Computations of atom spectra

This is a contribution to the theory of atoms in abelian categories recently developed in a series of papers by Kanda. We present a method that enables one to explicitly compute the atom spectrum of the module category over a wide range of non-commutative rings. We illustrate our method and results by several examples.Comment: 16 pages. Final version to appear in Mathematische Nachrichte

On the adjoint L-function of the p-adic GSp(4)

We explicitly compute the adjoint L-function of those L-packets of representations of the group GSp(4) over a p-adic field of characteristic zero that contain non-supercuspidal representations. As an application we verify a conjecture of Gross-Prasad and Rallis in this case. The conjecture states that the adjoint L-function has a pole at s=1 if and only if the L-packet contains a generic representation.Comment: small corrections and improvements; to appear in J. Number Theor

Torsion in the cohomology of congruence subgroups of SL(4,Z) and Galois representations

We report on the computation of torsion in certain homology theories of congruence subgroups of SL(4,Z). Among these are the usual group cohomology, the Tate-Farrell cohomology, and the homology of the sharbly complex. All of these theories yield Hecke modules. We conjecture that the Hecke eigenclasses in these theories have attached Galois representations. The interpretation of our computations at the torsion primes 2,3,5 is explained. We provide evidence for our conjecture in the 15 cases of odd torsion that we found in levels up to 31

A semidiscrete version of the Citti-Petitot-Sarti model as a plausible model for anthropomorphic image reconstruction and pattern recognition

In his beautiful book [66], Jean Petitot proposes a sub-Riemannian model for the primary visual cortex of mammals. This model is neurophysiologically justified. Further developments of this theory lead to efficient algorithms for image reconstruction, based upon the consideration of an associated hypoelliptic diffusion. The sub-Riemannian model of Petitot and Citti-Sarti (or certain of its improvements) is a left-invariant structure over the group $SE(2)$ of rototranslations of the plane. Here, we propose a semi-discrete version of this theory, leading to a left-invariant structure over the group $SE(2,N)$, restricting to a finite number of rotations. This apparently very simple group is in fact quite atypical: it is maximally almost periodic, which leads to much simpler harmonic analysis compared to $SE(2).$ Based upon this semi-discrete model, we improve on previous image-reconstruction algorithms and we develop a pattern-recognition theory that leads also to very efficient algorithms in practice.Comment: 123 pages, revised versio

The fundamental group of reductive Borel-Serre and Satake compactifications

Let $G$ be an almost simple, simply connected algebraic group defined over a number field $k$, and let $S$ be a finite set of places of $k$ including all infinite places. Let $X$ be the product over $v\in S$ of the symmetric spaces associated to $G(k_v)$, when $v$ is an infinite place, and the Bruhat-Tits buildings associated to $G(k_v)$, when $v$ is a finite place. The main result of this paper is an explicit computation of the fundamental group of the reductive Borel-Serre compactification of $\Gamma\backslash X$, where $\Gamma$ is an $S$-arithmetic subgroup of $G$. In the case that $\Gamma$ is neat, we show that this fundamental group is isomorphic to $\Gamma/E\Gamma$, where $E\Gamma$ is the subgroup generated by the elements of $\Gamma$ belonging to unipotent radicals of $k$-parabolic subgroups. Analogous computations of the fundamental group of the Satake compactifications are made. It is noteworthy that calculations of the congruence subgroup kernel $C(S,G)$ yield similar results.Comment: 21 pages, 1 figure, uses Xy-pic 3.8.6; in version 2, title changed to more accurately reflect main result, expository material on congruence subgroup problem removed, many small corrections and improvements in expositio

The su(2)_{-1/2} WZW model and the beta-gamma system

The bosonic beta-gamma ghost system has long been used in formal constructions of conformal field theory. It has become important in its own right in the last few years, as a building block of field theory approaches to disordered systems, and as a simple representative -- due in part to its underlying su(2)_{-1/2} structure -- of non-unitary conformal field theories. We provide in this paper the first complete, physical, analysis of this beta-gamma system, and uncover a number of striking features. We show in particular that the spectrum involves an infinite number of fields with arbitrarily large negative dimensions. These fields have their origin in a twisted sector of the theory, and have a direct relationship with spectrally flowed representations in the underlying su(2)_{-1/2} theory. We discuss the spectral flow in the context of the operator algebra and fusion rules, and provide a re-interpretation of the modular invariant consistent with the spectrum.Comment: 33 pages, 1 figure, LaTeX, v2: minor revision, references adde