119 research outputs found
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
The a-number of hyperelliptic curves
It is known that for a smooth hyperelliptic curve to have a large -number,
the genus must be small relative to the characteristic of the field, ,
over which the curve is defined. It was proven by Elkin that for a genus
hyperelliptic curve to have , the genus is bounded by
. In this paper, we show that this bound can be lowered to . 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 and .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-
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