1. CJM Online first
 Sugiyama, Shingo; Tsuzuki, Masao

Existence of Hilbert cusp forms with nonvanishing $L$values
We develop a derivative version of the relative trace formula
on $\operatorname{PGL}(2)$ studied in our previous work,
and derive an asymptotic formula of an average of central values
(derivatives)
of automorphic $L$functions for Hilbert cusp forms.
As an application, we prove the existence of Hilbert cusp forms
with nonvanishing central values (derivatives)
such that the absolute degrees of their Hecke fields are arbitrarily
large.
Keywords:automorphic representations, relative trace formulas, central $L$values, derivatives of $L$functions Categories:11F67, 11F72 

2. CJM 2014 (vol 67 pp. 404)
 Hua, Jiajie; Lin, Huaxin

Rotation Algebras and the Exel Trace Formula
We found that if $u$ and $v$ are any two unitaries in
a unital $C^*$algebra with $\uvvu\\lt 2$ and $uvu^*v^*$ commutes with
$u$ and $v,$ then the $C^*$subalgebra $A_{u,v}$ generated by $u$ and
$v$ is isomorphic to a quotient of some rotation algebra $A_\theta$
provided that $A_{u,v}$ has a unique tracial state.
We also found that the Exel trace formula holds in any unital
$C^*$algebra.
Let $\theta\in (1/2, 1/2)$ be a real number. We prove the
following:
For any $\epsilon\gt 0,$ there exists $\delta\gt 0$ satisfying the following:
if $u$ and $v$ are two unitaries in any unital simple $C^*$algebra
$A$ with tracial rank zero such that
\[
\uve^{2\pi i\theta}vu\\lt \delta
\text{ and }
{1\over{2\pi i}}\tau(\log(uvu^*v^*))=\theta,
\]
for all tracial state $\tau$ of $A,$ then there exists a pair
of unitaries $\tilde{u}$ and $\tilde{v}$ in $A$
such that
\[
\tilde{u}\tilde{v}=e^{2\pi i\theta} \tilde{v}\tilde{u},\,\,
\u\tilde{u}\\lt \epsilon
\text{ and }
\v\tilde{v}\\lt \epsilon.
\]
Keywords:rotation algebras, Exel trace formula Category:46L05 

3. CJM 2012 (vol 64 pp. 497)
 Li, WenWei

Le lemme fondamental pondÃ©rÃ© pour le groupe mÃ©taplectique
Dans cet article, on Ã©nonce une variante du lemme fondamental
pondÃ©rÃ© d'Arthur pour le groupe mÃ©taplectique de Weil, qui sera un
ingrÃ©dient indispensable de la stabilisation de la formule des
traces. Pour un corps de caractÃ©ristique rÃ©siduelle suffisamment
grande, on en donne une dÃ©monstration Ã l'aide de la mÃ©thode de
descente, qui est conditionnelle: on admet le lemme fondamental
pondÃ©rÃ© non standard sur les algÃ¨bres de Lie. Vu les travaux de
Chaudouard et Laumon, on s'attend Ã ce que cette condition soit
ultÃ©rieurement vÃ©rifiÃ©e.
Keywords:fundamental lemma, metaplectic group, endoscopy, trace formula Categories:11F70, 11F27, 22E50 

4. 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 selfadjoint
elements $H_0$ and $H$ in the algebra and relate it to the concept of
the de la HarpeSkandalis 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 BirmanSolomyak formula for
a general nonlinear perturbation, (ii) a universality of a spectral
averaging, and (iii) a generalization of the
DixmierFugledeKadison differentiation formula.
Keywords:perturbation determinant, trace formulae, von Neumann algebras Categories:47A55, 47C15, 47A53 

5. CJM 1999 (vol 51 pp. 952)
 Deitmar, Anton; Hoffmann, Werner

On Limit Multiplicities for Spaces of Automorphic Forms
Let $\Gamma$ be a rankone 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
squareintegrable $\Gamma$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 DeGeorgeWallach
and Delorme.
Keywords:limit multiplicities, automorphic forms, noncompact quotients, Selberg trace formula, functional calculus Categories:11F72, 22E30, 22E40, 43A85, 58G25 
