Plane maximal curves

Abstract

Some new results on plane F q 2 -maximal curves are stated and proved. By [32], the degree d of a plane F q 2 -maximal curve is less than or equal to q + 1 and equality holds if and only if the curve is F q 2 -isomorphic to the Hermitian curve. We show that d ≤ q + 1 can be improved to d ≤ (q + 2)/2 apart from the case d = q + 1 or q ≤ 5. This upper bound turns out to be sharp for q odd. In [4] it was pointed out that some Hurwitz curves are plane F q 2 -maximal curves. Here we prove that (1.3) is the necessary and sufficient condition for a Hurwitz curve to be F q 2 -maximal. We also show that this criterium holds true for the F q 2 -maximality of a wider family of curves.

Certain maximal curves and Cartier operators

“…We show that C(m) is maximal over F q 2 if and only if m divides q + 1. This generalizes [1,Corollary 3.5] which deals with the particular case when m belongs to the set of values of the polynomial T 2 − T + 1, and it also generalizes [9, Corollary 1] which deals with the case of q = p prime (see Remark 4.3).…”

“…The main tool is the Cartier operator, which is a nilpotent operator in the case of maximal (or minimal) curves over finite fields. We give generalizations of results from [1], [7], [9], [22] and [23].…”

“…As shown by J.-P. Serre, if there is a morphism defined over the field k between two curves f : C → D, then the L-polynomial of D divides the one of C. Hence a subcover D of a maximal curve C is also maximal (see [10]). So one way to construct explicit maximal curves is to find equations for subcovers of the Hermitian curve (see [1] and [4]). …”

The supersingularity of Hurwitz curves

“…A priori, if H n,ℓ is supersingular (or maximal or minimal) over F p then F m may not be because it has more normalized Weil numbers. Note that the key property used in [AKT01] is the existence of some positive integer j such that p j ≡ −1 mod m.…”

“…The curved arrows show that under appropriate conditions a Hurwitz or Fermat curve is supersingular if and only if it is minimal over some field extension. Corollaries 4.6 and 4.7 are under the condition that gcd(n, ℓ) = 1, while [AKT01] and Corollary 4.11 are under the condition that ℓ = 1, or gcd(n, ℓ) = 1 and m is prime.…”