The p-rank strata of the moduli space of hyperelliptic curves

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 g f 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 / ℓ -monodromy of every irreducible component of H g f is the symplectic group Sp 2 g ( Z / ℓ ) if g ⩾ 3 , and ℓ ≠ p is an odd prime (with mild hypotheses on ℓ 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.