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

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 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 is the symplectic group Sp2g(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.