137 research outputs found

    Shalika germs for tamely ramified elements in GLnGL_n

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    Degenerating the action of the elliptic Hall algebra on the Fock space, we give a combinatorial formula for the Shalika germs of tamely ramified regular semisimple elements γ\gamma of GLnGL_n over a nonarchimedean local field. As a byproduct, we compute the weight polynomials of affine Springer fibers in type A and orbital integrals of tamely ramified regular semisimple elements. We conjecture that the Shalika germs of γ\gamma correspond to residues of torus localization weights of a certain quasi-coherent sheaf Fγ\mathcal{F}_\gamma on the Hilbert scheme of points on A2\mathbb{A}^2, thereby finding a geometric interpretation for them. As corollaries, we obtain the polynomiality in qq of point-counts of compactified Jacobians of planar curves, as well as a virtual version of the Cherednik-Danilenko conjecture on their Betti numbers. Our results also provide further evidence for the ORS conjecture relating compactified Jacobians and HOMFLY-PT invariants of algebraic knots.Comment: 47 pages, added clarifications on the unramified case and an application to components of affine Springer fibers, fixed typos and reference

    Cycles of Quadratic Polynomials and Rational Points on a Genus-Two Curve

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    It has been conjectured that for NN sufficiently large, there are no quadratic polynomials in Q[z]\bold Q[z] with rational periodic points of period NN. Morton proved there were none with N=4N=4, by showing that the genus~22 algebraic curve that classifies periodic points of period~4 is birational to X1(16)X_1(16), whose rational points had been previously computed. We prove there are none with N=5N=5. Here the relevant curve has genus~1414, but it has a genus~22 quotient, whose rational points we compute by performing a~22-descent on its Jacobian and applying a refinement of the method of Chabauty and Coleman. We hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves. We also describe the three possible GalQ_{\bold Q}-stable 55-cycles, and show that there exist GalQ_{\bold Q}-stable NN-cycles for infinitely many NN. Furthermore, we answer a question of Morton by showing that the genus~1414 curve and its quotient are not modular. Finally, we mention some partial results for N=6N=6

    Simultaneous pp-orderings and minimising volumes in number fields

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    In the paper "On the interpolation of integer-valued polynomials" (Journal of Number Theory 133 (2013), pp. 4224--4232.) V. Volkov and F. Petrov consider the problem of existence of the so-called nn-universal sets (related to simultaneous pp-orderings of Bhargava) in the ring of Gaussian integers. We extend their results to arbitrary imaginary quadratic number fields and prove an existence theorem that provides a strong counterexample to a conjecture of Volkov-Petrov on minimal cardinality of nn-universal sets. Along the way, we discover a link with Euler-Kronecker constants and prove a lower bound on Euler-Kronecker constants which is of the same order of magnitude as the one obtained by Ihara.Comment: new version, substantial corrections in section 6, will appear in Journal of Number Theor

    Nilpoten Conjugacy Classes of Reductive p-adic Lie Algebras and Definability in Pas's Language

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    We will study the following question: Are nilpotent conjugacy classes of reductive Lie algebras over p-adic fields definable by a formula in Pas's language. We answer in the affirmative in three cases: special orthogonal Lie algebras so(n) for n odd, special linear Lie algebra sl(3) and the exceptional Lie algebra G2 over p-adic fields. The nilpotent conjugacy classes in these three cases have been parameterized by Waldspurger (so(n)) and S. DeBacker(sl(3), G2). For sl (3) and G2 we are required to extend Pas's language by a finite number of symbols

    Tropical totally positive matrices

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    We investigate the tropical analogues of totally positive and totally nonnegative matrices. These arise when considering the images by the nonarchimedean valuation of the corresponding classes of matrices over a real nonarchimedean valued field, like the field of real Puiseux series. We show that the nonarchimedean valuation sends the totally positive matrices precisely to the Monge matrices. This leads to explicit polyhedral representations of the tropical analogues of totally positive and totally nonnegative matrices. We also show that tropical totally nonnegative matrices with a finite permanent can be factorized in terms of elementary matrices. We finally determine the eigenvalues of tropical totally nonnegative matrices, and relate them with the eigenvalues of totally nonnegative matrices over nonarchimedean fields.Comment: The first author has been partially supported by the PGMO Program of FMJH and EDF, and by the MALTHY Project of the ANR Program. The second author is sported by the French Chateaubriand grant and INRIA postdoctoral fellowshi

    Complex Multiplication of Exactly Solvable Calabi-Yau Varieties

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    We propose a conceptual framework that leads to an abstract characterization for the exact solvability of Calabi-Yau varieties in terms of abelian varieties with complex multiplication. The abelian manifolds are derived from the cohomology of the Calabi-Yau manifold, and the conformal field theoretic quantities of the underlying string emerge from the number theoretic structure induced on the varieties by the complex multiplication symmetry. The geometric structure that provides a conceptual interpretation of the relation between geometry and the conformal field theory is discrete, and turns out to be given by the torsion points on the abelian varieties.Comment: 44 page
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