Tables of curves with many points

Abstract

These tables record results on curves with many points over finite fields. For relatively small genus (0 ≤ g ≤ 50) and q a small power of 2 or 3 we give in two tables the best presently known bounds for Nq(g), the maximum number of rational points on a smooth absolutely irreducible projective curve of genus g over a field Fq of cardinality q. In additional tables we list for a given pair (g, q) the type of construction of the best curve so far, and we give a reference to the literature where such a curve can be found.

Quotient curves of the Suzuki curve

“…Our contribution consists in proving the existence of non-tame quotient curves of S; for some of these curves we also provide a plane equation. A concrete application of our results provides new entries in the tables [9]: let q = 32 and r = 5; from Theorems 5.1(2) and 6.10(2) we have N 32 (24) ≥ 225 and N 32 (10) ≥ 113 respectively; cf. Section 7.…”

“…This value is in the interval from which the entries of the tables of curves with many rational points are taken for g ≤ 50, q ≤ 128 in van der Geer and van der Vlugt tables [9]. In Sections 4 and 5 we obtain an exhaustive list of tame quotient curves of S, namely quotients arising from subgroups of Aut(S) of odd order: indeed, we compute the genus as well as exhibit a plane model for such curves.…”

“…Let N q (g) be the maximum number of K-rational points that a curve over K of genus g > 0 can have. Our references for this section are [8] and [9].…”

“…We shall investigate the left hand side inequality in (7.1) with q = 2q 2 0 , and the genus g = g of a quotient curve S of the Suzuki curve S. Note that by (3.1), # S(K) = q + 2q 0 g + 1 ≥ ⌊b/ √ 2⌋; thus we might expect "many" K-rational points on the curve S (this is the case when q ≤ 128 and g ≤ 50 according to [9,Sect. 1]).…”

On the number of rational points of a plane algebraic curve

“…Nevertheless, formulas for N in terms of d and some other projective invariants of X are only known for few curves, see [5], [12]. For instance, N = q + 1 + (d − 1)(d − 2) √ q for the Fermat curve X d + Y d + 1 = 0 with √ q ≡ −1 (mod d), and q squared, but this formula does not hold true for q = q m 0 with m > 2 and q m−1 0 + .…”