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THE p-RANK STRATA OF THE MODULI SPACE OF HYPERELLIPTIC CURVES JEFFREY D. ACHTER AND RACHEL PRIES
 

Summary: THE p-RANK STRATA OF THE MODULI SPACE OF HYPERELLIPTIC CURVES
JEFFREY D. ACHTER AND RACHEL PRIES
ABSTRACT. We prove results about the intersection of the p-rank strata and the boundary of the
moduli space of hyperelliptic curves in characteristic p 3. This yields a strong technique that al-
lows us to analyze the stratum H
f
g of hyperelliptic curves of genus g and p-rank f. Using this, we
prove that the endomorphism ring of the Jacobian of a generic hyperelliptic curve of genus g and
p-rank f is isomorphic to Z if g 4. Furthermore, we prove that the Z/ -monodromy of every irre-
ducible component of H
f
g is the symplectic group Sp2g(Z/ ) if g 4 or f 1, and = p is an odd
prime (with mild hypotheses on when f = 0). These results yield numerous applications about
the generic behavior of hyperelliptic curves of given genus and p-rank over finite fields, including
applications about Newton polygons, absolutely simple Jacobians, class groups and zeta functions.
1. INTRODUCTION
Suppose C is a smooth connected projective hyperelliptic curve of genus g 1 over an alge-
braically closed field k of characteristic p 3. The Jacobian Pic0
(C) is a principally polarized
abelian variety of dimension g. The number of physical p-torsion points of Pic0

  

Source: Achter, Jeff - Department of Mathematics, Colorado State University

 

Collections: Environmental Sciences and Ecology; Mathematics