CURVES OVER FINITE FIELDS

The systematic study of curves with many rational points was initiated in the1980s by the fundamental work of J.P. Serre. Ever since, the topic has been under intensiveinvestigation both from theoretical and applicative points of view. Elementary tools were gradually replaced by more and more sophisticated techniques, as testified by the numerous contributions inthe literature. A deep theory, where many important results find their origins in Serre's pioneeringwork, grew up.

Automorphism groups. Let Aut(C) be the K-automorphism group, where K is the algebraic closure of the field Fq, of a projective, non-singular, geometrically irreducible, algebraic curve C of genus g2. It is well known that Aut(C) is finite and that the classical Hurwitz bound |Aut(C)| ≤ 84(g 1) holds provided that p does not divide |Aut(C)|. If p divides |Aut(C)| then the curve C may happen to have much larger K-automorphism group compared to its genus. This was first pointed out by Roquette. Later on, Stichtenoth proved that if

|Aut(C)| ≥ 16g^4, (1)

then C is birational equivalent to a Hermitian curve H(n), that is, to a non-singular plane curve with affine equation

Y^n+YX^n+1 = 0,

for some n = p^h 3.  The curves C with |Aut(C)| ≥ 8g^3 were classified by Henn. A corollary of Henn’s classification is a substantial improvement of (1): if

|Aut(C)| > 16g^3 + 24g^2 + g, (2)

then C is birationally equivalent to a Hermitian curve.

In [2013/1] we improve the bound (2) in the case where C is a non-singular plane curve.

Maximal curves. The number of rational places of an irreducible curve defined over a finite field Fq is bounded by the celebrated Hass-Weil Theorem. A curve attaining the upper bound is called a maximal curve. In general, maximal curves have large automorphism group and can be used to construct AG codes with good parameters. Maximal curves exist only for fields of square orders, and maximal curves C satisfy

|C(Fq^2)| = q^2 + 1 + 2qg,

where g denotes the genus of the curve C and C(Fq^2) denotes the set of rational places over the field Fq^2. It is sometimes attributed to Serre that any curve Fq^2-covered by the Hermitian curve Hq+1 : y^(q+1) = x^q + x is also Fq^2-maximal.

By a criterion going back to Tate and explicitly pointed out by Lachaud, a curve defined over Fp with genus g is Fp^2-maximal if and only if its Jacobian is Fp^2-isogenous to the g-th power of an Fp^2-maximal elliptic curve.

In [2020/7] explicit equations for algebraic curves with genus 4, 5, and 10 already studied in characteristic zero, are analyzed in positive characteristic p and  it is proved that they are either maximal or minimal over the finite field with p^2 elements for infinitely many p’s, via the  investigation of their Jacobian decomposition.

In [2021/12] explicit examples of maximal and minimal curves of Artin-Schreier type over finite fields in odd characteristic are presented which are closely related to quadratic forms from Fq^n to Fq

In [2021/14] it is shown that every Fp^2-maximal curve X of genus g ≥ 2 with |Aut(X)|>84(g-1) is Galois-covered by Hq+1. The hypothesis on |Aut(X)| is sharp, since there exists an Fp^2-maximal curve X for q = 71 of genus g = 7 with |Aut(X)|=84(7-1) which is not Galois-covered by the Hermitian curve H72.