778 research outputs found
Periodic harmonic functions on lattices and points count in positive characteristic
This survey addresses pluri-periodic harmonic functions on lattices with
values in a positive characteristic field. We mention, as a motivation, the
game "Lights Out" following the work of Sutner, Goldwasser-Klostermeyer-Ware,
Barua-Ramakrishnan-Sarkar, Hunzikel-Machiavello-Park e.a.; see also 2 previous
author's preprints for a more detailed account. Our approach explores harmonic
analysis and algebraic geometry over a positive characteristic field. The
Fourier transform allows us to interpret pluri-periods of harmonic functions on
lattices as torsion multi-orders of points on the corresponding affine
algebraic variety.Comment: These are notes on 13p. based on a talk presented during the meeting
"Analysis on Graphs and Fractals", the Cardiff University, 29 May-2 June 2007
(a sattelite meeting of the programme "Analysis on Graphs and its
Applications" at the Isaac Newton Institute from 8 January to 29 June 2007
The Zagier polynomials. Part II: Arithmetic properties of coefficients
The modified Bernoulli numbers \begin{equation*} B_{n}^{*} = \sum_{r=0}^{n}
\binom{n+r}{2r} \frac{B_{r}}{n+r}, \quad n > 0 \end{equation*} introduced by D.
Zagier in 1998 were recently extended to the polynomial case by replacing
by the Bernoulli polynomials . Arithmetic properties of the
coefficients of these polynomials are established. In particular, the 2-adic
valuation of the modified Bernoulli numbers is determined. A variety of
analytic, umbral, and asymptotic methods is used to analyze these polynomials
Discriminants of Chebyshev Radical Extensions
Let t be any integer and fix an odd prime ell. Let Phi(x) = T_ell^n(x)-t
denote the n-fold composition of the Chebyshev polynomial of degree ell shifted
by t. If this polynomial is irreducible, let K = bbq(theta), where theta is a
root of Phi. A theorem of Dedekind's gives a condition on t for which K is
monogenic. For other values of t, we apply the Montes algorithm to obtain a
formula for the discriminant of K and to compute basis elements for the ring of
integers O_K.Comment: This update contains proofs for the conjectures appearing in a
earlier version of this paper. This article draws heavily from
arXiv:0906.262
Polynomial fusion rings of W-extended logarithmic minimal models
The countably infinite number of Virasoro representations of the logarithmic
minimal model LM(p,p') can be reorganized into a finite number of
W-representations with respect to the extended Virasoro algebra symmetry W.
Using a lattice implementation of fusion, we recently determined the fusion
algebra of these representations and found that it closes, albeit without an
identity for p>1. Here, we provide a fusion-matrix realization of this fusion
algebra and identify a fusion ring isomorphic to it. We also consider various
extensions of it and quotients thereof, and introduce and analyze commutative
diagrams with morphisms between the involved fusion algebras and the
corresponding quotient polynomial fusion rings. One particular extension is
reminiscent of the fundamental fusion algebra of LM(p,p') and offers a natural
way of introducing the missing identity for p>1. Working out explicit fusion
matrices is facilitated by a further enlargement based on a pair of mutual
Moore-Penrose inverses intertwining between the W-fundamental and enlarged
fusion algebras.Comment: 48 page
Moments of the critical values of families of elliptic curves, with applications
We make conjectures on the moments of the central values of the family of all
elliptic curves and on the moments of the first derivative of the central
values of a large family of positive rank curves. In both cases the order of
magnitude is the same as that of the moments of the central values of an
orthogonal family of L-functions. Notably, we predict that the critical values
of all rank 1 elliptic curves is logarithmically larger than the rank 1 curves
in the positive rank family.
Furthermore, as arithmetical applications we make a conjecture on the
distribution of a_p's amongst all rank 2 elliptic curves, and also show how the
Riemann hypothesis can be deduced from sufficient knowledge of the first moment
of the positive rank family (based on an idea of Iwaniec).Comment: 24 page
Effective equidistribution and the Sato-Tate law for families of elliptic curves
Extending recent work of others, we provide effective bounds on the family of
all elliptic curves and one-parameter families of elliptic curves modulo p (for
p prime tending to infinity) obeying the Sato-Tate Law. We present two methods
of proof. Both use the framework of Murty-Sinha; the first involves only
knowledge of the moments of the Fourier coefficients of the L-functions and
combinatorics, and saves a logarithm, while the second requires a Sato-Tate
law. Our purpose is to illustrate how the caliber of the result depends on the
error terms of the inputs and what combinatorics must be done.Comment: Version 1.1, 24 pages: corrected the interpretation of Birch's moment
calculations, added to the literature review of previous results
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