For X a curve over a field of positive characteristic, we investigate when the canonical representation of Aut(X) on H 0 (X, Ω X ) is irreducible. Any curve with an irreducible canonical representation must either be superspecial or ordinary. Having a small automorphism group is an obstruction to having irreducible canonical representation; with this motivation, the bulk of the paper is spent bounding the size of automorphism groups of superspecial and ordinary curves. After proving that all automorphisms of an F q 2 -maximal curve are defined over F q 2 , we find all superspecial curves with g > 82 having an irreducible representation. In the ordinary case, we provide a bound on the size of the automorphism group of an ordinary curve that improves on a result of Nakajima. 1 3 2 X .Remark 1.2. From our proof, we may take c(p) to be on the order of p 2 . This does not yet imply reducibility of the canonical representation, but we have the following.Remark 1.3. An unpublished result of Guralnick and Zieve [6] states that for any prime p there is a positive constant c p so that, if X is an ordinary curve of genus g > 1 over an algebraically closed field of characteristic p, the group of automorphisms of X has order bounded by c p g 8/5 . Together with the superspecial results, this would imply that for a fixed characteristic, there do not exist arbitrarily high genus curves with irreducible canonical representation. We hope that the eventual published work will give even stronger ways of characterizing ordinary curves with irreducible canonical representations.

The star graph proposed by Akers et al. (Proc Int ConfParallel Process, University Park, PA, 1987, pp. 393-400) has many advantages over the n-cube. However, it suffers from having large gaps in the possible number of vertices. The arrangement graph was proposed by Day and Tripathi (Inf Process Lett 42 (1992), 235-241) to address this issue. Since it is a generalization of the star graph, it retains many of the nice properties of the star graph. In fact, it also generalizes the alternating group graph (Jwo et al., Networks 23 (1993), 315-326). There are many different measures of structural integrity of interconnection networks. In this article, we prove results of the following type for the arrangement graph: If h(r , n, k ) vertices are deleted from the arrangement graph A n,k , the resulting graph will either be connected or have a large component and small components having at most r − 1 vertices in total. Our result is tight for r ≤ 3, and it is asymptotically tight for r ≥ 4. Moreover, we also determine the cyclic vertex-connectivity of the arrangement graph. large component and some small components with at most r −1 vertices in total. In section 2, we introduce the necessary definitions, in section 3, we examine what happens when the number of deleted vertices is at most three times the common degree, and in section 4 we examine the case of deleting linearly many vertices. DEFINITIONS AND PRELIMINARIESA graph G = (V , E) with vertex set V and edge set E is r-regular if the degree of every vertex of G is r. If W ⊂ V is a set of vertices of G, then the graph obtained by deleting the vertices of W from G will be denoted by G − W . A noncomplete graph G is r-connected if deleting any set of fewer than r vertices results in a connected graph. A complete graph with r + 1 vertices is k-connected for k ≤ r. An rregular graph is maximally connected if it is r-connected.

The star graph proposed by [1] has many advantages over the n-cube. However it suffers from having large gaps in the number of possible vertices. The (n,k)-star graph was proposed in [18] to address this issue. Since it is a generalization of the star graph, it retains many of the nice properties of the star graph. There are many different measures of structural integrity of interconnection networks. In this paper, we prove results of the following type for the (n,k)-star graph. If n + (r - 1)k - g(r) vertices are deleted from an (n,k)-star graph, the resulting graph will either be connected or has a large component and small components having at most r - 1 vertices in total. Additional results on conditional vertex connectivity and cycle connectivity will also be given.

It is well known that for any prime $p\equiv 3$ (mod $4$), the class numbers of the quadratic fields $\mathbb{Q}(\sqrt{p})$ and $\mathbb{Q}(\sqrt{-p})$, $h(p)$ and $h(-p)$ respectively, are odd. It is natural to ask whether there is a formula for $h(p)/h(-p)$ modulo powers of $2$. We show the formula $h(p) \equiv h(-p) m(p)$ (mod $16$), where $m(p)$ is an integer defined using the "negative" continued fraction expansion of $\sqrt{p}$. Our result solves a conjecture of Richard Guy.Comment: 9 pages; additional background given in introduction concerning $h(p)$ and $h(-p)$ modulo small powers of

We give a fully faithful integral model for simply connected finite complexes in terms of E ∞ \mathbb {E}_{\infty } -ring spectra and the Nikolaus–Scholze Frobenius. The key technical input is the development of a homotopy coherent Frobenius action on a certain subcategory of p p -complete E ∞ \mathbb {E}_{\infty } -rings for each prime p p . Using this, we show that the data of a simply connected finite complex X X is the data of its Spanier-Whitehead dual, as an E ∞ \mathbb {E}_{\infty } -ring, together with a trivialization of the Frobenius action after completion at each prime. In producing the above Frobenius action, we explore two ideas which may be of independent interest. The first is a more general action of Frobenius in equivariant homotopy theory; we show that a version of Quillen’s Q Q -construction acts on the ∞ \infty -category of E ∞ \mathbb {E}_{\infty } -rings with “genuine equivariant multiplication,” which we call global algebras. The second is a “pre-group-completed” variant of algebraic K K -theory which we call partial K K -theory. We develop the notion of partial K K -theory and give a computation of the partial K K -theory of F p \mathbb {F}_p up to p p -completion.