790 research outputs found
Formalized linear algebra over Elementary Divisor Rings in Coq
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 and
well-founded strict divisibility
Elliptic curves and continued fractions
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
We study the Whittaker coefficients of the minimal parabolic Eisenstein
series on the -fold cover of the split odd orthogonal group . If
the degree of the cover is odd, then Beineke, Brubaker and Frechette have
conjectured that the -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 sufficiently
large, thereby confirming a conjecture of Brubaker, Bump and Friedberg.Comment: 62 page
The Spectrum of an Adelic Markov Operator
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
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