ( P ) , ..., j n ( P )) where j 0 ( P ) < ... < j n ( P ) , are called ( D , P ) orders.This can be obtained from { j 0 ( P ) , · · · , j n ( P ) } := { v P ( D ) : D ∈ D} . Herivelto Borges ICMC-USP-S˜ ao Carlos Bounds for the number of Rational points on curves over finite fields

Preliminaries and the St¨ ohr-Voloch Theory Variation of the St¨ ohr-Voloch approach Results Exemplos Exam Order sequence We define L i ( P ) to be the intersection of all hyperplanes H of P n ( K ) such that v P ( φ ∗ ( H )) ≥ j i +1 ( P ). Therefore, we have L 0 ( P ) ⊂ L 1 ( P ) ⊂ · · · ⊂ L n − 1 ( P ) . L i ( P ) is called i -th osculating space at P . Note that L 0 = { P } , L 1 ( P ) is the tangent line at P , etc. L n − 1 ( P ) is the osculating hyperplane. Herivelto Borges ICMC-USP-S˜ ao Carlos Bounds for the number of Rational points on curves over finite fields

Preliminaries and the St¨ ohr-Voloch Theory Variation of the St¨ ohr-Voloch approach Results Exemplos Exam Order sequence We define L i ( P ) to be the intersection of all hyperplanes H of P n ( K ) such that v P ( φ ∗ ( H )) ≥ j i +1 ( P ). Therefore, we have L 0 ( P ) ⊂ L 1 ( P ) ⊂ · · · ⊂ L n − 1 ( P ) . L i ( P ) is called i -th osculating space at P . Note that L 0 = { P } , L 1 ( P ) is the tangent line at P , etc. L n − 1 ( P ) is the osculating hyperplane. Herivelto Borges ICMC-USP-S˜ ao Carlos Bounds for the number of Rational points on curves over finite fields

Preliminaries and the St¨ ohr-Voloch Theory Variation of the St¨ ohr-Voloch approach Results Exemplos Exam Order sequence We define L i ( P ) to be the intersection of all hyperplanes H of P n ( K ) such that v P ( φ ∗ ( H )) ≥ j i +1 ( P ). Therefore, we have L 0 ( P ) ⊂ L 1 ( P ) ⊂ · · · ⊂ L n − 1 ( P ) . L i ( P ) is called i -th osculating space at P . Note that L 0 = { P } , L 1 ( P ) is the tangent line at P , etc. L n − 1 ( P ) is the osculating hyperplane. Herivelto Borges ICMC-USP-S˜ ao Carlos Bounds for the number of Rational points on curves over finite fields