Rational points on curves over finite fields and Drinfeld modular - PowerPoint PPT Presentation

Rational points on curves over finite fields and Drinfeld modular varieties Alp Bassa Sabancı University

Curves over Finite Fields Let C be smooth, projective, absolutely irreducible curve over F q . (alternatively F / F q an algebraic function field with full constant field F q ) C ( F q ) set of rational points of C . # 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? Trivial improvement # C ( F q ) ≤ q + 1 + ⌊ 2 √ q · g ( C ) ⌋ . Theorem (Serre) # C ( F q ) ≤ q + 1 + g ( C ) · ⌊ 2 √ q ⌋ . 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 = 2 ( √ 8 q + 1 − 1) ≤ √ 2 q ≤ 2 √ q ⇒ A ( q ) ≤ 1 Ihara = ⇒ A ( q ) ≤ √ q − 1 Drinfeld–Vladut =

Lower bounds for A ( q ) Serre (using class field towers): A ( q ) > 0 Ihara, Tsfasman–Vladut–Zink (modular curves): If q = ℓ 2 then A ( ℓ 2 ) ≥ √ 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)

A ( q ) for non-prime q B.–Beelen–Garcia–Stichtenoth ℓ prime power, n ≥ 2, q = ℓ n 2 A ( ℓ n ) ≥ 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

ℓ prime power, n = 2 k + 1 ≥ 3, q = ℓ n ≥ 2( ℓ k +1 − 1) 2 A ( ℓ 2 k +1 ) ≥ 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, lower bound ≈ 94% √ q − 1

How to obtain lower bounds for A ( q )? Find sequences F = ( C i ) i ≥ 0 with C i / F q and g ( C i ) → ∞ such that # C i ( F q ) λ ( F ) = lim is large. g ( C i ) i →∞ since 0 < λ ( F ) ≤ A ( q ) ≤ √ q − 1 λ ( F ) : limit of F = ( C i ) i ≥ 0 .

How to construct good sequences Various approaches: • Class field towers (over prime fields) • Modular curves (Elliptic, Shimura, Drinfeld) (over F q 2 ) • Explicit equations (recursively defined)

Modular towers X 0 ( N ) / Q modular curve parametrizing elliptic curves with a cyclic N -isogeny. Good reduction at primes p ∤ N . g ( X 0 ( N )) are known (formula). X 0 ( N ) / F p 2 has many F p 2 -rational points (why?).

Supersingular points Fact: E / k supersingular − → j ( E ) ∈ F p 2 , where p is the characteristic of k . isomorphism classes of supersingular elliptic curves give rise to F p 2 -rational points on X 0 ( N ) / k

Fix a prime p . ( 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 calculated # C N i ( F p 2 ) p 2 − 1 = p − 1 � → (Drinfeld-Vladut bound) g ( C N i )

Recursively defined towers Fix f ( U , V ) ∈ F q [ U , V ]. Let C n be the curve defined by f ( x 0 , x 1 ) = 0 f ( x 1 , x 2 ) = 0 · · · f ( x n − 1 , x n ) = 0 F = ( C n ) n ≥ 1 tower recursively defined by f . We obtain a covering of curves · · · → C n +1 → C n → · · · → C 1 → C 0 = P 1 .

Example There are several examples of recursively defined towers, with large limit (even optimal). Garcia–Stichtenoth, 1996, Norm-Trace tower q = ℓ 2 U ℓ +1 V ℓ + V = U ℓ + U λ = √ 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

Elkies has shown that all known optimal recursive towers are modular (Elliptic, Shimura, Drinfeld). Elkies: Fix s . The sequence X 0 ( s k ) is recursively defined. A point z ∈ Y 0 ( s k ) = X 0 ( s k ) − { cusps } represents an equivalence class of • the pairs ( E , C s k ) of elliptic curve E and cyclic subgroup C s k of order s k . • isogenies E → E / C s k

C s k is cyclic, so it has a unique filtration of the form C s k ⊃ C s k − 1 ⊃ · · · ⊃ C s ⊃ { e } In terms of isogenies: E 0 = E → E 1 = E / C s → · · · → E k = E / C s k . For i = 0 , 1 , . . . , k − 1, E i and E i +1 are related by a cyclic s -isogeny. So Φ s ( j ( E i ) , j ( E i +1 )) = 0 , where Φ s ( U , V ) is the modular polynomial of level s .

� � � � � � � � So we iterate the correspondence X 0 ( s ) X (1) X (1) X 0 ( s 2 ) X 0 ( s ) X 0 ( s ) X (1) X (1) X (1)

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

Drinfeld Modular Curves A = F ℓ [ T ], P a prime of A , F P = A / < P > = F ℓ d where d = deg P . F (2) P : The unique quadratic extension of F P . For N ∈ F ℓ [ T ] we have X 0 ( N ) an algebraic curve defined over F ℓ ( T ), Drinfeld modular curve, parametrizing rank 2 Drinfeld modules together with a cyclic N -isogeny. X 0 ( N ) has good reduction at all primes P ∤ N . X 0 ( N ) / F P

Many points on Drinfeld modular curves X 0 ( N ) / F P has many rational points over F (2) = F ℓ 2 d , where P d = deg P . Asymptotically: Theorem (Gekeler) P ∈ F ℓ [ T ] prime of degree d ( N k ) k ≥ 0 : sequence of polynomials in F ℓ [ T ] with • P ∤ N k • deg N k → ∞ Then the sequence of curves X 0 ( N k ) / F P attains the Drinfeld–Vladut bound over F (2) = F ℓ 2 d . P

Elkies: X 0 ( Q n ) recursive. Norm trace tower is related to (degree ℓ − 1 cover of) X 0 ( T n ) / F T − 1

elliptic modular curves → Shimura curves Drinfeld modular curves → modular curves of D -elliptic sheaves Papikian has shown that modular curves of D -elliptic sheaves attain the Drinfeld–Vladut bound.

Many points over non-quadratic fields Many points come from the supersingular points → defined over F (2) − P . In general: Theorem (Gekeler) Any supersingular Drinfeld module φ of rank r and characteristic P is isomorphic to one defined over L, where L is an extension F ( r ) of P F P of degree r. Idea: Look at space parametrizing rank r Drinfeld modules Problem: The corresponding space is higher dimensional (( r − 1)-dimensional), not a curve! Idea’: Look at curves on those spaces, passing through the many F ( r ) P -rational points.

Let φ be a normalized rank n Drinfeld Module of characteristic T − 1. φ T = τ n + g 1 τ n − 1 + g 2 τ n − 2 + · · · + g n − 1 τ + 1 . φ is supersingular if φ T − 1 is a purely inseparable map of degree ℓ n , i.e., φ T − 1 = τ n , i.e., g 1 = g 2 = · · · = g n − 1 = 0 . We want a • one dimensional sublocus, • passing through g 1 = g 2 = · · · = g n − 1 = 0 • invariant under isogenies (to obtain a recursive tower)

We call φ weakly supersingular, if φ T − 1 is a map of inseparability at least ℓ n − 1 , i.e., φ T − 1 = τ n + g 1 τ n − 1 , i.e., g 2 = · · · = g n − 1 = 0 . Note that the property of being weakly supersingular is invariant under isogenies! Look at the space of weakly supersingular normalized Drinfeld modes.

Isogenies Let λ : φ → ψ be an isogeny of the form τ − u whose kernel is annihilated by T . ∃ µ = τ n − 1 + a 2 τ n − 2 + · · · + a n − 1 τ + a n , s.t. µ · λ = φ T Then N n ( u ) + g 1 · N n − 1 ( u ) + 1 = 0 Notation: N k ( x ) = x 1+ ℓ + ··· + ℓ k − 2 + ℓ k − 1

Equations for the isogenous Drinfeld module λ : φ → ψ ψ T = τ n + h 1 · τ n − 1 + 1 Isogeny: λ · φ = ψ · λ ( τ − u ) · ( τ n + g 1 τ n − 1 + 1) = ( τ n + h 1 τ n − 1 + 1) · ( τ − u ) 1 − u = h 1 − u ℓ n g ℓ − ug 1 = − h 1 u ℓ n − 1

− g 1 = N n (1 / u ) + 1 , (1 / u ) ℓ n − 1 − h 1 = N n (1 / u ) + 1 1 / u Letting v 0 = 1 / u F q ( v 0 ) F q ( g 1 ) F q ( h 1 )

N n ( V ) + 1 = N n ( U ) + 1 . V ℓ n − 1 U