 
Summary: THE pRANK STRATA OF THE MODULI SPACE OF HYPERELLIPTIC CURVES
JEFFREY D. ACHTER AND RACHEL PRIES
ABSTRACT. We prove results about the intersection of the prank 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 prank f. Using this, we
prove that the endomorphism ring of the Jacobian of a generic hyperelliptic curve of genus g and
prank 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 prank 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 ptorsion points of Pic0
