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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
allows 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
irreducible component of H
f
g is the symplectic group Sp2g(Z/ ) if g 3, 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.
[MSC 2000]14H10, 11G20, 14D05
Keywords: p-rank, moduli, hyperelliptic, Jacobian, monodromy
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

  

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

 

Collections: Environmental Sciences and Ecology; Mathematics