393 research outputs found
Non-commutative Hilbert modular symbols
The main goal of this paper is to construct non-commutative Hilbert modular
symbols. However, we also construct commutative Hilbert modular symbols. Both
the commutative and the non-commutative Hilbert modular symbols are
generalizations of Manin's classical and non-commutative modular symbols. We
prove that many cases of (non-)commutative Hilbert modular symbols are periods
in the sense on Kontsevich-Zagier. Hecke operators act naturally on them.
Manin defines the non-commutative modilar symbol in terms of iterated path
integrals. In order to define non-commutative Hilbert modular symbols, we use a
generalization of iterated path integrals to higher dimensions, which we call
iterated integrals on membranes. Manin examines similarities between
non-commutative modular symbol and multiple zeta values both in terms of
infinite series and in terms of iterated path integrals. Here we examine
similarities in the formulas for non-commutative Hilbert modular symbol and
multiple Dedekind zeta values, recently defined by the author, both in terms of
infinite series and in terms of iterated integrals on membranes.Comment: 50 pages, 5 figures, substantial improvement of the article
arXiv:math/0611955 [math.NT], the portions compared to the previous version
are: Hecke operators, periods and some categorical construction
Strictly nilpotent elements and bispectral operators in the Weyl algebra
In this paper we give another characterization of the strictly nilpotent
elements in the Weyl algebra, which (apart from the polynomials) turn out to be
all bispectral operators with polynomial coefficients. This also allows to
reformulate in terms of bispectral operators the famous conjecture, that all
the endomorphisms of the Weyl algebra are automorphisms (Dixmier, Kirillov,
etc).Comment: 11 pages, to appear in Bull. Sci. Mat
Bispectral operators of prime order
The aim of this paper is to solve the bispectral problem for bispectral
operators whose order is a prime number. More precisely we give a complete list
of such bispectral operators. We use systematically the operator approach and
in particular - Dixmier ideas on the first Weyl algebra. When the order is 2
the main theorem is exactly the result of Duistermaat-Gr\"unbaum . On the other
hand our proofs seem to be simpler.Comment: 25 pages, to appear in CM
Automorphisms of algebras and Bochner`s property for discrete vector orthogonal polynomials
We construct new families of discrete vector orthogonal polynomials that have
the property to be eigenfunctions of some difference operator. They are
extensions of Charlier, Meixner and Kravchuk polynomial systems. The ideas
behind our approach lie in the studies of bispectral operators. We exploit
automorphisms of associative algebras which transform elementary (vector)
orthogonal polynomial systems which are eigenfunctions of a difference operator
into other systems of this type. While the extension of Charlier polynomilas is
well known it is obtained by different methods. The extension of Meixner
polynomial system is new.Comment: 14 pages. arXiv admin note: text overlap with arXiv:1512.0389
Non-abelian reciprocity laws on a Riemann surface
On a Riemann surface there are relations among the periods of holomorphic
differential forms, called Riemann's relations. If one looks carefully in
Riemann's proof, one notices that he uses iterated integrals. What I have done
in this paper is to generalize these relations to relations among generating
series of iterated integrals. Since the main result is formulated in terms of
generating series, it gives infinitely many relations - one for each
coefficient of the generating series. The lower order terms give the well known
classical relations. The new result is reciprocity for the higher degree terms,
which give non-trivial relations among iterated integrals on a Riemann surface.
As an application we refine the definition of Manin's noncommutative modular
symbol in order to include Eisenstein series. Finally, we have to point out
that this paper contains some constructions needed for multidimensional
reciprocity laws like a refinement of one of the Kato-Parshin reciprocity laws.Comment: 21 pages; Submitted to International Mathematics Research Notices
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