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 Fq. (alternatively F/Fq an algebraic function field with full constant field Fq) C(Fq) set of rational points of C. #C(Fq) is finite #C(Fq) =?

The Hasse–Weil bound

C − → ζC Zeta function of C

Theorem (Hasse–Weil)

The Riemann hypothesis holds for ζC.

Corollary (Hasse–Weil bound)

Let C/Fq be a curve of genus g(C). Then #C(Fq) ≤ q + 1 + 2√q · g(C).

How good is the Hasse–Weil bound?

Trivial improvement

#C(Fq) ≤ q + 1 + ⌊2√q · g(C)⌋.

Theorem (Serre)

#C(Fq) ≤ 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 Fq2 yq + y = xq+1.

Ihara, Manin: The Hasse–Weil bound can be improved if g(C) is

large (with respect to q).

Ihara’s constant

Ihara: A(q) = lim sup

g(C)→∞

#C(Fq) g(C) C runs over all absolutely irreducible, smooth, projective curves

• ver Fq.

Hasse–Weil bound = ⇒ A(q) ≤ 2√q Ihara = ⇒ A(q) ≤ 1

2(√8q + 1 − 1) ≤ √2q ≤ 2√q

Drinfeld–Vladut = ⇒ A(q) ≤ √q − 1

Lower bounds for A(q)

Serre (using class field towers): A(q) > 0 Ihara, Tsfasman–Vladut–Zink (modular curves): If q = ℓ2 then A(ℓ2) ≥ √q − 1 = ℓ − 1 In fact A(ℓ2) = ℓ − 1. Zink (Shimura surfaces): If q = p3, p a prime number, then A(p3) ≥ 2(p2 − 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 = ℓn A(ℓn) ≥ 2

1 ℓ−1 + 1 ℓn−1−1

• n = 2: ℓ − 1 → Drinfeld-Vladut bound
• n = 3: 2(ℓ2−1)

ℓ+2

→ Zink’s bound

ℓ prime power, n = 2k + 1 ≥ 3, q = ℓn A(ℓ2k+1) ≥ 2

1 ℓk−1 + 1 ℓk+1−1

≥ 2(ℓk+1 − 1) ℓ + 1 + ǫ with ǫ = ℓ − 1 ℓk − 1. Note: ℓk+ 1

2 − 1 ≥ A(ℓ2k+1) ≥

2

1 ℓk−1 + 1 ℓk+1−1

. 215 (23)5 (25)3 q = 2k,k large, lower bound √q − 1 ≈ 94%

How to obtain lower bounds for A(q)?

Find sequences F = (Ci)i≥0 with Ci/Fq and g(Ci) → ∞ such that λ(F) = lim

i→∞

#Ci(Fq) g(Ci) is large. since 0 < λ(F) ≤ A(q) ≤ √q − 1 λ(F) : limit of F = (Ci)i≥0.

How to construct good sequences

Various approaches:

• Class field towers (over prime fields)
• Modular curves (Elliptic, Shimura, Drinfeld) (over Fq2)
• Explicit equations (recursively defined)

Modular towers

X0(N)/Q modular curve parametrizing elliptic curves with a cyclic N-isogeny. Good reduction at primes p ∤ N. g(X0(N)) are known (formula). X0(N)/Fp2 has many Fp2-rational points (why?).

Supersingular points

Fact: E/k supersingular − → j(E) ∈ Fp2, where p is the characteristic of k. isomorphism classes of supersingular elliptic curves give rise to Fp2-rational points on X0(N)/k

Fix a prime p. (Ni)i≥0 with Ni → ∞, p ∤ Ni. CNi = (X0(Ni) (mod p))

• #CNi(Fp2) is large (supersingular points)
• g(CNi) can be calculated

#CNi(Fp2) g(CNi) →

• p2 − 1 = p − 1

(Drinfeld-Vladut bound)

Recursively defined towers

Fix f (U, V ) ∈ Fq[U, V ]. Let Cn be the curve defined by f (x0, x1) = 0 f (x1, x2) = 0 · · · f (xn−1, xn) = 0 F = (Cn)n≥1 tower recursively defined by f . We obtain a covering of curves · · · → Cn+1 → Cn → · · · → C1 → C0 = P1.

Example

There are several examples of recursively defined towers, with large limit (even optimal). Garcia–Stichtenoth, 1996, Norm-Trace tower q = ℓ2 V ℓ + V = Uℓ+1 Uℓ + U λ = √q − 1 Attains the Drinfeld–Vladut bound. Genus computation is difficult (wild ramification) Why many rational points?

q = ℓ2 V ℓ + V = Uℓ+1 Uℓ + U X ℓ

n+Xn =

X ℓ+1

n−1

X ℓ

n−1 + Xn−1

, . . . , X ℓ

3+X3 =

X ℓ+1

2

X ℓ

2 + X2

, X ℓ

2+X2 =

X ℓ+1

1

X ℓ

1 + X1

X1 = a1 ∈ Fq s.t. TrFq/Fℓ(a1) = 0 (ℓ2 − ℓ choices) X2 = a2 with aℓ

2 + a2 =

aℓ+1

1

aℓ

1 + a1

∈ Fℓ\{0} ℓ choices with a2 ∈ Fq, TrFq/Fℓ(a2) = 0) X3 = a3 with aℓ

3 + a3 =

aℓ+1

2

aℓ

2 + a2

∈ Fℓ\{0} ℓ choices with a3 ∈ Fq, TrFq/Fℓ(a3) = 0) · · · · · · so #Cn(Fq) ≥ (ℓ2 − ℓ)ℓn−1

Elkies has shown that all known optimal recursive towers are modular (Elliptic, Shimura, Drinfeld).

Elkies: Fix s. The sequence X0(sk) is recursively defined.

A point z ∈ Y0(sk) = X0(sk) − {cusps} represents an equivalence class of

• the pairs (E, Csk) of elliptic curve E and cyclic subgroup Csk
• f order sk.
• isogenies E → E/Csk

Csk is cyclic, so it has a unique filtration of the form Csk ⊃ Csk−1 ⊃ · · · ⊃ Cs ⊃ {e} In terms of isogenies: E0 = E → E1 = E/Cs → · · · → Ek = E/Csk. For i = 0, 1, . . . , k − 1, Ei and Ei+1 are related by a cyclic s-isogeny. So Φs(j(Ei), j(Ei+1)) = 0, where Φs(U, V ) is the modular polynomial of level s.

So we iterate the correspondence X0(s)

• X(1)

X(1) X0(s2)

• X0(s)
• X(1)

X0(s)

• X(1)

X(1)

Drinfeld Modular Varieties

C∞ C ¯ k∞ R k∞ Q Fℓ(T) Z Fℓ[T] Z-lattices inside C → rank 1 or 2 Fℓ[T]-lattices inside C∞ → arbitrary high rank possible

Drinfeld Modular Curves

A = Fℓ[T], P a prime of A, FP = A/ < P >= Fℓd where d = deg P. F(2)

P : The unique quadratic extension of FP.

For N ∈ Fℓ[T] we have X0(N) an algebraic curve defined over Fℓ(T), Drinfeld modular curve, parametrizing rank 2 Drinfeld modules together with a cyclic N-isogeny. X0(N) has good reduction at all primes P ∤ N. X0(N)/FP

Many points on Drinfeld modular curves

X0(N)/FP has many rational points over F(2)

P

= Fℓ2d, where d = deg P. Asymptotically:

Theorem (Gekeler)

P ∈ Fℓ[T] prime of degree d (Nk)k≥0: sequence of polynomials in Fℓ[T] with

• P ∤ Nk
• deg Nk → ∞

Then the sequence of curves X0(Nk)/FP attains the Drinfeld–Vladut bound over F(2)

P

= Fℓ2d.

Elkies: X0(Qn) recursive.

Norm trace tower is related to (degree ℓ − 1 cover of) X0(T n)/FT−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)

P

• f

FP 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. φT = τ n + g1τ n−1 + g2τ n−2 + · · · + gn−1τ + 1. φ is supersingular if φT−1 is a purely inseparable map of degree ℓn, i.e., φT−1 = τ n, i.e., g1 = g2 = · · · = gn−1 = 0. We want a

• one dimensional sublocus,
• passing through g1 = g2 = · · · = gn−1 = 0
• invariant under isogenies (to obtain a recursive tower)

We call φ weakly supersingular, if φT−1 is a map of inseparability at least ℓn−1, i.e., φT−1 = τ n + g1τ n−1, i.e., g2 = · · · = gn−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 + a2τ n−2 + · · · + an−1τ + an, s.t. µ · λ = φT Then Nn(u) + g1 · Nn−1(u) + 1 = 0 Notation: Nk(x) = x1+ℓ+···+ℓk−2+ℓk−1

Equations for the isogenous Drinfeld module

λ : φ → ψ ψT = τ n + h1 · τ n−1 + 1 Isogeny: λ · φ = ψ · λ (τ − u) · (τ n + g1τ n−1 + 1) = (τ n + h1τ n−1 + 1) · (τ − u) gℓ

1 − u = h1 − uℓn

−ug1 = −h1uℓn−1

−g1 = Nn(1/u) + 1 (1/u)ℓn−1 , −h1 = Nn(1/u) + 1 1/u Letting v0 = 1/u Fq(v0) Fq(h1) Fq(g1)

Nn(V ) + 1 V ℓn−1 = Nn(U) + 1 U .

A new family of towers over all non-prime fields

B.–Beelen–Garcia–Stichtenoth F5 over Fℓn, n ≥ 2: Notation:Trn(t) = t + tℓ + · · · + tℓn−1, Nn(t) = t1+ℓ+ℓ2+...+ℓn−1 Nn(V ) + 1 V ℓn−1 = Nn(U) + 1 U . Splitting: Nn(α) = −1 λ(F5) ≥ 2

1 ℓ−1 + 1 ℓn−1−1

• n = 2: ℓ − 1 → Drinfeld-Vladut bound
• n = 3: 2(ℓ2−1)

ℓ+2

→ Zink’s bound

F6/Fq, q = ℓn, n = 2k + 1 ≥ 3 Trk(V ) − 1 (Trk+1(V ) − 1)ℓk = (Trk(U) − 1)ℓk+1 (Trk+1(U) − 1) V ℓn − V V ℓk = −(1/U)ℓn − (1/U) Uℓk+1

F6/Fq, q = ℓn, n = 2k + 1 λ(F6) ≥ 2

1 ℓk−1 + 1 ℓk+1−1

≥ 2(ℓk+1 − 1) ℓ + 1 + ǫ with ǫ = ℓ − 1 ℓk − 1. Note: ℓk+ 1

2 − 1 ≥ A(ℓ2k+1) ≥

2

1 ℓk−1 + 1 ℓk+1−1

. 215 (23)5 (25)3 q = 2k,k large, λ(F5) √q − 1 ≈ 94%