How many rational points can a high genus curve over a finite field - PowerPoint PPT Presentation

How many rational points can a high genus curve over a finite field have? Alp Bassa Sabancı University

Affine plane curves k a perfect field (e.g. Q , R , C , F q ...) ¯ k a fixed algebraic closure of k Let f ( X , Y ) ∈ k [ X , Y ]. The affine plane curve defined by f ( X , Y ): C f := { ( x , y ) ∈ ¯ k × ¯ k | f ( x , y ) = 0 } C f is defined over k . The set of k -rational points of C f : C f ( k ) := { ( x , y ) ∈ k × k | f ( x , y ) = 0 }

An example k = R f ( X , Y ) = Y 2 − X · ( X − 1) · ( X + 1) . C f ( R )

Curves in n -space Can generalize this to curves in higher dimensional space: C ⊂ ¯ k n f 1 , f 2 , . . . f n − 1 ∈ k [ X 1 , X 2 , . . . , X n ]. Affine curve: C := { ( a 1 , . . . , a n ) ∈ ¯ k n | f i ( a 1 , . . . , a n ) = 0 for i = 1 , 2 , . . . n − 1 } The set of k -rational points of C : C ( k ) := { ( a 1 , . . . , a n ) ∈ k n | f i ( a 1 , . . . , a n ) = 0 for i = 1 , 2 , . . . n − 1 }

From now on we assume that C is a • absolutely irreducible • smooth • projective curve defined over k .

The genus Invariant g ( C ) : a nonnegative integer C is a line/conic − → genus 0 C is an elliptic curve − → genus 1

Curves over Finite Fields From now on k = F q n for some n ∈ N C / F q → C ⊂ ¯ F q C ( F q ) ⊂ F n q So # C ( F q ) is finite # C ( F q ) =?

The Hasse–Weil bound C − → ζ C Zeta function of C Theorem (Hasse–Weil) The Riemann hypothesis holds for ζ C . Corollary (Hasse–Weil bound) Let C / F q be a curve of genus g ( C ) . Then # C ( F q ) ≤ q + 1 + 2 √ q · g ( C ) .

How good is the Hasse–Weil bound? Various improvements, but: If the genus g ( C ) is small (with respect to q ) − → Hasse–Weil bound is good. It can be attained, maximal curves , for example over F q 2 y q + y = x q +1 . Ihara, Manin: The Hasse–Weil bound can be improved if g ( C ) is large (with respect to q ).

Ihara’s constant Ihara: # C ( F q ) A ( q ) = lim sup g ( C ) g ( C ) →∞ C runs over all absolutely irreducible, smooth, projective curves over F q . ⇒ A ( q ) ≤ 2 √ q Hasse–Weil bound = ⇒ A ( q ) ≤ √ 2 q Ihara = ⇒ A ( q ) ≤ √ q − 1 Drinfeld–Vladut =

Lower bounds for A ( q ) Serre (using class field towers): A ( q ) > 0 Ihara (modular curves): If q = ℓ 2 then A ( q ) ≥ √ q − 1 = ℓ − 1 In fact A ( ℓ 2 ) = ℓ − 1. Zink (Shimura surfaces): If q = p 3 , p a prime number, then A ( p 3 ) ≥ 2( p 2 − 1) p + 2 (generalized by Bezerra–Garcia–Stichtenoth to all cubic finite fields)

How to obtain lower bounds for A ( q )? Find sequences C i / F q such that g ( C i ) → ∞ and # C i ( F q ) lim is large. g ( C i ) i →∞ Many ways to construct good sequences: • Modular curves (Elliptic, Shimura, Drinfeld) (over F q 2 ) • Class field towers (over prime fields) • Explicit equations (recursively defined)

Recursively defined towers f 1 , f 2 , . . . f n − 1 ∈ F q [ X 1 , X 2 , . . . , X n ] n | f i ( a 1 , . . . , a n ) = 0 for i = 1 , 2 , . . . n − 1 } C := { ( a 1 , . . . , a n ) ∈ ¯ F q Recursively defined tower: Fix F ( U , V ) ∈ F q [ U , V ]. Define f 1 = F ( X 1 , X 2 ) f 2 = F ( X 2 , X 3 ) · · · f n − 1 = F ( X n − 1 , X n ) n | f 1 = f 2 = · · · = f n − 1 = 0 } C n := { ( a 1 , . . . , a n ) ∈ ¯ F q F = ( C n ) n ≥ 1 tower recursively defined by F .

�� �� Recursively defined by f ( U , V ) ∈ F q [ U , V ] 4 C 4 = { ( a 1 , a 2 , a 3 , a 4 ) | F ( a 1 , a 2 ) = F ( a 2 , a 3 ) = F ( a 3 , a 4 ) = 0 } ⊆ ¯ F q 3 C 3 = { ( a 1 , a 2 , a 3 ) | F ( a 1 , a 2 ) = 0 , F ( a 2 , a 3 ) = 0 } ⊆ ¯ F q 2 C 2 = { ( a 1 , a 2 ) | F ( a 1 , a 2 ) = 0 } ⊆ ¯ F q

Limit of a tower Limit of the tower F = ( C n ) n ≥ 1 over F q # C n ( F q ) ≤ A ( q ) ≤ √ q − 1 λ ( F ) = lim g ( C n ) n →∞ exists λ ( F ) = 0 − → asymptotically bad λ ( F ) > 0 − → asymptotically good

Example Garcia–Stichtenoth, 1996, Norm-Trace tower F 1 q = ℓ 2 U ℓ +1 V ℓ + V = U ℓ + U λ ( F 1 ) = √ q − 1 Attains the Drinfeld–Vladut bound. Genus computation is difficult (wild ramification) Why many rational points?

U ℓ +1 V ℓ + V = q = ℓ 2 U ℓ + U X ℓ +1 X ℓ +1 X ℓ +1 X ℓ n − 1 , . . . , X ℓ , X ℓ 2 1 n + X n = 3 + X 3 = 2 + X 2 = X ℓ X ℓ X ℓ n − 1 + X n − 1 2 + X 2 1 + X 1 X 1 = a 1 ∈ F q s.t. Tr F q / F ℓ ( a 1 ) � = 0 ( ℓ 2 − ℓ choices) a ℓ +1 X 2 = a 2 with a ℓ 1 2 + a 2 = ∈ F ℓ \{ 0 } a ℓ 1 + a 1 ℓ choices with a 2 ∈ F q , Tr F q / F ℓ ( a 2 ) � = 0) a ℓ +1 X 3 = a 3 with a ℓ 2 3 + a 3 = ∈ F ℓ \{ 0 } a ℓ 2 + a 2 ℓ choices with a 3 ∈ F q , Tr F q / F ℓ ( a 3 ) � = 0) · · · · · · so # C n ( F q ) ≥ ( ℓ 2 − ℓ ) ℓ n − 1

Towers over cubic finite fields q = 2 3 = 8 , F 2 / F q • van der Geer–van der Vlugt, V 2 + V = U + 1 + 1 / U Attains Zink’s bound for p = 2. q = ℓ 3 , F 3 / F q • Bezerra–Garcia–Stichtenoth, = U ℓ + U + 1 λ ( F 3 ) ≥ 2( ℓ 2 − 1) 1 − V . V ℓ U ℓ + 2 Generalizes Zink’s bound. q = ℓ 3 , F 4 / F q • B.–Garcia–Stichtenoth, λ ( F 4 ) ≥ 2( ℓ 2 − 1) − U ℓ ( ℓ − 1) ( V ℓ − V ) ℓ − 1 + 1 = . ( U ℓ − 1 − 1) ℓ − 1 ℓ + 2

A new family of towers over all non-prime fields B.–Beelen–Garcia–Stichtenoth F 5 over F ℓ n , n ≥ 2: Notation: Tr n ( t ) = t + t ℓ + · · · + t ℓ n − 1 , N n ( t ) = t 1+ ℓ + ℓ 2 + ... + ℓ n − 1 N n ( V ) + 1 = N n ( U ) + 1 . V ℓ n − 1 U Splitting: N n ( α ) = − 1 2 λ ( F 5 ) ≥ 1 1 ℓ − 1 + ℓ n − 1 − 1 • n = 2: ℓ − 1 → Drinfeld-Vladut bound • n = 3: 2( ℓ 2 − 1) → Zink’s bound ℓ +2

F 6 / F q , q = ℓ n , n = 2 k + 1 ≥ 3 ( Tr k +1 ( V ) − 1) ℓ k = ( Tr k ( U ) − 1) ℓ k +1 Tr k ( V ) − 1 ( Tr k +1 ( U ) − 1) V ℓ n − V = − (1 / U ) ℓ n − (1 / U ) V ℓ k U ℓ k +1

F 6 / F q , q = ℓ n , n = 2 k + 1 ≥ 2( ℓ k +1 − 1) 2 λ ( F 6 ) ≥ 1 1 ℓ + 1 + ǫ ℓ k − 1 + ℓ k +1 − 1 with ǫ = ℓ − 1 ℓ k − 1 . Note: 2 ℓ k + 1 2 − 1 ≥ A ( ℓ 2 k +1 ) ≥ . 1 1 ℓ k − 1 + ℓ k +1 − 1 2 15 (2 3 ) 5 (2 5 ) 3 q = 2 k , k large, λ ( F 5 ) √ q − 1 ≈ 94%

Elliptic Curves E / k , char ( k ) � = 2 , 3 E : Y 2 = X 3 + A · X + B , where 4 A 3 + 27 B 2 � = 0.

Elliptic Curves over C k = C Λ = ω 1 Z ⊕ ω 2 Z . C / Λ topologically a torus inherits a complex structure from C . Complex manifold → E ( C )

The group law Points in E inherit a group structure from C :

The group law Points in E inherit a group structure from C :

The group law Points in E inherit a group structure from C :

The group law Points in E inherit a group structure from C :

The group law Points in E inherit a group structure from C :

Isogenies A morphism ϕ : E 1 → E 2 , which is a group homomorphism is called an isogeny . Example: E elliptic curve, N ∈ N [ N ] : E → E P �→ P + P + . . . P � �� � N times # ker ( ϕ ) is finite. # ker ( ϕ ) = N → ϕ is an N -isogeny → ker ( ϕ ) ⊂ ker ([ N ]).

Torsion ker ([ N ]) = { P ∈ E | N · P = 0 } =: E [ N ] → N -torsion points E [ N ] ∼ if char ( k ) ∤ N = Z / n Z × Z / n Z { 0 } → supersingular E [ p ] ∼ if char ( k ) = p = or Z / p Z → ordinary

Isomorphism classes of elliptic curves C / Λ 1 and C / Λ 2 are isomorphic ⇐ ⇒ Λ 1 and Λ 2 are homothetic, i.e. Λ 1 = α Λ 2 , α ∈ C × . Let H = { τ ∈ C | Im ( τ ) > 0 } . Every lattice is homothetic to a lattice of the form Λ τ = Z + Z τ with τ ∈ H . When are Λ τ and Λ τ ′ the same lattice?

When are Λ τ and Λ τ ′ the same lattice? SL 2 ( Z ) acts on H by fractional linear transformations: � a � · τ = a τ + b b c τ + d . c d Λ τ and Λ τ ′ are the same lattice ⇐ ⇒ τ and τ ′ are in the same orbit under the action of SL 2 ( Z ).

Isomorphism classes of elliptic curves Elliptic curves / isomorphism ← → lattices in C / homothety ← → H / SL 2 ( Z ) → X (1)

The j -Function There exists a holomorphic function j : H → C , which is invariant under SL 2 ( Z ). j : H / SL 2 ( Z ) → C is a bijection!

− → j -line [ E ] − → j -invariant Fact: E supersingular − → j ( E ) ∈ F p 2 , where p is the characteristic. ∴ j -line parametrizes isomorphism classes of Elliptic curves → has designated F p 2 -rational points.

Enhanced Elliptic Curves Elliptic curves with some additional structure ( E , C ) E : Elliptic Curve C : cyclic subgroup of order N / N -isogeny ( E , C ) ∼ ( E ′ , C ′ ) isomorphism takes C → C ′ . X 0 ( N ) modular curve parametrizing ( E , C ).

� � A ⊂ X 0 ( N )( F p 2 ) X 0 ( N ) forget ⊃ supersingular ⊂ X (1)( F p 2 ) X (1)

( N i ) i ≥ 0 with N i → ∞ , p ∤ N i . C N i = ( X 0 ( N i ) (mod p )) • # C N i ( F p 2 ) is large (supersingular points) • g ( C N i ) can be estimated # C N i ( F p 2 ) � p 2 − 1 = p − 1 → (Drinfeld-Vladut bound) g ( C N i ) Elkies: X 0 ( ℓ n ) recursive.

Drinfeld Modular Varieties C ∞ C K R R F q ( T ) Q F q [ T ] Z Z -lattices inside C → rank 1 or 2 F q [ T ]-lattices inside C ∞ → arbitrary high rank possible