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1. CJM 2014 (vol 66 pp. 1201)

Adler, Jeffrey D.; Lansky, Joshua M.
 Lifting Representations of Finite Reductive Groups I: Semisimple Conjugacy Classes Suppose that $\tilde{G}$ is a connected reductive group defined over a field $k$, and $\Gamma$ is a finite group acting via $k$-automorphisms of $\tilde{G}$ satisfying a certain quasi-semisimplicity condition. Then the identity component of the group of $\Gamma$-fixed points in $\tilde{G}$ is reductive. We axiomatize the main features of the relationship between this fixed-point group and the pair $(\tilde{G},\Gamma)$, and consider any group $G$ satisfying the axioms. If both $\tilde{G}$ and $G$ are $k$-quasisplit, then we can consider their duals $\tilde{G}^*$ and $G^*$. We show the existence of and give an explicit formula for a natural map from the set of semisimple stable conjugacy classes in $G^*(k)$ to the analogous set for $\tilde{G}^*(k)$. If $k$ is finite, then our groups are automatically quasisplit, and our result specializes to give a map of semisimple conjugacy classes. Since such classes parametrize packets of irreducible representations of $G(k)$ and $\tilde{G}(k)$, one obtains a mapping of such packets. Keywords:reductive group, lifting, conjugacy class, representation, Lusztig seriesCategories:20G15, 20G40, 20C33, 22E35

2. CJM 2014 (vol 67 pp. 424)

Samart, Detchat
 Mahler Measures as Linear Combinations of $L$-values of Multiple Modular Forms We study the Mahler measures of certain families of Laurent polynomials in two and three variables. Each of the known Mahler measure formulas for these families involves $L$-values of at most one newform and/or at most one quadratic character. In this paper, we show, either rigorously or numerically, that the Mahler measures of some polynomials are related to $L$-values of multiple newforms and quadratic characters simultaneously. The results suggest that the number of modular $L$-values appearing in the formulas significantly depends on the shape of the algebraic value of the parameter chosen for each polynomial. As a consequence, we also obtain new formulas relating special values of hypergeometric series evaluated at algebraic numbers to special values of $L$-functions. Keywords:Mahler measures, Eisenstein-Kronecker series, $L$-functions, hypergeometric seriesCategories:11F67, 33C20

3. CJM 2014 (vol 66 pp. 1078)

Lanphier, Dominic; Skogman, Howard
 Values of Twisted Tensor $L$-functions of Automorphic Forms Over Imaginary Quadratic Fields Let $K$ be a complex quadratic extension of $\mathbb{Q}$ and let $\mathbb{A}_K$ denote the adeles of $K$. We find special values at all of the critical points of twisted tensor $L$-functions attached to cohomological cuspforms on $GL_2(\mathbb{A}_K)$, and establish Galois equivariance of the values. To investigate the values, we determine the archimedean factors of a class of integral representations of these $L$-functions, thus proving a conjecture due to Ghate. We also investigate analytic properties of these $L$-functions, such as their functional equations. Keywords:twisted tensor $L$-function, cuspform, hypergeometric seriesCategories:11F67, 11F37

4. CJM 2013 (vol 66 pp. 284)

Eikrem, Kjersti Solberg
 Random Harmonic Functions in Growth Spaces and Bloch-type Spaces Let $h^\infty_v(\mathbf D)$ and $h^\infty_v(\mathbf B)$ be the spaces of harmonic functions in the unit disk and multi-dimensional unit ball which admit a two-sided radial majorant $v(r)$. We consider functions $v$ that fulfill a doubling condition. In the two-dimensional case let $u (re^{i\theta},\xi) = \sum_{j=0}^\infty (a_{j0} \xi_{j0} r^j \cos j\theta +a_{j1} \xi_{j1} r^j \sin j\theta)$ where $\xi =\{\xi_{ji}\}$ is a sequence of random subnormal variables and $a_{ji}$ are real; in higher dimensions we consider series of spherical harmonics. We will obtain conditions on the coefficients $a_{ji}$ which imply that $u$ is in $h^\infty_v(\mathbf B)$ almost surely. Our estimate improves previous results by Bennett, Stegenga and Timoney, and we prove that the estimate is sharp. The results for growth spaces can easily be applied to Bloch-type spaces, and we obtain a similar characterization for these spaces, which generalizes results by Anderson, Clunie and Pommerenke and by Guo and Liu. Keywords:harmonic functions, random series, growth space, Bloch-type spaceCategories:30B20, 31B05, 30H30, 42B05

5. CJM 2013 (vol 66 pp. 241)

Broussous, P.
 Transfert du pseudo-coefficient de Kottwitz et formules de caractÃ¨re pour la sÃ©rie discrÃ¨te de $\mathrm{GL}(N)$ sur un corps local Soit $G$ le groupe $\mathrm{GL}(N,F)$, oÃ¹ $F$ est un corps localement compact et non archimÃ©dien. En utilisant la thÃ©orie des types simples de Bushnell et Kutzko, ainsi qu'une idÃ©e originale d'Henniart, nous construisons des pseudo-coefficients explicites pour les reprÃ©sentations de la sÃ©rie discrÃ¨te de $G$. Comme application, nous en dÃ©duisons des formules inÃ©dites pour la valeur du charactÃ¨re d'Harish-Chandra de certaines telles reprÃ©sentations en certains Ã©lÃ©ments elliptiques rÃ©guliers. Keywords:reductive p-adic groups , discrete series, Harish-Chandra character, pseudo-coefficientCategory:22E50

6. CJM 2012 (vol 65 pp. 241)

Aguiar, Marcelo; Lauve, Aaron
 Lagrange's Theorem for Hopf Monoids in Species Following Radford's proof of Lagrange's theorem for pointed Hopf algebras, we prove Lagrange's theorem for Hopf monoids in the category of connected species. As a corollary, we obtain necessary conditions for a given subspecies $\mathbf k$ of a Hopf monoid $\mathbf h$ to be a Hopf submonoid: the quotient of any one of the generating series of $\mathbf h$ by the corresponding generating series of $\mathbf k$ must have nonnegative coefficients. Other corollaries include a necessary condition for a sequence of nonnegative integers to be the dimension sequence of a Hopf monoid in the form of certain polynomial inequalities, and of a set-theoretic Hopf monoid in the form of certain linear inequalities. The latter express that the binomial transform of the sequence must be nonnegative. Keywords:Hopf monoids, species, graded Hopf algebras, Lagrange's theorem, generating series, PoincarÃ©-Birkhoff-Witt theorem, Hopf kernel, Lie kernel, primitive element, partition, composition, linear order, cyclic order, derangementCategories:05A15, 05A20, 05E99, 16T05, 16T30, 18D10, 18D35

7. 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

8. CJM 2011 (vol 64 pp. 935)

McIntosh, Richard J.
 The H and K Families of Mock Theta Functions In his last letter to Hardy, Ramanujan defined 17 functions $F(q)$, $|q|\lt 1$, which he called mock $\theta$-functions. He observed that as $q$ radially approaches any root of unity $\zeta$ at which $F(q)$ has an exponential singularity, there is a $\theta$-function $T_\zeta(q)$ with $F(q)-T_\zeta(q)=O(1)$. Since then, other functions have been found that possess this property. These functions are related to a function $H(x,q)$, where $x$ is usually $q^r$ or $e^{2\pi i r}$ for some rational number $r$. For this reason we refer to $H$ as a universal'' mock $\theta$-function. Modular transformations of $H$ give rise to the functions $K$, $K_1$, $K_2$. The functions $K$ and $K_1$ appear in Ramanujan's lost notebook. We prove various linear relations between these functions using Appell-Lerch sums (also called generalized Lambert series). Some relations (mock theta conjectures'') involving mock $\theta$-functions of even order and $H$ are listed. Keywords:mock theta function, $q$-series, Appell-Lerch sum, generalized Lambert seriesCategories:11B65, 33D15

9. CJM 2010 (vol 63 pp. 200)

Rahman, Mizan
 An Explicit Polynomial Expression for a $q$-Analogue of the 9-$j$ Symbols Using standard transformation and summation formulas for basic hypergeometric series we obtain an explicit polynomial form of the $q$-analogue of the 9-$j$ symbols, introduced by the author in a recent publication. We also consider a limiting case in which the 9-$j$ symbol factors into two Hahn polynomials. The same factorization occurs in another limit case of the corresponding $q$-analogue. Keywords:6-$j$ and 9-$j$ symbols, $q$-analogues, balanced and very-well-poised basic hypergeometric series, orthonormal polynomials in one and two variables, Racah and $q$-Racah polynomials and their extensionsCategories:33D45, 33D50

10. CJM 2009 (vol 62 pp. 94)

Kuo, Wentang
 The Langlands Correspondence on the Generic Irreducible Constituents of Principal Series Let $G$ be a connected semisimple split group over a $p$-adic field. We establish the explicit link between principal nilpotent orbits and the irreducible constituents of principal series in terms of $L$-group objects. Keywords:Langlands correspondence, nilpotent orbits, principal seriesCategories:22E50, 22E35

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 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, traceCategories:11M41, 11F30, , 11F55, 11H06, 11R47

13. CJM 2007 (vol 59 pp. 85)

Foster, J. H.; Serbinowska, Monika
 On the Convergence of a Class of Nearly Alternating Series Let $C$ be the class of convex sequences of real numbers. The quadratic irrational numbers can be partitioned into two types as follows. If $\alpha$ is of the first type and $(c_k) \in C$, then $\sum (-1)^{\lfloor k\alpha \rfloor} c_k$ converges if and only if $c_k \log k \rightarrow 0$. If $\alpha$ is of the second type and $(c_k) \in C$, then $\sum (-1)^{\lfloor k\alpha \rfloor} c_k$ converges if and only if $\sum c_k/k$ converges. An example of a quadratic irrational of the first type is $\sqrt{2}$, and an example of the second type is $\sqrt{3}$. The analysis of this problem relies heavily on the representation of $\alpha$ as a simple continued fraction and on properties of the sequences of partial sums $S(n)=\sum_{k=1}^n (-1)^{\lfloor k\alpha \rfloor}$ and double partial sums $T(n)=\sum_{k=1}^n S(k)$. Keywords:Series, convergence, almost alternating, convex, continued fractionsCategories:40A05, 11A55, 11B83

14. CJM 2000 (vol 52 pp. 961)

Ismail, Mourad E. H.; Pitman, Jim
 Algebraic Evaluations of Some Euler Integrals, Duplication Formulae for Appell's Hypergeometric Function $F_1$, and Brownian Variations Explicit evaluations of the symmetric Euler integral $\int_0^1 u^{\alpha} (1-u)^{\alpha} f(u) \,du$ are obtained for some particular functions $f$. These evaluations are related to duplication formulae for Appell's hypergeometric function $F_1$ which give reductions of $F_1 (\alpha, \beta, \beta, 2 \alpha, y, z)$ in terms of more elementary functions for arbitrary $\beta$ with $z = y/(y-1)$ and for $\beta = \alpha + \half$ with arbitrary $y$, $z$. These duplication formulae generalize the evaluations of some symmetric Euler integrals implied by the following result: if a standard Brownian bridge is sampled at time $0$, time $1$, and at $n$ independent random times with uniform distribution on $[0,1]$, then the broken line approximation to the bridge obtained from these $n+2$ values has a total variation whose mean square is $n(n+1)/(2n+1)$. Keywords:Brownian bridge, Gauss's hypergeometric function, Lauricella's multiple hypergeometric series, uniform order statistics, Appell functionsCategories:33C65, 60J65

15. CJM 1998 (vol 50 pp. 794)

Louboutin, Stéphane
 Upper bounds on $|L(1,\chi)|$ and applications We give upper bounds on the modulus of the values at $s=1$ of Artin $L$-functions of abelian extensions unramified at all the infinite places. We also explain how we can compute better upper bounds and explain how useful such computed bounds are when dealing with class number problems for $\CM$-fields. For example, we will reduce the determination of all the non-abelian normal $\CM$-fields of degree $24$ with Galois group $\SL_2(F_3)$ (the special linear group over the finite field with three elements) which have class number one to the computation of the class numbers of $23$ such $\CM$-fields. Keywords:Dedekind zeta function, Dirichlet series, $\CM$-field, relative class numberCategories:11M20, 11R42, 11Y35, 11R29

16. CJM 1997 (vol 49 pp. 1224)

Ørsted, Bent; Zhang, Genkai
 Tensor products of analytic continuations of holomorphic discrete series We give the irreducible decomposition of the tensor product of an analytic continuation of the holomorphic discrete series of $\SU(2, 2)$ with its conjugate. Keywords:Holomorphic discrete series, tensor product, spherical function, Clebsch-Gordan coefficient, Plancherel theoremCategories:22E45, 43A85, 32M15, 33A65

17. CJM 1997 (vol 49 pp. 543)

Ismail, Mourad E. H.; Rahman, Mizan; Suslov, Sergei K.
 Some summation theorems and transformations for $q$-series We introduce a double sum extension of a very well-poised series and extend to this the transformations of Bailey and Sears as well as the ${}_6\f_5$ summation formula of F.~H.~Jackson and the $q$-Dixon sum. We also give $q$-integral representations of the double sum. Generalizations of the Nassrallah-Rahman integral are also found. Keywords:Basic hypergeometric series, balanced series,, very well-poised series, integral representations,, Al-Salam-Chihara polynomials.Categories:33D20, 33D60
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