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1. CJM 2012 (vol 64 pp. 497)

Li, Wen-Wei
Le lemme fondamental pondéré pour le groupe métaplectique
Dans cet article, on énonce une variante du lemme fondamental pondéré d'Arthur pour le groupe métaplectique de Weil, qui sera un ingrédient indispensable de la stabilisation de la formule des traces. Pour un corps de caractéristique résiduelle suffisamment grande, on en donne une démonstration à l'aide de la méthode de descente, qui est conditionnelle: on admet le lemme fondamental pondéré non standard sur les algèbres de Lie. Vu les travaux de Chaudouard et Laumon, on s'attend à ce que cette condition soit ultérieurement vérifiée.

Keywords:fundamental lemma, metaplectic group, endoscopy, trace formula
Categories:11F70, 11F27, 22E50

2. CJM 2009 (vol 62 pp. 133)

Makarov, Konstantin A.; Skripka, Anna
Some Applications of the Perturbation Determinant in Finite von Neumann Algebras
In the finite von Neumann algebra setting, we introduce the concept of a perturbation determinant associated with a pair of self-adjoint elements $H_0$ and $H$ in the algebra and relate it to the concept of the de la Harpe--Skandalis homotopy invariant determinant associated with piecewise $C^1$-paths of operators joining $H_0$ and $H$. We obtain an analog of Krein's formula that relates the perturbation determinant and the spectral shift function and, based on this relation, we derive subsequently (i) the Birman--Solomyak formula for a general non-linear perturbation, (ii) a universality of a spectral averaging, and (iii) a generalization of the Dixmier--Fuglede--Kadison differentiation formula.

Keywords:perturbation determinant, trace formulae, von Neumann algebras
Categories:47A55, 47C15, 47A53

3. CJM 2007 (vol 59 pp. 673)

Ash, Avner; Friedberg, Solomon
Hecke $L$-Functions and the Distribution of Totally Positive Integers
Let $K$ be a totally real number field of degree $n$. We show that the number of totally positive integers (or more generally the number of totally positive elements of a given fractional ideal) of given trace is evenly distributed around its expected value, which is obtained from geometric considerations. This result depends on unfolding an integral over a compact torus.

Keywords:Eisenstein series, toroidal integral, Fourier series, Hecke $L$-function, totally positive integer, trace
Categories:11M41, 11F30, , 11F55, 11H06, 11R47

4. CJM 2006 (vol 58 pp. 1229)

Henniart, Guy; Lemaire, Bertrand
Intégrales orbitales tordues sur $\GL(n,F)$ et corps locaux proches\,: applications
Soient $F$ un corps commutatif localement compact non archim\'edien, $G=\GL (n,F)$ pour un entier $n\geq 2$, et $\kappa$ un caract\`ere de $F^\times$ trivial sur $(F^\times)^n$. On prouve une formule pour les $\kappa$-int\'egrales orbitales r\'eguli\`eres sur $G$ permettant, si $F$ est de caract\'eristique $>0$, de les relever \`a la caract\'eristique nulle. On en d\'eduit deux r\'esultats nouveaux en caract\'eristique $>0$\,: le ``lemme fondamental'' pour l'induction automorphe, et une version simple de la formule des traces tordue locale d'Arthur reliant $\kappa$-int\'egrales orbitales elliptiques et caract\`eres $\kappa$-tordus. Cette formule donne en particulier, pour une s\'erie $\kappa$-discr\`ete de $G$, les $\kappa$-int\'egrales orbitales elliptiques d'un pseudo-coefficient comme valeurs du caract\`ere $\kappa$-tordu.

Keywords:corps local, représentation lisse, intégrale orbitale tordue, induction automorphe, lemme fondamental, formule des traces locale, pseudo-coefficient

5. CJM 2004 (vol 56 pp. 225)

Blower, Gordon; Ransford, Thomas
Complex Uniform Convexity and Riesz Measure
The norm on a Banach space gives rise to a subharmonic function on the complex plane for which the distributional Laplacian gives a Riesz measure. This measure is calculated explicitly here for Lebesgue $L^p$ spaces and the von~Neumann-Schatten trace ideals. Banach spaces that are $q$-uniformly $\PL$-convex in the sense of Davis, Garling and Tomczak-Jaegermann are characterized in terms of the mass distribution of this measure. This gives a new proof that the trace ideals $c^p$ are $2$-uniformly $\PL$-convex for $1\leq p\leq 2$.

Keywords:subharmonic functions, Banach spaces, Schatten trace ideals
Categories:46B20, 46L52

6. CJM 2002 (vol 54 pp. 736)

Kearnes, K. A.; Kiss, E. W.; Szendrei, Á.; Willard, R. D.
Chief Factor Sizes in Finitely Generated Varieties
Let $\mathbf{A}$ be a $k$-element algebra whose chief factor size is $c$. We show that if $\mathbf{B}$ is in the variety generated by $\mathbf{A}$, then any abelian chief factor of $\mathbf{B}$ that is not strongly abelian has size at most $c^{k-1}$. This solves Problem~5 of {\it The Structure of Finite Algebras}, by D.~Hobby and R.~McKenzie. We refine this bound to $c$ in the situation where the variety generated by $\mathbf{A}$ omits type $\mathbf{1}$. As a generalization, we bound the size of multitraces of types~$\mathbf{1}$, $\mathbf{2}$, and $\mathbf{3}$ by extending coordinatization theory. Finally, we exhibit some examples of bad behavior, even in varieties satisfying a congruence identity.

Keywords:tame congruence theory, chief factor, multitrace

7. CJM 2001 (vol 53 pp. 631)

Walters, Samuel G.
K-Theory of Non-Commutative Spheres Arising from the Fourier Automorphism
For a dense $G_\delta$ set of real parameters $\theta$ in $[0,1]$ (containing the rationals) it is shown that the group $K_0 (A_\theta \rtimes_\sigma \mathbb{Z}_4)$ is isomorphic to $\mathbb{Z}^9$, where $A_\theta$ is the rotation C*-algebra generated by unitaries $U$, $V$ satisfying $VU = e^{2\pi i\theta} UV$ and $\sigma$ is the Fourier automorphism of $A_\theta$ defined by $\sigma(U) = V$, $\sigma(V) = U^{-1}$. More precisely, an explicit basis for $K_0$ consisting of nine canonical modules is given. (A slight generalization of this result is also obtained for certain separable continuous fields of unital C*-algebras over $[0,1]$.) The Connes Chern character $\ch \colon K_0 (A_\theta \rtimes_\sigma \mathbb{Z}_4) \to H^{\ev} (A_\theta \rtimes_\sigma \mathbb{Z}_4)^*$ is shown to be injective for a dense $G_\delta$ set of parameters $\theta$. The main computational tool in this paper is a group homomorphism $\vtr \colon K_0 (A_\theta \rtimes_\sigma \mathbb{Z}_4) \to \mathbb{R}^8 \times \mathbb{Z}$ obtained from the Connes Chern character by restricting the functionals in its codomain to a certain nine-dimensional subspace of $H^{\ev} (A_\theta \rtimes_\sigma \mathbb{Z}_4)$. The range of $\vtr$ is fully determined for each $\theta$. (We conjecture that this subspace is all of $H^{\ev}$.)

Keywords:C*-algebras, K-theory, automorphisms, rotation algebras, unbounded traces, Chern characters
Categories:46L80, 46L40, 19K14

8. CJM 2000 (vol 52 pp. 633)

Walters, Samuel G.
Chern Characters of Fourier Modules
Let $A_\theta$ denote the rotation algebra---the universal $C^\ast$-algebra generated by unitaries $U,V$ satisfying $VU=e^{2\pi i\theta}UV$, where $\theta$ is a fixed real number. Let $\sigma$ denote the Fourier automorphism of $A_\theta$ defined by $U\mapsto V$, $V\mapsto U^{-1}$, and let $B_\theta = A_\theta \rtimes_\sigma \mathbb{Z}/4\mathbb{Z}$ denote the associated $C^\ast$-crossed product. It is shown that there is a canonical inclusion $\mathbb{Z}^9 \hookrightarrow K_0(B_\theta)$ for each $\theta$ given by nine canonical modules. The unbounded trace functionals of $B_\theta$ (yielding the Chern characters here) are calculated to obtain the cyclic cohomology group of order zero $\HC^0(B_\theta)$ when $\theta$ is irrational. The Chern characters of the nine modules---and more importantly, the Fourier module---are computed and shown to involve techniques from the theory of Jacobi's theta functions. Also derived are explicit equations connecting unbounded traces across strong Morita equivalence, which turn out to be non-commutative extensions of certain theta function equations. These results provide the basis for showing that for a dense $G_\delta$ set of values of $\theta$ one has $K_0(B_\theta)\cong\mathbb{Z}^9$ and is generated by the nine classes constructed here.

Keywords:$C^\ast$-algebras, unbounded traces, Chern characters, irrational rotation algebras, $K$-groups
Categories:46L80, 46L40

9. CJM 1999 (vol 51 pp. 952)

Deitmar, Anton; Hoffmann, Werner
On Limit Multiplicities for Spaces of Automorphic Forms
Let $\Gamma$ be a rank-one arithmetic subgroup of a semisimple Lie group~$G$. For fixed $K$-Type, the spectral side of the Selberg trace formula defines a distribution on the space of infinitesimal characters of~$G$, whose discrete part encodes the dimensions of the spaces of square-integrable $\Gamma$-automorphic forms. It is shown that this distribution converges to the Plancherel measure of $G$ when $\Ga$ shrinks to the trivial group in a certain restricted way. The analogous assertion for cocompact lattices $\Gamma$ follows from results of DeGeorge-Wallach and Delorme.

Keywords:limit multiplicities, automorphic forms, noncompact quotients, Selberg trace formula, functional calculus
Categories:11F72, 22E30, 22E40, 43A85, 58G25

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