The field contains the infinite sequence of uniformly bounded continued fractions and similar patterns can be found in for any d>0. This paper studies the broader structure underlying these patterns, and develops related results and conjectures for closed geodesics on arithmetic manifolds, packing constants of ideals, class numbers and heights.

We prove that for any positive integers x,d and k with gcd (x,d)=1 and 3<k<35, the product x(x+d)⋯(x+(k−1)d) cannot be a perfect power. This yields a considerable extension of previous results of Győry et al. and Bennett et al., which covered the cases where k≤11. We also establish more general theorems for the case where x can also be a negative integer and where the product yields an almost perfect power. As in the proofs of the earlier theorems, for fixed k we reduce the problem to systems of ternary equations. However, our results do not follow as a mere computational sharpening of the approach utilized previously; instead, they require the introduction of fundamentally new ideas. For k>11, a large number of new ternary equations arise, which we solve by combining the Frey curve and Galois representation approach with local and cyclotomic considerations. Furthermore, the number of systems of equations grows so rapidly with k that, in contrast with the previous proofs, it is practically impossible to handle the various cases in the usual manner. The main novelty of this paper lies in the development of an algorithm for our proofs, which enables us to use a computer. We apply an efficient, iterated combination of our procedure for solving the new ternary equations that arise with several sieves based on the ternary equations already solved. In this way, we are able to exclude the solvability of the enormous number of systems of equations under consideration. Our general algorithm seems to work for larger values of k as well, although there is, of course, a computational time constraint.

The aim of this article is to classify two-dimensional split trianguline representations of p-adic fields. This is a generalization of a result of Colmez who classified two-dimensional split trianguline representations of for p≠2 by using (φ,Γ)-modules over a Robba ring. In this article, for any prime p and for any p-adic field K, we classify two-dimensional split trianguline representations of using B-pairs as defined by Berger.

This paper provides congruences between unstable and stable automorphic forms for the symplectic similitude group GSp(4). More precisely, we raise the level of certain CAP representations Π arising from classical modular forms. We first transfer Π to π on a suitable inner form G; this is achieved by θ-lifting. For π, we prove a precise level-raising result that is inspired by the work of Bellaiche and Clozel and which relies on computations of Schmidt. We thus obtain a congruent to π, with a local component that is irreducibly induced from an unramified twist of the Steinberg representation of the Klingen parabolic. To transfer back to GSp(4), we use Arthur’s stable trace formula. Since has a local component of the above type, all endoscopic error terms vanish. Indeed, by results due to Weissauer, we only need to show that such a component does not participate in the θ-correspondence with any GO(4); this is an exercise in using Kudla’s filtration of the Jacquet modules of the Weil representation. We therefore obtain a cuspidal automorphic representation of GSp(4), congruent to Π, which is neither CAP nor endoscopic. It is crucial for our application that we can arrange for to have vectors fixed by the non-special maximal compact subgroups at all primes dividing N. Since G is necessarily ramified at some prime r, we have to show a non-special analogue of the fundamental lemma at r. Finally, we give an application of our main result to the Bloch–Kato conjecture, assuming a conjecture of Skinner and Urban on the rank of the monodromy operators at the primes dividing N.

For a fixed parabolic subalgebra 𝔭 of we prove that the centre of the principal block 𝒪0𝔭 of the parabolic category 𝒪 is naturally isomorphic to the cohomology ring H*(ℬ𝔭) of the corresponding Springer fibre. We give a diagrammatic description of 𝒪0𝔭 for maximal parabolic 𝔭 and give an explicit isomorphism to Braden’s description of the category PervB(G(k,n)) of Schubert-constructible perverse sheaves on Grassmannians. As a consequence Khovanov’s algebra ℋn is realised as the endomorphism ring of some object from PervB(G(n,n)) which corresponds under localisation and the Riemann–Hilbert correspondence to a full projective–injective module in the corresponding category 𝒪0𝔭. From there one can deduce that Khovanov’s tangle invariants are obtained from the more general functorial invariants in [C. Stroppel, Categorification of the Temperley Lieb category, tangles, and cobordisms via projective functors, Duke Math. J. 126(3) (2005), 547–596] by restriction.

The face ring of a homology manifold (without boundary) modulo a generic system of parameters is studied. Its socle is computed and it is verified that a particular quotient of this ring is Gorenstein. This fact is used to prove that the algebraic g-conjecture for spheres implies all enumerative consequences of its far-reaching generalization (due to Kalai) to manifolds. A special case of Kalai’s conjecture is established for homology manifolds that have a codimension-two face whose link contains many vertices.

We define and study virtual representation spaces for vectors having both positive and negative dimensions at the vertices of a quiver without oriented cycles. We consider the natural semi-invariants on these spaces which we call virtual semi-invariants and prove that they satisfy the three basic theorems: the first fundamental theorem, the saturation theorem and the canonical decomposition theorem. In the special case of Dynkin quivers with n vertices, this gives the fundamental interrelationship between supports of the semi-invariants and the tilting triangulation of the (n−1)-sphere.

We investigate cluster-tilting objects (and subcategories) in triangulated 2-Calabi–Yau and related categories. In particular, we construct a new class of such categories related to preprojective algebras of non-Dynkin quivers associated with elements in the Coxeter group. This class of 2-Calabi–Yau categories contains, as special cases, the cluster categories and the stable categories of preprojective algebras of Dynkin graphs. For these 2-Calabi–Yau categories, we construct cluster-tilting objects associated with each reduced expression. The associated quiver is described in terms of the reduced expression. Motivated by the theory of cluster algebras, we formulate the notions of (weak) cluster structure and substructure, and give several illustrations of these concepts. We discuss connections with cluster algebras and subcluster algebras related to unipotent groups, in both the Dynkin and non-Dynkin cases.