126 research outputs found
On Level One Cuspidal Bianchi Modular Forms
In this paper, we present the outcome of vast computer calculations, locating
several of the very rare instances of level one cuspidal Bianchi modular forms
that are not lifts of elliptic modular forms.Comment: final versio
The transfer matrix: a geometrical perspective
We present a comprehensive and self-contained discussion of the use of the
transfer matrix to study propagation in one-dimensional lossless systems,
including a variety of examples, such as superlattices, photonic crystals, and
optical resonators. In all these cases, the transfer matrix has the same
algebraic properties as the Lorentz group in a (2+1)-dimensional spacetime, as
well as the group of unimodular real matrices underlying the structure of the
abcd law, which explains many subtle details. We elaborate on the geometrical
interpretation of the transfer-matrix action as a mapping on the unit disk and
apply a simple trace criterion to classify the systems into three types with
very different geometrical and physical properties. This approach is applied to
some practical examples and, in particular, an alternative framework to deal
with periodic (and quasiperiodic) systems is proposed.Comment: 50 pages, 24 figure
On the cohomology of linear groups over imaginary quadratic fields
Let Gamma be the group GL_N (OO_D), where OO_D is the ring of integers in the
imaginary quadratic field with discriminant D<0. In this paper we investigate
the cohomology of Gamma for N=3,4 and for a selection of discriminants: D >=
-24 when N=3, and D=-3,-4 when N=4. In particular we compute the integral
cohomology of Gamma up to p-power torsion for small primes p. Our main tool is
the polyhedral reduction theory for Gamma developed by Ash and Koecher. Our
results extend work of Staffeldt, who treated the case n=3, D=-4. In a sequel
to this paper, we will apply some of these results to the computations with the
K-groups K_4 (OO_{D}), when D=-3,-4
On Fermat's equation over some quadratic imaginary number fields
Assuming a deep but standard conjecture in the Langlands programme, we prove
Fermat's Last Theorem over . Under the same assumption, we also
prove that, for all prime exponents , Fermat's equation
does not have non-trivial solutions over
and .Comment: The present is a revised version, including suggestions from
referees, that was accepted for publication in Research in Number Theory; 16
page
Higher torsion in the Abelianization of the full Bianchi groups
Consider the Bianchi groups, namely the SL_2 groups over rings of imaginary
quadratic integers. In the literature, there has been so far no example of
p-torsion in the integral homology of the full Bianchi groups, for p a prime
greater than the order of elements of finite order in the Bianchi group, which
is at most 6. However, extending the scope of the computations, we can observe
examples of torsion in the integral homology of the quotient space, at prime
numbers as high as for instance p = 80737 at the discriminant -1747.Comment: contains sections formerly from arXiv:1104.530
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