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.

In this paper, we show that there exist families of curves (defined over an algebraically closed field k of characteristic p > 2) whose Jacobians have interesting p-torsion. For example, for every 0 ≤ f ≤ g, we find the dimension of the locus of hyperelliptic curves of genus g with p-rank at most f . We also produce families of curves so that the p-torsion of the Jacobian of each fibre contains multiple copies of the group scheme α p . The method is to study curves which admit an action by (Z/2) n so that the quotient is a projective line. As a result, some of these families intersect the hyperelliptic locus H g .

We survey results and open questions about the p-ranks and Newton polygons of Jacobians of curves in positive characteristic p. We prove some geometric results about the p-rank stratification of the moduli space of (hyperelliptic) curves. For example, if 0 ≤ f ≤ g − 1, we prove that every component of the p-rank f + 1 stratum of M g contains a component of the p-rank f stratum in its closure. We prove that the p-rank f stratum of M g is connected. For all primes p and all g ≥ 4, we demonstrate the existence of a Jacobian of a smooth curve, defined over F p , whose Newton polygon has slopes {0, 1 4 , 3 4 , 1}. We include partial results about the generic Newton polygons of curves of given genus g and p-rank f .

In this paper, I investigate wildly ramified G -Galois covers of curves [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] branched at exactly one point over an algebraically closed field k of characteristic p . I answer a question of M. Raynaud by showing that proper families of such covers of a twisted projective line are isotrivial. The method is to construct an affine moduli space for covers whose inertia group is of the form [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]. There are two other applications of this space in the case that [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /]. The first uses formal patching to compute the dimension of the space of nonisotrivial deformations of φ in terms of the lower jump of the filtration of higher inertia groups. The second gives necessary criteria for good reduction of families of such covers. These results will be used in a future paper to prove the existence of such covers φ with specified ramification data.