We construct examples of curves defined over the finite field F q 6 which are covered by the GK-curve. Thus such curves are maximal over F q 6 although they cannot be covered by the Hermitian curve for q > 2. We also give examples of maximal curves that cannot be Galois covered by the Hermitian curve over the finite field F q 2n with n > 3 odd and q > 2. We… (More)
We show that a maximal curve over F q 2 defined by the affine equation y n = f (x), where f (x) ∈ F q 2 [x] has degree coprime to n, is such that n is a divisor of q + 1 if and only if f (x) has a root in F q 2. In this case, all the roots of f (x) belong to F q 2 ; cf. we characterize certain maximal curves defined by equations of type y n = x m + x over… (More)
We characterize certain maximal curves over finite fields whose plane models are of Hurwitz type, namely x m y a + y n + x b = 0. We also consider maximal hyperelliptic curves of maximal genus. Finally, we discuss maximal curves of type y q + y = x m via class field theory.
Let C be a projective, geometrically irreducible and non-singular algebraic curve defined over the finite field F q 2 with q 2 elements. We say that C is maximal over F q 2 if the number of its rational points attains the Hasse-Weil's upper bound: #C(F q 2) = 1 + q 2 + 2gq. Here we consider maximal Picard curves over a finite field with q 2 elements of… (More)
For a field k, we consider the following category: objects: abelian varieties over k; morphisms: Mor(A, B) := Hom(A, B) ⊗ Q. This is called the category of abelian varieties up to isogeny, Isab(k), over k because two abelian varieties become isomorphic in Isab(k) if and only if they are isogenous. It is a Q-linear category (i.e., it is additive and the… (More)