 
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
allows 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
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 prank over finite fields, including applications
about Newton polygons, absolutely simple Jacobians, class groups and zeta functions.
[MSC 2000]14H10, 11G20, 14D05
Keywords: prank, 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
