5 research outputs found
Explicit construction of a plane sextic model for genus-five Howe curves, II
A Howe curve is defined as the normalization of the fiber product over a
projective line of two hyperelliptic curves. Howe curves are very useful to
produce important classes of curves over fields of positive characteristic,
e.g., maximal, superspecial, or supersingular ones. Determining their feasible
equations explicitly is a basic problem, and it has been solved in the
hyperelliptic case and in the non-hyperelliptic case with genus not greater
than . In this paper, we construct an explicit plane sextic model for
non-hyperelliptic Howe curves of genus . We also determine the number and
type of singularities on our sextic model, and prove that the singularities are
generically double points. Our results together with Moriya-Kudo's recent
ones imply that for each , there exists a non-hyperellptic
curve of genus with such that
its associated plane sextic has double points.Comment: 19 pages, Comments are welcome
Explicit construction of a plane sextic model for genus-five Howe curves, I
In the past several years, Howe curves have been studied actively in the
field of algebraic curves over fields of positive characteristic. Here, a Howe
curve is defined as the desingularization of the fiber product over a
projective line of two hyperelliptic curves. In this paper, we construct an
explicit plane sextic model for non-hyperelliptic Howe curves of genus five. We
also determine singularities of our sextic model.Comment: Comments are welcome
New examples of maximal curves with low genus
We investigate the Jacobian decomposition of some algebraic curves over
finite fields with genus , and . As a corollary, explicit equations
for curves that are either maximal or minimal over the finite field with
elements are obtained for infinitely many 's. Lists of small 's for which
maximality holds are provided. In some cases we describe the automorphism group
of the curve