144 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
Generic Newton polygons for curves of given p-rank
We survey results and open questions about the -ranks and Newton polygons
of Jacobians of curves in positive characteristic . We prove some geometric
results about the -rank stratification of the moduli space of
(hyperelliptic) curves. For example, if , we prove that
every component of the -rank stratum of contains a
component of the -rank stratum in its closure. We prove that the
-rank stratum of is connected. For all primes
and all , we demonstrate the existence of a Jacobian of a smooth
curve, defined over , whose Newton polygon has slopes
. We include partial results about the
generic Newton polygons of curves of given genus and -rank .Comment: 16 pages, to appear in Algebraic Curves and Finite Fields: Codes,
Cryptography, and other emergent applications, edited by H. Niederreiter, A.
Ostafe, D. Panario, and A. Winterho
- …