574 research outputs found

    Remarks on endomorphisms and rational points

    Full text link
    Let X be a variety over a number field and let f: X --> X be an "interesting" rational self-map with a fixed point q. We make some general remarks concerning the possibility of using the behaviour of f near q to produce many rational points on X. As an application, we give a simplified proof of the potential density of rational points on the variety of lines of a cubic fourfold (originally obtained by Claire Voisin and the first author in 2007).Comment: LaTeX, 22 pages. v2: some minor observations added, misprints corrected, appendix modified

    Distance-regular Cayley graphs with small valency

    Full text link
    We consider the problem of which distance-regular graphs with small valency are Cayley graphs. We determine the distance-regular Cayley graphs with valency at most 44, the Cayley graphs among the distance-regular graphs with known putative intersection arrays for valency 55, and the Cayley graphs among all distance-regular graphs with girth 33 and valency 66 or 77. We obtain that the incidence graphs of Desarguesian affine planes minus a parallel class of lines are Cayley graphs. We show that the incidence graphs of the known generalized hexagons are not Cayley graphs, and neither are some other distance-regular graphs that come from small generalized quadrangles or hexagons. Among some ``exceptional'' distance-regular graphs with small valency, we find that the Armanios-Wells graph and the Klein graph are Cayley graphs.Comment: 19 pages, 4 table

    Singer quadrangles

    Get PDF
    [no abstract available

    The quantum H3H_3 integrable system

    Full text link
    The quantum H3H_3 integrable system is a 3D system with rational potential related to the non-crystallographic root system H3H_3. It is shown that the gauge-rotated H3H_3 Hamiltonian as well as one of the integrals, when written in terms of the invariants of the Coxeter group H3H_3, is in algebraic form: it has polynomial coefficients in front of derivatives. The Hamiltonian has infinitely-many finite-dimensional invariant subspaces in polynomials, they form the infinite flag with the characteristic vector \vec \al\ =\ (1,2,3). One among possible integrals is found (of the second order) as well as its algebraic form. A hidden algebra of the H3H_3 Hamiltonian is determined. It is an infinite-dimensional, finitely-generated algebra of differential operators possessing finite-dimensional representations characterized by a generalized Gauss decomposition property. A quasi-exactly-solvable integrable generalization of the model is obtained. A discrete integrable model on the uniform lattice in a space of H3H_3-invariants "polynomially"-isospectral to the quantum H3H_3 model is defined.Comment: 32 pages, 3 figure

    Hilbert modular forms and p-adic Hodge theory

    Full text link
    We consider the p-adic Galois representation associated to a Hilbert modular form. We show the compatibility with the local Langlands correspondence at a place divising p under a certain assumption. We also prove the monodromy-weight conjecture. The prime-to-p case is established by Carayol.Comment: 45 pages: page size adjuste
    • …
    corecore