Superspecial rank of supersingular abelian varieties and Jacobians

An abelian variety defined over an algebraically closed field k of positive characteristic is supersingular if it is isogenous to a product of supersingular elliptic curves and is superspecial if it is isomorphic to a product of supersingular elliptic curves. In this paper, the superspecial condition is generalized by defining the superspecial rank of an abelian variety, which is an invariant of its p-torsion. The main results in this paper are about the superspecial rank of supersingular abelian varieties and Jacobians of curves. For example, it turns out that the superspecial rank determines information about the decomposition of a supersingular abelian variety up to isomorphism; namely it is a bound for the maximal number of supersingular elliptic curves appearing in such a decomposition.Comment: V2: New coauthor, major rewrit

Counting isomorphism classes of superspecial curves (Theory and Applications of Supersingular Curves and Supersingular Abelian Varieties)

A superspecial curve is a (non-singular) curve over a field of positive characteristic whose Jacobian variety is isomorphic to a product of supersingular elliptic curves over the algebraic closure. It is known that for given genus and characteristic, there exist only finitely many superspecial curves, up to isomorphism over an algebraically closed field. In this article, we give a brief survey on results of counting isomorphism classes of superspecial curves. In particular, this article summarizes some recent results in the case of genera four and five, obtained by the author and S. Harashita. We also survey results obtained in a joint work with Harashita and E. W. Howe, on the enumeration of superspecial curves in a certain class of non-hyperelliptic curves of genus four

The a-number of hyperelliptic curves

It is known that for a smooth hyperelliptic curve to have a large $a$-number, the genus must be small relative to the characteristic of the field, $p>0$, over which the curve is defined. It was proven by Elkin that for a genus $g$ hyperelliptic curve $C$ to have $a_C=g-1$, the genus is bounded by $g<\frac{3p}{2}$. In this paper, we show that this bound can be lowered to $g <p$. The method of proof is to force the Cartier-Manin matrix to have rank one and examine what restrictions that places on the affine equation defining the hyperelliptic curve. We then use this bound to summarize what is known about the existence of such curves when $p=3,5$ and $7$.Comment: 7 pages. v2: revised and improved the proof of the main theorem based on suggestions from the referee. To appear in the proceedings volume of Women in Numbers Europe-

Optimal curves of genus 1,2 and 3

In this survey, we discuss the problem of the maximum number of points of curves of genus 1,2 and 3 over finite fieldsComment: 18 pages. To appear in "Publications Mathematiques de Besancon(PMB)