191 research outputs found
Configurations of Extremal Even Unimodular Lattices
We extend the results of Ozeki on the configurations of extremal even
unimodular lattices. Specifically, we show that if L is such a lattice of rank
56, 72, or 96, then L is generated by its minimal-norm vectors.Comment: 8 pages. To appear, International Journal of Number Theor
An elementary approach to toy models for D. H. Lehmer's conjecture
In 1947, Lehmer conjectured that the Ramanujan's tau function
never vanishes for all positive integers , where is the -th
Fourier coefficient of the cusp form of weight 12. The theory of
spherical -design is closely related to Lehmer's conjecture because it is
shown, by Venkov, de la Harpe, and Pache, that is equivalent to
the fact that the shell of norm of the -lattice is a spherical
8-design. So, Lehmer's conjecture is reformulated in terms of spherical
-design.
Lehmer's conjecture is difficult to prove, and still remains open. However,
Bannai-Miezaki showed that none of the nonempty shells of the integer lattice
\ZZ^2 in \RR^2 is a spherical 4-design, and that none of the nonempty
shells of the hexagonal lattice is a spherical 6-design. Moreover, none
of the nonempty shells of the integer lattices associated to the algebraic
integers of imaginary quadratic fields whose class number is either 1 or 2,
except for \QQ(\sqrt{-1}) and \QQ(\sqrt{-3}) is a spherical 2-design. In
the proof, the theory of modular forms played an important role.
Recently, Yudin found an elementary proof for the case of \ZZ^{2}-lattice
which does not use the theory of modular forms but uses the recent results of
Calcut. In this paper, we give the elementary (i.e., modular form free) proof
and discuss the relation between Calcut's results and the theory of imaginary
quadratic fields.Comment: 18 page
Construction of spherical cubature formulas using lattices
We construct cubature formulas on spheres supported by homothetic images of
shells in some Euclidian lattices. Our analysis of these cubature formulas uses
results from the theory of modular forms. Examples are worked out on the sphere
of dimension n-1 for n=4, 8, 12, 14, 16, 20, 23, and 24, and the sizes of the
cubature formulas we obtain are compared with the lower bounds given by Linear
Programming
Spherical designs and lattices
In this article we prove that integral lattices with minimum <= 7 (or <= 9)
whose set of minimal vectors form spherical 9-designs (or 11-designs
respectively) are extremal, even and unimodular. We furthermore show that there
does not exist an integral lattice with minimum <=11 which yields a 13-design.Comment: The final publication is available at
http://link.springer.com/article/10.1007%2Fs13366-013-0155-
Quantum Fields in Hyperbolic Space-Times with Finite Spatial Volume
The one-loop effective action for a massive self-interacting scalar field is
investigated in -dimensional ultrastatic space-time ,
being a non-compact hyperbolic manifold with finite volume. Making
use of the Selberg trace formula, the -function related to the small
disturbance operator is constructed. For an arbitrary gravitational coupling,
it is found that has a simple pole at . The one-loop effective
action is analysed by means of proper-time regularisations and the one-loop
divergences are explicitly found. It is pointed out that, in this special case,
also -function regularisation requires a divergent counterterm, which
however is not necessary in the free massless conformal invariant coupling
case. Finite temperature effects are studied and the high-temperature expansion
is presented. A possible application to the problem of the divergences of the
entanglement entropy for a free massless scalar field in a Rindler-like
space-time is briefly discussed.Comment: 13 pages, LaTex. The contribution of hyperbolic elements has been
added. Other minor corrections and reference
The trace of the heat kernel on a compact hyperbolic 3-orbifold
The heat coefficients related to the Laplace-Beltrami operator defined on the
hyperbolic compact manifold H^3/\Ga are evaluated in the case in which the
discrete group \Ga contains elliptic and hyperbolic elements. It is shown
that while hyperbolic elements give only exponentially vanishing corrections to
the trace of the heat kernel, elliptic elements modify all coefficients of the
asymptotic expansion, but the Weyl term, which remains unchanged. Some physical
consequences are briefly discussed in the examples.Comment: 11 page
Support varieties for selfinjective algebras
Support varieties for any finite dimensional algebra over a field were
introduced by Snashall-Solberg using graded subalgebras of the Hochschild
cohomology. We mainly study these varieties for selfinjective algebras under
appropriate finite generation hypotheses. Then many of the standard results
from the theory of support varieties for finite groups generalize to this
situation. In particular, the complexity of the module equals the dimension of
its corresponding variety, all closed homogeneous varieties occur as the
variety of some module, the variety of an indecomposable module is connected,
periodic modules are lines and for symmetric algebras a generalization of
Webb's theorem is true
Selberg Supertrace Formula for Super Riemann Surfaces III: Bordered Super Riemann Surfaces
This paper is the third in a sequel to develop a super-analogue of the
classical Selberg trace formula, the Selberg supertrace formula. It deals with
bordered super Riemann surfaces. The theory of bordered super Riemann surfaces
is outlined, and the corresponding Selberg supertrace formula is developed. The
analytic properties of the Selberg super zeta-functions on bordered super
Riemann surfaces are discussed, and super-determinants of Dirac-Laplace
operators on bordered super Riemann surfaces are calculated in terms of Selberg
super zeta-functions.Comment: 43 pages, amste
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