790 research outputs found

    Formalized linear algebra over Elementary Divisor Rings in Coq

    Get PDF
    This paper presents a Coq formalization of linear algebra over elementary divisor rings, that is, rings where every matrix is equivalent to a matrix in Smith normal form. The main results are the formalization that these rings support essential operations of linear algebra, the classification theorem of finitely presented modules over such rings and the uniqueness of the Smith normal form up to multiplication by units. We present formally verified algorithms computing this normal form on a variety of coefficient structures including Euclidean domains and constructive principal ideal domains. We also study different ways to extend B\'ezout domains in order to be able to compute the Smith normal form of matrices. The extensions we consider are: adequacy (i.e. the existence of a gdco operation), Krull dimension ≤1\leq 1 and well-founded strict divisibility

    Elliptic curves and continued fractions

    Full text link
    We detail the continued fraction expansion of the square root of the general monic quartic polynomial, noting that each line of the expansion corresponds to addition of the divisor at infinity. We analyse the data yielded by the general expansion. In that way we obtain `elliptic sequences' satisfying Somos relations. I mention several new results on such sequences. The paper includes a detailed `reminder exposition' on continued fractions of quadratic irrationals in function fields.Comment: v1; final -- I hop

    Eisenstein Series on Covers of Odd Orthogonal Groups

    Full text link
    We study the Whittaker coefficients of the minimal parabolic Eisenstein series on the nn-fold cover of the split odd orthogonal group SO2r+1SO_{2r+1}. If the degree of the cover is odd, then Beineke, Brubaker and Frechette have conjectured that the pp-power contributions to the Whittaker coefficients may be computed using the theory of crystal graphs of type C, by attaching to each path component a Gauss sum or a degenerate Gauss sum depending on the fine structure of the path. We establish their conjecture using a combination of automorphic and combinatorial-representation-theoretic methods. Surprisingly, we must make use of the type A theory, and the two different crystal graph descriptions of Brubaker, Bump and Friedberg available for type A based on different factorizations of the long word into simple reflections. We also establish a formula for the Whittaker coefficients in the even degree cover case, again based on crystal graphs of type C. As a further consequence, we establish a Lie-theoretic description of the coefficients for nn sufficiently large, thereby confirming a conjecture of Brubaker, Bump and Friedberg.Comment: 62 page

    The Spectrum of an Adelic Markov Operator

    Full text link
    With the help of the representation of SL(2,Z) on the rank two free module over the integer adeles, we define the transition operator of a Markov chain. The real component of its spectrum exhibits a gap, whereas the non-real component forms a circle of radius 1/\sqrt{2}.Comment: 38 pages, 5 figure
    • …
    corecore