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1. CJM Online first

Rotger, Victor; de Vera-Piquero, Carlos
 Galois Representations Over Fields of Moduli and Rational Points on Shimura Curves The purpose of this note is introducing a method for proving the existence of no rational points on a coarse moduli space $X$ of abelian varieties over a given number field $K$, in cases where the moduli problem is not fine and points in $X(K)$ may not be represented by an abelian variety (with additional structure) admitting a model over the field $K$. This is typically the case when the abelian varieties that are being classified have even dimension. The main idea, inspired on the work of Ellenberg and Skinner on the modularity of $\mathbb{Q}$-curves, is that to a point $P=[A]\in X(K)$ represented by an abelian variety $A/\bar K$ one may still attach a Galois representation of $\operatorname{Gal}(\bar K/K)$ with values in the quotient group $\operatorname{GL}(T_\ell(A))/\operatorname{Aut}(A)$, provided $\operatorname{Aut}(A)$ lies in the centre of $\operatorname{GL}(T_\ell(A))$. We exemplify our method in the cases where $X$ is a Shimura curve over an imaginary quadratic field or an Atkin-Lehner quotient over $\mathbb{Q}$. Keywords:Shimura curves, rational points, Galois representations, Hasse principle, Brauer-Manin obstructionCategories:11G18, 14G35, 14G05

2. CJM Online first

Choiy, Kwangho
 Transfer of Plancherel Measures for Unitary Supercuspidal Representations between $p$-adic Inner Forms Let $F$ be a $p$-adic field of characteristic $0$, and let $M$ be an $F$-Levi subgroup of a connected reductive $F$-split group such that $\Pi_{i=1}^{r} SL_{n_i} \subseteq M \subseteq \Pi_{i=1}^{r} GL_{n_i}$ for positive integers $r$ and $n_i$. We prove that the Plancherel measure for any unitary supercuspidal representation of $M(F)$ is identically transferred under the local Jacquet-Langlands type correspondence between $M$ and its $F$-inner forms, assuming a working hypothesis that Plancherel measures are invariant on a certain set. This work extends the result of MuiÄ and Savin (2000) for Siegel Levi subgroups of the groups $SO_{4n}$ and $Sp_{4n}$ under the local Jacquet-Langlands correspondence. It can be applied to a simply connected simple $F$-group of type $E_6$ or $E_7$, and a connected reductive $F$-group of type $A_{n}$, $B_{n}$, $C_n$ or $D_n$. Keywords:Plancherel measure, inner form, local to global global argument, cuspidal automorphic representation, Jacquet-Langlands correspondenceCategories:22E50, 11F70, 22E55, 22E35

3. CJM Online first

He, Jianxun; Xiao, Jinsen
 Inversion of the Radon Transform on the Free Nilpotent Lie Group of Step Two Let $F_{2n,2}$ be the free nilpotent Lie group of step two on $2n$ generators, and let $\mathbf P$ denote the affine automorphism group of $F_{2n,2}$. In this article the theory of continuous wavelet transform on $F_{2n,2}$ associated with $\mathbf P$ is developed, and then a type of radial wavelets is constructed. Secondly, the Radon transform on $F_{2n,2}$ is studied and two equivalent characterizations of the range for Radon transform are given. Several kinds of inversion Radon transform formulae are established. One is obtained from the Euclidean Fourier transform, the others are from group Fourier transform. By using wavelet transform we deduce an inversion formula of the Radon transform, which does not require the smoothness of functions if the wavelet satisfies the differentiability property. Specially, if $n=1$, $F_{2,2}$ is the $3$-dimensional Heisenberg group $H^1$, the inversion formula of the Radon transform is valid which is associated with the sub-Laplacian on $F_{2,2}$. This result cannot be extended to the case $n\geq 2$. Keywords:Radon transform, wavelet transform, free nilpotent Lie group, unitary representation, inversion formula, sub-LaplacianCategories:43A85, 44A12, 52A38

4. CJM Online first

Abdesselam, Abdelmalek; Chipalkatti, Jaydeep
 On Hilbert Covariants Let $F$ denote a binary form of order $d$ over the complex numbers. If $r$ is a divisor of $d$, then the Hilbert covariant $\mathcal{H}_{r,d}(F)$ vanishes exactly when $F$ is the perfect power of an order $r$ form. In geometric terms, the coefficients of $\mathcal{H}$ give defining equations for the image variety $X$ of an embedding $\mathbf{P}^r \hookrightarrow \mathbf{P}^d$. In this paper we describe a new construction of the Hilbert covariant; and simultaneously situate it into a wider class of covariants called the GÃ¶ttingen covariants, all of which vanish on $X$. We prove that the ideal generated by the coefficients of $\mathcal{H}$ defines $X$ as a scheme. Finally, we exhibit a generalisation of the GÃ¶ttingen covariants to $n$-ary forms using the classical Clebsch transfer principle. Keywords:binary forms, covariants, $SL_2$-representationsCategories:14L30, 13A50

5. CJM 2012 (vol 64 pp. 721)

Achab, Dehbia; Faraut, Jacques
 Analysis of the Brylinski-Kostant Model for Spherical Minimal Representations We revisit with another view point the construction by R. Brylinski and B. Kostant of minimal representations of simple Lie groups. We start from a pair $(V,Q)$, where $V$ is a complex vector space and $Q$ a homogeneous polynomial of degree 4 on $V$. The manifold $\Xi$ is an orbit of a covering of ${\rm Conf}(V,Q)$, the conformal group of the pair $(V,Q)$, in a finite dimensional representation space. By a generalized Kantor-Koecher-Tits construction we obtain a complex simple Lie algebra $\mathfrak g$, and furthermore a real form ${\mathfrak g}_{\mathbb R}$. The connected and simply connected Lie group $G_{\mathbb R}$ with ${\rm Lie}(G_{\mathbb R})={\mathfrak g}_{\mathbb R}$ acts unitarily on a Hilbert space of holomorphic functions defined on the manifold $\Xi$. Keywords:minimal representation, Kantor-Koecher-Tits construction, Jordan algebra, Bernstein identity, Meijer $G$-functionCategories:17C36, 22E46, 32M15, 33C80

6. CJM 2011 (vol 64 pp. 669)

Pantano, Alessandra; Paul, Annegret; Salamanca-Riba, Susana A.
 The Genuine Omega-regular Unitary Dual of the Metaplectic Group We classify all genuine unitary representations of the metaplectic group whose infinitesimal character is real and at least as regular as that of the oscillator representation. In a previous paper we exhibited a certain family of representations satisfying these conditions, obtained by cohomological induction from the tensor product of a one-dimensional representation and an oscillator representation. Our main theorem asserts that this family exhausts the genuine omega-regular unitary dual of the metaplectic group. Keywords:Metaplectic group, oscillator representation, bottom layer map, cohomological induction, Parthasarathy's Dirac Operator Inequality, pseudospherical principal seriesCategory:22E46

7. CJM 2011 (vol 64 pp. 409)

Rainer, Armin
 Lifting Quasianalytic Mappings over Invariants Let $\rho \colon G \to \operatorname{GL}(V)$ be a rational finite dimensional complex representation of a reductive linear algebraic group $G$, and let $\sigma_1,\dots,\sigma_n$ be a system of generators of the algebra of invariant polynomials $\mathbb C[V]^G$. We study the problem of lifting mappings $f\colon \mathbb R^q \supseteq U \to \sigma(V) \subseteq \mathbb C^n$ over the mapping of invariants $\sigma=(\sigma_1,\dots,\sigma_n) \colon V \to \sigma(V)$. Note that $\sigma(V)$ can be identified with the categorical quotient $V /\!\!/ G$ and its points correspond bijectively to the closed orbits in $V$. We prove that if $f$ belongs to a quasianalytic subclass $\mathcal C \subseteq C^\infty$ satisfying some mild closedness properties that guarantee resolution of singularities in $\mathcal C$, e.g., the real analytic class, then $f$ admits a lift of the same class $\mathcal C$ after desingularization by local blow-ups and local power substitutions. As a consequence we show that $f$ itself allows for a lift that belongs to $\operatorname{SBV}_{\operatorname{loc}}$, i.e., special functions of bounded variation. If $\rho$ is a real representation of a compact Lie group, we obtain stronger versions. Keywords:lifting over invariants, reductive group representation, quasianalytic mappings, desingularization, bounded variationCategories:14L24, 14L30, 20G20, 22E45

8. CJM 2011 (vol 64 pp. 455)

Sherman, David
 On Cardinal Invariants and Generators for von Neumann Algebras We demonstrate how most common cardinal invariants associated with a von Neumann algebra $\mathcal M$ can be computed from the decomposability number, $\operatorname{dens}(\mathcal M)$, and the minimal cardinality of a generating set, $\operatorname{gen}(\mathcal M)$. Applications include the equivalence of the well-known generator problem, Is every separably-acting von Neumann algebra singly-generated?", with the formally stronger questions, Is every countably-generated von Neumann algebra singly-generated?" and Is the $\operatorname{gen}$ invariant monotone?" Modulo the generator problem, we determine the range of the invariant $\bigl( \operatorname{gen}(\mathcal M), \operatorname{dens}(\mathcal M) \bigr)$, which is mostly governed by the inequality $\operatorname{dens}(\mathcal M) \leq \mathfrak C^{\operatorname{gen}(\mathcal M)}$. Keywords:von Neumann algebra, cardinal invariant, generator problem, decomposability number, representation densityCategory:46L10

9. CJM 2011 (vol 63 pp. 1107)

Liu, Baiying
 Genericity of Representations of p-Adic $Sp_{2n}$ and Local Langlands Parameters Let $G$ be the $F$-rational points of the symplectic group $Sp_{2n}$, where $F$ is a non-Archimedean local field of characteristic $0$. Cogdell, Kim, Piatetski-Shapiro, and Shahidi constructed local Langlands functorial lifting from irreducible generic representations of $G$ to irreducible representations of $GL_{2n+1}(F)$. Jiang and Soudry constructed the descent map from irreducible supercuspidal representations of $GL_{2n+1}(F)$ to those of $G$, showing that the local Langlands functorial lifting from the irreducible supercuspidal generic representations is surjective. In this paper, based on above results, using the same descent method of studying $SO_{2n+1}$ as Jiang and Soudry, we will show the rest of local Langlands functorial lifting is also surjective, and for any local Langlands parameter $\phi \in \Phi(G)$, we construct a representation $\sigma$ such that $\phi$ and $\sigma$ have the same twisted local factors. As one application, we prove the $G$-case of a conjecture of Gross-Prasad and Rallis, that is, a local Langlands parameter $\phi \in \Phi(G)$ is generic, i.e., the representation attached to $\phi$ is generic, if and only if the adjoint $L$-function of $\phi$ is holomorphic at $s=1$. As another application, we prove for each Arthur parameter $\psi$, and the corresponding local Langlands parameter $\phi_{\psi}$, the representation attached to $\phi_{\psi}$ is generic if and only if $\phi_{\psi}$ is tempered. Keywords:generic representations, local Langlands parametersCategories:22E50, 11S37

10. CJM 2011 (vol 63 pp. 616)

Lee, Edward
 A Modular Quintic Calabi-Yau Threefold of Level 55 In this note we search the parameter space of Horrocks-Mumford quintic threefolds and locate a Calabi-Yau threefold that is modular, in the sense that the $L$-function of its middle-dimensional cohomology is associated with a classical modular form of weight 4 and level 55. Keywords: Calabi-Yau threefold, non-rigid Calabi-Yau threefold, two-dimensional Galois representation, modular variety, Horrocks-Mumford vector bundleCategories:14J15, 11F23, 14J32, 11G40

11. CJM 2009 (vol 62 pp. 34)

Campbell, Peter S.; Nevins, Monica
 Branching Rules for Ramified Principal Series Representations of $\mathrm{GL}(3)$ over a $p$-adic Field We decompose the restriction of ramified principal series representations of the $p$-adic group $\mathrm{GL}(3,\mathrm{k})$ to its maximal compact subgroup $K=\mathrm{GL}(3,R)$. Its decomposition is dependent on the degree of ramification of the inducing characters and can be characterized in terms of filtrations of the Iwahori subgroup in $K$. We establish several irreducibility results and illustrate the decomposition with some examples. Keywords:principal series representations, branching rules, maximal compact subgroups, representations of $p$-adic groupsCategories:20G25, 20G05

12. CJM 2009 (vol 62 pp. 439)

Sundhäll, Marcus; Tchoundja, Edgar
 On Hankel Forms of Higher Weights: The Case of Hardy Spaces In this paper we study bilinear Hankel forms of higher weights on Hardy spaces in several dimensions. (The Schatten class Hankel forms of higher weights on weighted Bergman spaces have already been studied by Janson and Peetre for one dimension and by SundhÃ¤ll for several dimensions). We get a full characterization of Schatten class Hankel forms in terms of conditions for the symbols to be in certain Besov spaces. Also, the Hankel forms are bounded and compact if and only if the symbols satisfy certain Carleson measure criteria and vanishing Carleson measure criteria, respectively. Keywords:Hankel forms, Schattenâvon Neumann classes, Bergman spaces, Hardy spaces, Besov spaces, transvectant, unitary representations, MÃ¶bius groupCategories:32A25, 32A35, 32A37, 47B35

13. CJM 2008 (vol 60 pp. 1067)

Kariyama, Kazutoshi
 On Types for Unramified $p$-Adic Unitary Groups Let $F$ be a non-archimedean local field of residue characteristic neither 2 nor 3 equipped with a galois involution with fixed field $F_0$, and let $G$ be a symplectic group over $F$ or an unramified unitary group over $F_0$. Following the methods of Bushnell--Kutzko for $\GL(N,F)$, we define an analogue of a simple type attached to a certain skew simple stratum, and realize a type in $G$. In particular, we obtain an irreducible supercuspidal representation of $G$ like $\GL(N,F)$. Keywords:$p$-adic unitary group, type, supercuspidal representation, Hecke algebraCategories:22E50, 22D99

14. CJM 2008 (vol 60 pp. 208)

Ramakrishna, Ravi
 Constructing Galois Representations with Very Large Image Starting with a 2-dimensional mod $p$ Galois representation, we construct a deformation to a power series ring in infinitely many variables over the $p$-adics. The image of this representation is full in the sense that it contains $\SL_2$ of this power series ring. Furthermore, all ${\mathbb Z}_p$ specializations of this deformation are potentially semistable at $p$. Keywords:Galois representation, deformationCategory:11f80

15. CJM 2007 (vol 59 pp. 332)

Leuschke, Graham J.
 Endomorphism Rings of Finite Global Dimension For a commutative local ring $R$, consider (noncommutative) $R$-algebras $\Lambda$ of the form $\Lambda = \operatorname{End}_R(M)$ where $M$ is a reflexive $R$-module with nonzero free direct summand. Such algebras $\Lambda$ of finite global dimension can be viewed as potential substitutes for, or analogues of, a resolution of singularities of $\operatorname{Spec} R$. For example, Van den Bergh has shown that a three-dimensional Gorenstein normal $\mathbb{C}$-algebra with isolated terminal singularities has a crepant resolution of singularities if and only if it has such an algebra $\Lambda$ with finite global dimension and which is maximal Cohen--Macaulay over $R$ (a noncommutative crepant resolution of singularities''). We produce algebras $\Lambda=\operatorname{End}_R(M)$ having finite global dimension in two contexts: when $R$ is a reduced one-dimensional complete local ring, or when $R$ is a Cohen--Macaulay local ring of finite Cohen--Macaulay type. If in the latter case $R$ is Gorenstein, then the construction gives a noncommutative crepant resolution of singularities in the sense of Van den Bergh. Keywords:representation dimension, noncommutative crepant resolution, maximal Cohen--Macaulay modulesCategories:16G50, 16G60, 16E99

16. CJM 2006 (vol 58 pp. 23)

Dabbaghian-Abdoly, Vahid
 Constructing Representations of Finite Simple Groups and Covers Let $G$ be a finite group and $\chi$ be an irreducible character of $G$. An efficient and simple method to construct representations of finite groups is applicable whenever $G$ has a subgroup $H$ such that $\chi_H$ has a linear constituent with multiplicity $1$. In this paper we show (with a few exceptions) that if $G$ is a simple group or a covering group of a simple group and $\chi$ is an irreducible character of $G$ of degree less than 32, then there exists a subgroup $H$ (often a Sylow subgroup) of $G$ such that $\chi_H$ has a linear constituent with multiplicity $1$. Keywords:group representations, simple groups, central covers, irreducible representationsCategories:20C40, 20C15

17. CJM 2005 (vol 57 pp. 897)

Berezhnoĭ, Evgenii I.; Maligranda, Lech
 Representation of Banach Ideal Spaces and Factorization of Operators Representation theorems are proved for Banach ideal spaces with the Fatou property which are built by the Calder{\'o}n--Lozanovski\u\i\ construction. Factorization theorems for operators in spaces more general than the Lebesgue $L^{p}$ spaces are investigated. It is natural to extend the Gagliardo theorem on the Schur test and the Rubio de~Francia theorem on factorization of the Muckenhoupt $A_{p}$ weights to reflexive Orlicz spaces. However, it turns out that for the scales far from $L^{p}$-spaces this is impossible. For the concrete integral operators it is shown that factorization theorems and the Schur test in some reflexive Orlicz spaces are not valid. Representation theorems for the Calder{\'o}n--Lozanovski\u\i\ construction are involved in the proofs. Keywords:Banach ideal spaces, weighted spaces, weight functions,, CalderÃ³n--Lozanovski\u\i\ spaces, Orlicz spaces, representation of, spaces, uniqueness problem, positive linear operators, positive sublinear, operators, Schur test, factorization of operators, fCategories:46E30, 46B42, 46B70

18. CJM 2005 (vol 57 pp. 648)

Nevins, Monica
 Branching Rules for Principal Series Representations of $SL(2)$ over a $p$-adic Field We explicitly describe the decomposition into irreducibles of the restriction of the principal series representations of $SL(2,k)$, for $k$ a $p$-adic field, to each of its two maximal compact subgroups (up to conjugacy). We identify these irreducible subrepresentations in the Kirillov-type classification of Shalika. We go on to explicitly describe the decomposition of the reducible principal series of $SL(2,k)$ in terms of the restrictions of its irreducible constituents to a maximal compact subgroup. Keywords:representations of $p$-adic groups, $p$-adic integers, orbit method, $K$-typesCategories:20G25, 22E35, 20H25

19. CJM 2005 (vol 57 pp. 598)

Kornelson, Keri A.
 Local Solvability of Laplacian Difference Operators Arising from the Discrete Heisenberg Group Differential operators $D_x$, $D_y$, and $D_z$ are formed using the action of the $3$-dimensional discrete Heisenberg group $G$ on a set $S$, and the operators will act on functions on $S$. The Laplacian operator $L=D_x^2 + D_y^2 + D_z^2$ is a difference operator with variable differences which can be associated to a unitary representation of $G$ on the Hilbert space $L^2(S)$. Using techniques from harmonic analysis and representation theory, we show that the Laplacian operator is locally solvable. Keywords:discrete Heisenberg group,, unitary representation,, local solvability,, difference operatorCategories:43A85, 22D10, 39A70

20. CJM 2002 (vol 54 pp. 634)

Weber, Eric
 Frames and Single Wavelets for Unitary Groups We consider a unitary representation of a discrete countable abelian group on a separable Hilbert space which is associated to a cyclic generalized frame multiresolution analysis. We extend Robertson's theorem to apply to frames generated by the action of the group. Within this setup we use Stone's theorem and the theory of projection valued measures to analyze wandering frame collections. This yields a functional analytic method of constructing a wavelet from a generalized frame multi\-resolution analysis in terms of the frame scaling vectors. We then explicitly apply our results to the action of the integers given by translations on $L^2({\mathbb R})$. Keywords:wavelet, multiresolution analysis, unitary group representation, frameCategories:42C40, 43A25, 42C15, 46N99

21. CJM 2000 (vol 52 pp. 1121)

Ballantine, Cristina M.
 Ramanujan Type Buildings We will construct a finite union of finite quotients of the affine building of the group $\GL_3$ over the field of $p$-adic numbers $\mathbb{Q}_p$. We will view this object as a hypergraph and estimate the spectrum of its underlying graph. Keywords:automorphic representations, buildingsCategory:11F70

22. CJM 2000 (vol 52 pp. 1221)

Hopenwasser, Alan; Peters, Justin R.; Power, Stephen C.
 Nest Representations of TAF Algebras A nest representation of a strongly maximal TAF algebra $A$ with diagonal $D$ is a representation $\pi$ for which $\lat \pi(A)$ is totally ordered. We prove that $\ker \pi$ is a meet irreducible ideal if the spectrum of $A$ is totally ordered or if (after an appropriate similarity) the von Neumann algebra $\pi(D)''$ contains an atom. Keywords:nest representation, meet irreducible ideal, strongly maximal TAF algebraCategories:47L40, 47L35

23. CJM 1999 (vol 51 pp. 850)

Muhly, Paul S.; Solel, Baruch
 Tensor Algebras, Induced Representations, and the Wold Decomposition Our objective in this sequel to \cite{MSp96a} is to develop extensions, to representations of tensor algebras over $C^{*}$-correspondences, of two fundamental facts about isometries on Hilbert space: The Wold decomposition theorem and Beurling's theorem, and to apply these to the analysis of the invariant subspace structure of certain subalgebras of Cuntz-Krieger algebras. Keywords:tensor algebras, correspondence, induced representation, Wold decomposition, Beurling's theoremCategories:46L05, 46L40, 46L89, 47D15, 47D25, 46M10, 46M99, 47A20, 47A45, 47B35

24. CJM 1998 (vol 50 pp. 1253)

López-Bautista, Pedro Ricardo; Villa-Salvador, Gabriel Daniel
 Integral representation of $p$-class groups in ${\Bbb Z}_p$-extensions and the Jacobian variety For an arbitrary finite Galois $p$-extension $L/K$ of $\zp$-cyclotomic number fields of $\CM$-type with Galois group $G = \Gal(L/K)$ such that the Iwasawa invariants $\mu_K^-$, $\mu_L^-$ are zero, we obtain unconditionally and explicitly the Galois module structure of $\clases$, the minus part of the $p$-subgroup of the class group of $L$. For an arbitrary finite Galois $p$-extension $L/K$ of algebraic function fields of one variable over an algebraically closed field $k$ of characteristic $p$ as its exact field of constants with Galois group $G = \Gal(L/K)$ we obtain unconditionally and explicitly the Galois module structure of the $p$-torsion part of the Jacobian variety $J_L(p)$ associated to $L/k$. Keywords:${\Bbb Z}_p$-extensions, Iwasawa's theory, class group, integral representation, fields of algebraic functions, Jacobian variety, Galois module structureCategories:11R33, 11R23, 11R58, 14H40

25. CJM 1997 (vol 49 pp. 722)

Elder, G. Griffith; Madan, Manohar L.
 Galois module structure of the integers in wildly ramified $C_p\times C_p$ extensions Let $L/K$ be a finite Galois extension of local fields which are finite extensions of $\bQ_p$, the field of $p$-adic numbers. Let $\Gal (L/K)=G$, and $\euO_L$ and $\bZ_p$ be the rings of integers in $L$ and $\bQ_p$, respectively. And let $\euP_L$ denote the maximal ideal of $\euO_L$. We determine, explicitly in terms of specific indecomposable $\bZ_p[G]$-modules, the $\bZ_p[G]$-module structure of $\euO_L$ and $\euP_L$, for $L$, a composite of two arithmetically disjoint, ramified cyclic extensions of $K$, one of which is only weakly ramified in the sense of Erez \cite{erez}. Keywords:Galois module structure---integral representation.Categories:11S15, 20C32
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