On the number of rational points of a plane curve

On the number of rational points of a plane curve

Masaaki HOMMA and Seon Jeong KIM

1 Notation

F^q: the finite field of q elements.

P²: projective plane over F^q.

P² ⊃ C: a curve of degree d over F^q without an F^q-linear component.

N_q(C): the number of F^q points of C.

i.e., Let F (X, Y, Z) = 0 be an equation of C over F^q. Then

C(F^q) = {(α, β, γ) ∈ P²(F^q)|F (α, β, γ) = 0}. N_q(C) = ^#C(F^q)

2 Sziklai Conjecture

P. Sziklai, A bound on the number of points of a plane curve, Finite Fields Appl. 14 (2008) 41–43

Conjecture N_q(C) ≤ (d − 1)q + 1

A counter example d = q = 4

K : X⁴ + Y ⁴ + Z⁴ + X²Y ² + Y ²Z² + Z²X² + X²Y Z + XY ²Z + XY Z² = 0

N₄(K) = 14(> (4 − 1)4 + 1)

3 Modification

Conjecture

Unless C is a curve over F⁴ which is projectively equivalent to K over F⁴, we might have

N_q(C) ≤ (d − 1)q + 1.

An evidence(Homma and Kim, FFA 15 (2009), 468-474.)

C: of degree 4 over F⁴. ⇒ N₄(C) ≤ 14, and if N₄(C) = 14, then C is projectively equivalent to the curve K over F⁴.

4 ComparisonWithKnownBounds

S(d) = (d − 1)q + 1 (Sziklai’s hypothetical bound) N_q(C) ≤ T = q² + q + 1 (for any C)

N_q(C) ≤ HW (d) = q + 1 + (d − 1)(d − 2)√ q (for an irreducible C)

N_q(C) ≤ SV (d) = ¹

2 d(d + q − 1) (for an irreducible classical C)

N_q(C) = HV (d) = d(d + q − 1)

(for an irreducible nonclassical nonsingular C) T ≤ S(d) ⇔ q + 2 ≤ d

HW (d) ≤ S(d) ⇔ 2 ≤ d ≤ √q + 1

max{SV (d), HV (d)} ≤ S(d) ⇔ 2 ≤ d ≤ q − 1

5 Main Theorem

Theorem For a plane curve over F^q of degree d = q or q + 1 without an F^q-linear component, the modified Sziklai conjecture holds true.

Proof: Homma-Kim;“FFA 15 (2009), 468-474”

and “arXiv:0907.1325”

Corollary If C is a nonsingular plane curve of degree d over F^q, then N_q(C) ≤ (d − 1)q + 1 unless C ' K over F⁴.

Muito Obrigado!