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1. CJM Online first

Kaftal, Victor; Ng, Ping Wong; Zhang, Shuang
Strict comparison of positive elements in multiplier algebras
Main result: If a C*-algebra $\mathcal{A}$ is simple, $\sigma$-unital, has finitely many extremal traces, and has strict comparison of positive elements by traces, then its multiplier algebra $\operatorname{\mathcal{M}}(\mathcal{A})$ also has strict comparison of positive elements by traces. The same results holds if ``finitely many extremal traces" is replaced by ``quasicontinuous scale". A key ingredient in the proof is that every positive element in the multiplier algebra of an arbitrary $\sigma$-unital C*-algebra can be approximated by a bi-diagonal series. An application of strict comparison: If $\mathcal{A}$ is a simple separable stable C*-algebra with real rank zero, stable rank one, and strict comparison of positive elements by traces, then whether a positive element is a positive linear combination of projections is determined by the trace values of its range projection.

Keywords:strict comparison, bi-diagonal form, positive combinations
Categories:46L05, 46L35, 46L45, 47C15

2. CJM Online first

Izumi, Masaki; Morrison, Scott; Penneys, David
Quotients of $A_2 * T_2$
We study unitary quotients of the free product unitary pivotal category $A_2*T_2$. We show that such quotients are parametrized by an integer $n\geq 1$ and an $2n$-th root of unity $\omega$. We show that for $n=1,2,3$, there is exactly one quotient and $\omega=1$. For $4\leq n\leq 10$, we show that there are no such quotients. Our methods also apply to quotients of $T_2*T_2$, where we have a similar result. The essence of our method is a consistency check on jellyfish relations. While we only treat the specific cases of $A_2 * T_2$ and $T_2 * T_2$, we anticipate that our technique can be extended to a general method for proving nonexistence of planar algebras with a specified principal graph. During the preparation of this manuscript, we learnt of Liu's independent result on composites of $A_3$ and $A_4$ subfactor planar algebras (arxiv:1308.5691). In 1994, Bisch-Haagerup showed that the principal graph of a composite of $A_3$ and $A_4$ must fit into a certain family, and Liu has classified all such subfactor planar algebras. We explain the connection between the quotient categories and the corresponding composite subfactor planar algebras. As a corollary of Liu's result, there are no such quotient categories for $n\geq 4$. This is an abridged version of arxiv:1308.5723.

Keywords:pivotal category, free product, quotient, subfactor, intermediate subfactor
Category:46L37

3. CJM Online first

Günther, Christian; Schmidt, Kai-Uwe
$L^q$ norms of Fekete and related polynomials
A Littlewood polynomial is a polynomial in $\mathbb{C}[z]$ having all of its coefficients in $\{-1,1\}$. There are various old unsolved problems, mostly due to Littlewood and Erdős, that ask for Littlewood polynomials that provide a good approximation to a function that is constant on the complex unit circle, and in particular have small $L^q$ norm on the complex unit circle. We consider the Fekete polynomials \[ f_p(z)=\sum_{j=1}^{p-1}(j\,|\,p)\,z^j, \] where $p$ is an odd prime and $(\,\cdot\,|\,p)$ is the Legendre symbol (so that $z^{-1}f_p(z)$ is a Littlewood polynomial). We give explicit and recursive formulas for the limit of the ratio of $L^q$ and $L^2$ norm of $f_p$ when $q$ is an even positive integer and $p\to\infty$. To our knowledge, these are the first results that give these limiting values for specific sequences of nontrivial Littlewood polynomials and infinitely many $q$. Similar results are given for polynomials obtained by cyclically permuting the coefficients of Fekete polynomials and for Littlewood polynomials whose coefficients are obtained from additive characters of finite fields. These results vastly generalise earlier results on the $L^4$ norm of these polynomials.

Keywords:character polynomial, Fekete polynomial, $L^q$ norm, Littlewood polynomial
Categories:11B83, 42A05, 30C10

4. CJM Online first

Banks, Jessica; Rathbun, Matt
Monodromy action on unknotting tunnels in fiber surfaces
In \cite{RatTOFL}, the second author showed that a tunnel of a tunnel number one, fibered link in $S^3$ can be isotoped to lie as a properly embedded arc in the fiber surface of the link. In this paper, we observe that this is true for fibered links in any 3-manifold, we analyze how the arc behaves under the monodromy action, and we show that the tunnel arc is nearly clean, with the possible exception of twisting around the boundary of the fiber.

Keywords:fibered, monodromy, tunnel, clean
Category:57M25

5. CJM Online first

Greenberg, Matthew; Seveso, Marco
$p$-adic families of cohomological modular forms for indefinite quaternion algebras and the Jacquet-Langlands correspondence
We use the method of Ash and Stevens to prove the existence of small slope $p$-adic families of cohomological modular forms for an indefinite quaternion algebra $B$. We prove that the Jacquet-Langlands correspondence relating modular forms on $\textbf{GL}_2/\mathbb{Q}$ and cohomomological modular forms for $B$ is compatible with the formation of $p$-adic families. This result is an analogue of a theorem of Chenevier concerning definite quaternion algebras.

Keywords:modular forms, p-adic families, Jacquet-Langlands correspondence, Shimura curves, eigencurves
Categories:11F11, 11F67, 11F85

6. CJM Online first

Nikolidakis, Eleftherios Nikolaos
Extremal sequences for the Bellman function of the dyadic maximal operator and applications to the Hardy operator
We prove that the extremal sequences for the Bellman function of the dyadic maximal operator behave approximately as eigenfunctions of this operator for a specific eigenvalue. We use this result to prove the analogous one with respect to the Hardy operator.

Keywords:Bellman function, dyadic, Hardy operator, maximal
Category:42B25

7. CJM Online first

Brandes, Julia; Parsell, Scott T.
Simultaneous additive equations: Repeated and differing degrees
We obtain bounds for the number of variables required to establish Hasse principles, both for existence of solutions and for asymptotic formulæ, for systems of additive equations containing forms of differing degree but also multiple forms of like degree. Apart from the very general estimates of Schmidt and Browning--Heath-Brown, which give weak results when specialized to the diagonal situation, this is the first result on such "hybrid" systems. We also obtain specialised results for systems of quadratic and cubic forms, where we are able to take advantage of some of the stronger methods available in that setting. In particular, we achieve essentially square root cancellation for systems consisting of one cubic and $r$ quadratic equations.

Keywords:equations in many variables, counting solutions of Diophantine equations, applications of the Hardy-Littlewood method
Categories:11D72, 11D45, 11P55

8. CJM Online first

Saanouni, Tarek
Global and non global solutions for some fractional heat equations with pure power nonlinearity
The initial value problem for a semi-linear fractional heat equation is investigated. In the focusing case, global well-posedness and exponential decay are obtained. In the focusing sign, global and non global existence of solutions are discussed via the potential well method.

Keywords:nonlinear fractional heat equation, global Existence, decay, blow-up
Category:35Q55

9. CJM Online first

Andrade, Jaime; Dávila, Nestor; Pérez-Chavela, Ernesto; Vidal, Claudio
Dynamics and regularization of the Kepler problem on surfaces of constant curvature
We classify and analyze the orbits of the Kepler problem on surfaces of constant curvature (both positive and negative, $\mathbb S^2$ and $\mathbb H^2$, respectively) as function of the angular momentum and the energy. Hill's region are characterized and the problem of time-collision is studied. We also regularize the problem in Cartesian and intrinsic coordinates, depending on the constant angular momentum and we describe the orbits of the regularized vector field. The phase portrait both for $\mathbb S^2$ and $\mathbb H^2$ are pointed out.

Keywords:Kepler problem on surfaces of constant curvature, Hill's region, singularities, regularization, qualitative analysis of ODE
Categories:70F16, 70G60

10. CJM Online first

Kamgarpour, Masoud
On the notion of conductor in the local geometric Langlands correspondence
Under the local Langlands correspondence, the conductor of an irreducible representation of $\operatorname{Gl}_n(F)$ is greater than the Swan conductor of the corresponding Galois representation. In this paper, we establish the geometric analogue of this statement by showing that the conductor of a categorical representation of the loop group is greater than the irregularity of the corresponding meromorphic connection.

Keywords:local geometric Langlands, connections, cyclic vectors, opers, conductors, Segal-Sugawara operators, Chervov-Molev operators, critical level, smooth representations, affine Kac-Moody algebra, categorical representations
Categories:17B67, 17B69, 22E50, 20G25

11. CJM Online first

Hartglass, Michael
Free product C*-algebras associated to graphs, free differentials, and laws of loops
We study a canonical C$^*$-algebra, $\mathcal{S}(\Gamma, \mu)$, that arises from a weighted graph $(\Gamma, \mu)$, specific cases of which were previously studied in the context of planar algebras. We discuss necessary and sufficient conditions of the weighting which ensure simplicity and uniqueness of trace of $\mathcal{S}(\Gamma, \mu)$, and study the structure of its positive cone. We then study the $*$-algebra, $\mathcal{A}$, generated by the generators of $\mathcal{S}(\Gamma, \mu)$, and use a free differential calculus and techniques of Charlesworth and Shlyakhtenko, as well as Mai, Speicher, and Weber to show that certain ``loop" elements have no atoms in their spectral measure. After modifying techniques of Shlyakhtenko and Skoufranis to show that self adjoint elements $x \in M_{n}(\mathcal{A})$ have algebraic Cauchy transform, we explore some applications to eigenvalues of polynomials in Wishart matrices and to diagrammatic elements in von Neumann algebras initially considered by Guionnet, Jones, and Shlyakhtenko.

Keywords:free probability, C*-algebra
Category:46L09

12. CJM Online first

Fischer, Vera; Mejia, Diego Alejandro
Splitting, Bounding, and Almost Disjointness can be quite Different
We prove the consistency of $\operatorname{add}(\mathcal{N})\lt \operatorname{v}(\mathcal{N})\lt \mathfrak{p}=\mathfrak{s}=\mathfrak{g}\lt \operatorname{add}(\mathcal{M})=\operatorname{f}(\mathcal{M})\lt \mathfrak{a}=\mathfrak{r}=\operatorname{non}(\mathcal{N})=\mathfrak{c}$ with $\mathrm{ZFC}$, where each of these cardinal invariants assume arbitrary uncountable regular values.

Keywords:cardinal characteristics of the continuum, splitting, bounding number, maximal almost-disjoint families, template forcing iterations, isomorphism-of-names
Categories:03E17, 03E35, 03E40

13. CJM Online first

Brasca, Riccardo
Eigenvarieties for cuspforms over PEL type Shimura varieties with dense ordinary locus
Let $p \gt 2$ be a prime and let $X$ be a compactified PEL Shimura variety of type (A) or (C) such that $p$ is an unramified prime for the PEL datum and such that the ordinary locus is dense in the reduction of $X$. Using the geometric approach of Andreatta, Iovita, Pilloni, and Stevens we define the notion of families of overconvergent locally analytic $p$-adic modular forms of Iwahoric level for $X$. We show that the system of eigenvalues of any finite slope cuspidal eigenform of Iwahoric level can be deformed to a family of systems of eigenvalues living over an open subset of the weight space. To prove these results, we actually construct eigenvarieties of the expected dimension that parameterize finite slope systems of eigenvalues appearing in the space of families of cuspidal forms.

Keywords:$p$-adic modular forms, eigenvarieties, PEL-type Shimura varieties
Categories:11F55, 11F33

14. CJM Online first

Marquis, Timothée; Neeb, Karl-Hermann
Isomorphisms of twisted Hilbert loop algebras
The closest infinite dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e. real Hilbert spaces with a Lie algebra structure for which the scalar product is invariant. Locally affine Lie algebras (LALAs) correspond to double extensions of (twisted) loop algebras over simple Hilbert-Lie algebras $\mathfrak{k}$, also called affinisations of $\mathfrak{k}$. They possess a root space decomposition whose corresponding root system is a locally affine root system of one of the $7$ families $A_J^{(1)}$, $B_J^{(1)}$, $C_J^{(1)}$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ and $BC_J^{(2)}$ for some infinite set $J$. To each of these types corresponds a ``minimal" affinisation of some simple Hilbert-Lie algebra $\mathfrak{k}$, which we call standard. In this paper, we give for each affinisation $\mathfrak{g}$ of a simple Hilbert-Lie algebra $\mathfrak{k}$ an explicit isomorphism from $\mathfrak{g}$ to one of the standard affinisations of $\mathfrak{k}$. The existence of such an isomorphism could also be derived from the classification of locally affine root systems, but for representation theoretic purposes it is crucial to obtain it explicitly as a deformation between two twists which is compatible with the root decompositions. We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of $\mathfrak{g}$. In subsequent work, the present paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of $\mathfrak{k}$.

Keywords:locally affine Lie algebra, Hilbert-Lie algebra, positive energy representation
Categories:17B65, 17B70, 17B22, 17B10

15. CJM Online first

Hartz, Michael
On the isomorphism problem for multiplier algebras of Nevanlinna-Pick spaces
We continue the investigation of the isomorphism problem for multiplier algebras of reproducing kernel Hilbert spaces with the complete Nevanlinna-Pick property. In contrast to previous work in this area, we do not study these spaces by identifying them with restrictions of a universal space, namely the Drury-Arveson space. Instead, we work directly with the Hilbert spaces and their reproducing kernels. In particular, we show that two multiplier algebras of Nevanlinna-Pick spaces on the same set are equal if and only if the Hilbert spaces are equal. Most of the article is devoted to the study of a special class of complete Nevanlinna-Pick spaces on homogeneous varieties. We provide a complete answer to the question of when two multiplier algebras of spaces of this type are algebraically or isometrically isomorphic. This generalizes results of Davidson, Ramsey, Shalit, and the author.

Keywords:non-selfadjoint operator algebras, reproducing kernel Hilbert spaces, multiplier algebra, Nevanlinna-Pick kernels, isomorphism problem
Categories:47L30, 46E22, 47A13

16. CJM Online first

Chen, Xianghong; Seeger, Andreas
Convolution powers of Salem measures with applications
We study the regularity of convolution powers for measures supported on Salem sets, and prove related results on Fourier restriction and Fourier multipliers. In particular we show that for $\alpha$ of the form ${d}/{n}$, $n=2,3,\dots$ there exist $\alpha$-Salem measures for which the $L^2$ Fourier restriction theorem holds in the range $p\le \frac{2d}{2d-\alpha}$. The results rely on ideas of Körner. We extend some of his constructions to obtain upper regular $\alpha$-Salem measures, with sharp regularity results for $n$-fold convolutions for all $n\in \mathbb{N}$.

Keywords:convolution powers, Fourier restriction, Salem sets, Salem measures, random sparse sets, Fourier multipliers of Bochner-Riesz type
Categories:42A85, 42B99, 42B15, 42A61

17. CJM Online first

Adamus, Janusz; Seyedinejad, Hadi
Finite determinacy and stability of flatness of analytic mappings
It is proved that flatness of an analytic mapping germ from a complete intersection is determined by its sufficiently high jet. As a consequence, one obtains finite determinacy of complete intersections. It is also shown that flatness and openness are stable under deformations.

Keywords:finite determinacy, stability, flatness, openness, complete intersection
Categories:58K40, 58K25, 32S05, 58K20, 32S30, 32B99, 32C05, 13B40

18. CJM Online first

Grinberg, Darij
Dual immaculate creation operators and a dendriform algebra structure on the quasisymmetric functions
The dual immaculate functions are a basis of the ring $\operatorname*{QSym}$ of quasisymmetric functions, and form one of the most natural analogues of the Schur functions. The dual immaculate function corresponding to a composition is a weighted generating function for immaculate tableaux in the same way as a Schur function is for semistandard Young tableaux; an " immaculate tableau" is defined similarly to be a semistandard Young tableau, but the shape is a composition rather than a partition, and only the first column is required to strictly increase (whereas the other columns can be arbitrary; but each row has to weakly increase). Dual immaculate functions have been introduced by Berg, Bergeron, Saliola, Serrano and Zabrocki in arXiv:1208.5191, and have since been found to possess numerous nontrivial properties. In this note, we prove a conjecture of Mike Zabrocki which provides an alternative construction for the dual immaculate functions in terms of certain "vertex operators". The proof uses a dendriform structure on the ring $\operatorname*{QSym}$; we discuss the relation of this structure to known dendriform structures on the combinatorial Hopf algebras $\operatorname*{FQSym}$ and $\operatorname*{WQSym}$.

Keywords:combinatorial Hopf algebras, quasisymmetric functions, dendriform algebras, immaculate functions, Young tableaux
Category:05E05

19. CJM 2016 (vol 68 pp. 784)

Doran, Charles F.; Harder, Andrew
Toric Degenerations and Laurent Polynomials Related to Givental's Landau-Ginzburg Models
For an appropriate class of Fano complete intersections in toric varieties, we prove that there is a concrete relationship between degenerations to specific toric subvarieties and expressions for Givental's Landau-Ginzburg models as Laurent polynomials. As a result, we show that Fano varieties presented as complete intersections in partial flag manifolds admit degenerations to Gorenstein toric weak Fano varieties, and their Givental Landau-Ginzburg models can be expressed as corresponding Laurent polynomials. We also use this to show that all of the Laurent polynomials obtained by Coates, Kasprzyk and Prince by the so called Przyjalkowski method correspond to toric degenerations of the corresponding Fano variety. We discuss applications to geometric transitions of Calabi-Yau varieties.

Keywords:Fano varieties, Landau-Ginzburg models, Calabi-Yau varieties, toric varieties
Categories:14M25, 14J32, 14J33, 14J45

20. CJM 2016 (vol 68 pp. 908)

Sugiyama, Shingo; Tsuzuki, Masao
Existence of Hilbert Cusp Forms with Non-vanishing $L$-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 non-vanishing 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

21. CJM 2016 (vol 68 pp. 571)

Gras, Georges
Les $\theta$-régulateurs locaux d'un nombre algébrique : Conjectures $p$-adiques
Let $K/\mathbb{Q}$ be Galois and let $\eta\in K^\times$ be such that $\operatorname{Reg}_\infty (\eta) \ne 0$. We define the local $\theta$-regulators $\Delta_p^\theta(\eta) \in \mathbb{F}_p$ for the $\mathbb{Q}_p\,$-irreducible characters $\theta$ of $G=\operatorname{Gal}(K/\mathbb{Q})$. A linear representation ${\mathcal L}^\theta\simeq \delta \, V_\theta$ is associated with $\Delta_p^\theta (\eta)$ whose nullity is equivalent to $\delta \geq 1$. Each $\Delta_p^\theta (\eta)$ yields $\operatorname{Reg}_p^\theta (\eta)$ modulo $p$ in the factorization $\prod_{\theta}(\operatorname{Reg}_p^\theta (\eta))^{\varphi(1)}$ of $\operatorname{Reg}_p^G (\eta) := \frac{ \operatorname{Reg}_p(\eta)}{p^{[K : \mathbb{Q}\,]} }$ (normalized $p$-adic regulator). From $\operatorname{Prob}\big (\Delta_p^\theta(\eta) = 0 \ \& \ {\mathcal L}^\theta \simeq \delta \, V_\theta\big ) \leq p^{- f \delta^2}$ ($f \geq 1$ is a residue degree) and the Borel-Cantelli heuristic, we conjecture that, for $p$ large enough, $\operatorname{Reg}_p^G (\eta)$ is a $p$-adic unit or that $p^{\varphi(1)} \parallel \operatorname{Reg}_p^G (\eta)$ (a single $\theta$ with $f=\delta=1$); this obstruction may be lifted assuming the existence of a binomial probability law confirmed through numerical studies (groups $C_3$, $C_5$, $D_6$). This conjecture would imply that, for all $p$ large enough, Fermat quotients, normalized $p$-adic regulators are $p$-adic units and that number fields are $p$-rational. We recall some deep cohomological results that may strengthen such conjectures.

Keywords:$p$-adic regulators, Leopoldt-Jaulent conjecture, Frobenius group determinants, characters, Fermat quotient, Abelian $p$-ramification, probabilistic number theory
Categories:11F85, 11R04, 20C15, 11C20, 11R37, 11R27, 11Y40

22. CJM Online first

Runde, Volker; Viselter, Ami
On positive definiteness over locally compact quantum groups
The notion of positive-definite functions over locally compact quantum groups was recently introduced and studied by Daws and Salmi. Based on this work, we generalize various well-known results about positive-definite functions over groups to the quantum framework. Among these are theorems on "square roots" of positive-definite functions, comparison of various topologies, positive-definite measures and characterizations of amenability, and the separation property with respect to compact quantum subgroups.

Keywords:bicrossed product, locally compact quantum group, non-commutative $L^p$-space, positive-definite function, positive-definite measure, separation property
Categories:20G42, 22D25, 43A35, 46L51, 46L52, 46L89

23. CJM Online first

Ovchinnikov, Alexey; Wibmer, Michael
Tannakian categories with semigroup actions
Ostrowski's theorem implies that $\log(x),\log(x+1),\dots$ are algebraically independent over $\mathbb{C}(x)$. More generally, for a linear differential or difference equation, it is an important problem to find all algebraic dependencies among a non-zero solution $y$ and particular transformations of $y$, such as derivatives of $y$ with respect to parameters, shifts of the arguments, rescaling, etc. In the present paper, we develop a theory of Tannakian categories with semigroup actions, which will be used to attack such questions in full generality, as each linear differential equation gives rise to a Tannakian category. Deligne studied actions of braid groups on categories and obtained a finite collection of axioms that characterizes such actions to apply it to various geometric constructions. In this paper, we find a finite set of axioms that characterizes actions of semigroups that are finite free products of semigroups of the form $\mathbb{N}^n\times \mathbb{Z}/{n_1}\mathbb{Z}\times\cdots\times\mathbb{Z}/{n_r}\mathbb{Z}$ on Tannakian categories. This is the class of semigroups that appear in many applications.

Keywords:semigroup actions on categories, Tannakian categories, difference algebraic groups, differential and difference equations with parameters
Categories:18D10, 12H10, 20G05, 33C05, 33C80, 34K06

24. CJM Online first

Garbagnati, Alice
On K3 surface quotients of K3 or Abelian surfaces
The aim of this paper is to prove that a K3 surface is the minimal model of the quotient of an Abelian surface by a group $G$ (respectively of a K3 surface by an Abelian group $G$) if and only if a certain lattice is primitively embedded in its Néron-Severi group. This allows one to describe the coarse moduli space of the K3 surfaces which are (rationally) $G$-covered by Abelian or K3 surfaces (in the latter case $G$ is an Abelian group). If either $G$ has order 2 or $G$ is cyclic and acts on an Abelian surface, this result was already known, so we extend it to the other cases. Moreover, we prove that a K3 surface $X_G$ is the minimal model of the quotient of an Abelian surface by a group $G$ if and only if a certain configuration of rational curves is present on $X_G$. Again this result was known only in some special cases, in particular if $G$ has order 2 or 3.

Keywords:K3 surfaces, Kummer surfaces, Kummer type lattice, quotient surfaces
Categories:14J28, 14J50, 14J10

25. CJM Online first

Xiao, Jie; Ye, Deping
Anisotropic Sobolev Capacity with Fractional Order
In this paper, we introduce the anisotropic Sobolev capacity with fractional order and develop some basic properties for this new object. Applications to the theory of anisotropic fractional Sobolev spaces are provided. In particular, we give geometric characterizations for a nonnegative Radon measure $\mu$ that naturally induces an embedding of the anisotropic fractional Sobolev class $\dot{\Lambda}_{\alpha,K}^{1,1}$ into the $\mu$-based-Lebesgue-space $L^{n/\beta}_\mu$ with $0\lt \beta\le n$. Also, we investigate the anisotropic fractional $\alpha$-perimeter. Such a geometric quantity can be used to approximate the anisotropic Sobolev capacity with fractional order. Estimation on the constant in the related Minkowski inequality, which is asymptotically optimal as $\alpha\rightarrow 0^+$, will be provided.

Keywords:sharpness, isoperimetric inequality, Minkowski inequality, fractional Sobolev capacity, fractional perimeter
Categories:52A38, 53A15, 53A30
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