We compute the Z/ℓ and Z ℓ monodromy of every irreducible component of the moduli spaces of hyperelliptic and trielliptic curves. In particular, we provide a proof that the Z/ℓ monodromy of the moduli space of hyperelliptic curves of genus g is the symplectic group Sp 2g (Z/ℓ). We prove that the Z/ℓ monodromy of the moduli space of trielliptic curves with signature (r, s) is the special unitary group SU (r,s) (Z/ℓ ⊗ Z[ζ 3 ]).

ABSTRACT. We determine the Z/ℓ-monodromy and Z ℓ -monodromy of every irreducible component of the stratum M f g of curves of genus g and p-rank f in characteristic p. In particular, we prove that the Z/ℓ-monodromy of every component of M f g is the symplectic group Sp 2g (Z/ℓ) if g ≥ 3 and if ℓ is a prime distinct from p. The method involves results on the intersection of M f g with the boundary of M g . We give applications to the generic behavior of automorphism groups, Jacobians, class groups, and zeta functions of curves of given genus and p-rank.

Suppose X is a hyperelliptic curve of genus g defined over an algebraically closed field k of characteristic p = 2. We prove that the de Rham cohomology of X decomposes into pieces indexed by the branch points of the hyperelliptic cover. This allows us to compute the isomorphism class of the 2-torsion group scheme J X [2] of the Jacobian of X in terms of the Ekedahl-Oort type. The interesting feature is that J X [2] depends only on some discrete invariants of X , namely, on the ramification invariants associated with the branch points. We give a complete classification of the group schemes that occur as the 2-torsion group schemes of Jacobians of hyperelliptic k-curves of arbitrary genus, showing that only relatively few of the possible group schemes actually do occur. Theorem 1.3. Let X be a hyperelliptic k-curve with affine equation y 2 − y = f (x) defined over an algebraically closed field of characteristic 2 as described in Notation 1.1. Then the 2-torsion group scheme of the Jacobian variety of X isand the a-number of X is a X = (g + 1 − #{α ∈ B | d α ≡ 1 mod 4})/2.

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/ℓ-monodromy of every irreducible component of H f g is the symplectic group Sp 2g (Z/ℓ) if g ≥ 4 or f ≥ 1, 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.1

We study a moduli space A S g for Artin-Schreier curves of genus g over an algebraically closed field k of characteristic p. We study the stratification of A S g by p-rank into strata A S g.s of Artin-Schreier curves of genus g with p-rank exactly s. We enumerate the irreducible components of A S g,s and find their dimensions. As an application, when p = 2, we prove that every irreducible component of the moduli space of hyperelliptic k-curves with genus g and 2-rank s has dimension g − 1 + s. We also determine all pairs (p, g) for which A S g is irreducible. Finally, we study deformations of Artin-Schreier curves with varying p-rank.