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 .

Let G be a finite, connected graph. An arithmetical structure on G is a pair of positive integer vectors d, r such that (diag(d) − A)r = 0, where A is the adjacency matrix of G. We investigate the combinatorics of arithmetical structures on path and cycle graphs, as well as the associated critical groups (the torsion part of the cokernels of the matrices (diag(d)−A)). For paths, we prove that arithmetical structures are enumerated by the Catalan numbers, and we obtain refined enumeration results related to ballot sequences. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients 2n−1 n−1 , and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles.

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An arithmetical structure on a finite, connected graph G is a pair of vectors (d, r) with positive integer entries for which (diag(d) − A)r = 0, where A is the adjacency matrix of G and where the entries of r have no common factor. The critical group of an arithmetical structure is the torsion part of the cokernel of (diag(d) − A). In this paper, we study arithmetical structures and their critical groups on bidents, which are graphs consisting of a path with two "prongs" at one end. We give a process for determining the number of arithmetical structures on the bident with n vertices and show that this number grows at the same rate as the Catalan numbers as n increases. We also completely characterize the groups that occur as critical groups of arithmetical structures on bidents.Let G be a finite, connected graph with n vertices, and let A be the adjacency matrix of G. An arithmetical structure on G is given by a pair of vectors (d, r) ∈ (Z >0 ) n × (Z >0 ) n for which the

In this paper we consider the critical group of finite connected graphs which admit harmonic actions by the dihedral group D n . In particular, we show that if the orbits of the D n -action all have either n or 2n points then the critical group of such a graph can be decomposed in terms of the critical groups of the quotients of the graph by certain subgroups of the automorphism group. This is analogous to a theorem of Kani and Rosen which decomposes the Jacobians of algebraic curves with a D n -action. arXiv:1304.6011v1 [math.CO]