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1. CJM Online first

Böcherer, Siegfried; Kikuta, Toshiyuki; Takemori, Sho
Weights of the mod $p$ kernel of the theta operators
Let $\Theta ^{[j]}$ be an analogue of the Ramanujan theta operator for Siegel modular forms. For a given prime $p$, we give the weights of elements of mod $p$ kernel of $\Theta ^{[j]}$, where the mod $p$ kernel of $\Theta ^{[j]}$ is the set of all Siegel modular forms $F$ such that $\Theta ^{[j]}(F)$ is congruent to zero modulo $p$. In order to construct examples of the mod $p$ kernel of $\Theta ^{[j]}$ from any Siegel modular form, we introduce new operators $A^{(j)}(M)$ and show the modularity of $F|A^{(j)}(M)$ when $F$ is a Siegel modular form. Finally, we give some examples of the mod $p$ kernel of $\Theta ^{[j]}$ and the filtrations of some of them.

Keywords:Siegel modular form, congruences for modular forms, Fourier coefficients, Ramanujan's operator, filtration
Categories:11F33, 11F46

2. CJM 2012 (vol 64 pp. 1395)

Rodney, Scott
Existence of Weak Solutions of Linear Subelliptic Dirichlet Problems With Rough Coefficients
This article gives an existence theory for weak solutions of second order non-elliptic linear Dirichlet problems of the form \begin{align*} \nabla'P(x)\nabla u +{\bf HR}u+{\bf S'G}u +Fu &= f+{\bf T'g} \text{ in }\Theta \\ u&=\varphi\text{ on }\partial \Theta. \end{align*} The principal part $\xi'P(x)\xi$ of the above equation is assumed to be comparable to a quadratic form ${\mathcal Q}(x,\xi) = \xi'Q(x)\xi$ that may vanish for non-zero $\xi\in\mathbb{R}^n$. This is achieved using techniques of functional analysis applied to the degenerate Sobolev spaces $QH^1(\Theta)=W^{1,2}(\Theta,Q)$ and $QH^1_0(\Theta)=W^{1,2}_0(\Theta,Q)$ as defined in previous works. Sawyer and Wheeden give a regularity theory for a subset of the class of equations dealt with here.

Keywords:degenerate quadratic forms, linear equations, rough coefficients, subelliptic, weak solutions
Categories:35A01, 35A02, 35D30, 35J70, 35H20

3. CJM 2011 (vol 63 pp. 634)

Lü, Guangshi
On Higher Moments of Fourier Coefficients of Holomorphic Cusp Forms
Let $S_{k}(\Gamma)$ be the space of holomorphic cusp forms of even integral weight $k$ for the full modular group. Let $\lambda_f(n)$ and $\lambda_g(n)$ be the $n$-th normalized Fourier coefficients of two holomorphic Hecke eigencuspforms $f(z), g(z) \in S_{k}(\Gamma)$, respectively. In this paper we are able to show the following results about higher moments of Fourier coefficients of holomorphic cusp forms.\newline (i) For any $\varepsilon>0$, we have \begin{equation*} \sum_{n\leq x}\lambda_f^5(n) \ll_{f,\varepsilon}x^{\frac{15}{16}+\varepsilon} \quad\text{and}\quad\sum_{n\leq x}\lambda_f^7(n) \ll_{f,\varepsilon}x^{\frac{63}{64}+\varepsilon}. \end{equation*} (ii) If $\operatorname{sym}^3\pi_f \ncong \operatorname{sym}^3\pi_g$, then for any $\varepsilon>0$, we have \begin{equation*} \sum_{n \leq x}\lambda_f^3(n)\lambda_g^3(n)\ll_{f,\varepsilon}x^{\frac{31}{32}+\varepsilon}; \end{equation*} If $\operatorname{sym}^2\pi_f \ncong \operatorname{sym}^2\pi_g$, then for any $\varepsilon>0$, we have \[ \sum_{n \leq x}\lambda_f^4(n)\lambda_g^2(n)=cx\log x+c'x+O_{f,\varepsilon}\bigl(x^{\frac{31}{32}+\varepsilon}\bigr); \] If $\operatorname{sym}^2\pi_f \ncong \operatorname{sym}^2\pi_g$ and $\operatorname{sym}^4\pi_f \ncong \operatorname{sym}^4\pi_g$, then for any $\varepsilon>0$, we have \[ \sum_{n \leq x}\lambda_f^4(n)\lambda_g^4(n)=xP(\log x)+O_{f,\varepsilon}\bigl(x^{\frac{127}{128}+\varepsilon}\bigr), \] where $P(x)$ is a polynomial of degree $3$.

Keywords: Fourier coefficients of cusp forms, symmetric power $L$-function
Categories:11F30, , , , 11F11, 11F66

4. CJM 2010 (vol 63 pp. 181)

Ismail, Mourad E. H.; Obermaier, Josef
Characterizations of Continuous and Discrete $q$-Ultraspherical Polynomials
We characterize the continuous $q$-ultraspherical polynomials in terms of the special form of the coefficients in the expansion $\mathcal{D}_q P_n(x)$ in the basis $\{P_n(x)\}$, $\mathcal{D}_q$ being the Askey--Wilson divided difference operator. The polynomials are assumed to be symmetric, and the connection coefficients are multiples of the reciprocal of the square of the $L^2$ norm of the polynomials. A similar characterization is given for the discrete $q$-ultraspherical polynomials. A new proof of the evaluation of the connection coefficients for big $q$-Jacobi polynomials is given.

Keywords:continuous $q$-ultraspherical polynomials, big $q$-Jacobi polynomials, discrete $q$-ultra\-spherical polynomials, Askey--Wilson operator, $q$-difference operator, recursion coefficients
Categories:33D45, 42C05

5. CJM 2009 (vol 61 pp. 762)

D'Cruz, Clare; Puthenpurakal, Tony J.
The Hilbert Coefficients of the Fiber Cone and the $a$-Invariant of the Associated Graded Ring
Let $(A,\m)$ be a Noetherian local ring with infinite residue field and let $I$ be an ideal in $A$ and let $F(I) = \bigoplus_{n \geq 0}I^n/\m I^n$ be the fiber cone of $I$. We prove certain relations among the Hilbert coefficients $f_0(I),f_1(I), f_2(I)$ of $F(I)$ when the $a$-invariant of the associated graded ring $G(I)$ is negative.

Keywords:fiber cone, $a$-invariant, Hilbert coefficients of fiber cone
Categories:13A30, 13D40

6. CJM 2009 (vol 61 pp. 351)

Graham, William; Hunziker, Markus
Multiplication of Polynomials on Hermitian Symmetric spaces and Littlewood--Richardson Coefficients
Let $K$ be a complex reductive algebraic group and $V$ a representation of $K$. Let $S$ denote the ring of polynomials on $V$. Assume that the action of $K$ on $S$ is multiplicity-free. If $\lambda$ denotes the isomorphism class of an irreducible representation of $K$, let $\rho_\lambda\from K \rightarrow GL(V_{\lambda})$ denote the corresponding irreducible representation and $S_\lambda$ the $\lambda$-isotypic component of $S$. Write $S_\lambda \cdot S_\mu$ for the subspace of $S$ spanned by products of $S_\lambda$ and $S_\mu$. If $V_\nu$ occurs as an irreducible constituent of $V_\lambda\otimes V_\mu$, is it true that $S_\nu\subseteq S_\lambda\cdot S_\mu$? In this paper, the authors investigate this question for representations arising in the context of Hermitian symmetric pairs. It is shown that the answer is yes in some cases and, using an earlier result of Ruitenburg, that in the remaining classical cases, the answer is yes provided that a conjecture of Stanley on the multiplication of Jack polynomials is true. It is also shown how the conjecture connects multiplication in the ring $S$ to the usual Littlewood--Richardson rule.

Keywords:Hermitian symmetric spaces, multiplicity free actions, Littlewood--Richardson coefficients, Jack polynomials
Categories:14L30, 22E46

7. CJM 2008 (vol 60 pp. 685)

Savu, Anamaria
Closed and Exact Functions in the Context of Ginzburg--Landau Models
For a general vector field we exhibit two Hilbert spaces, namely the space of so called \emph{closed functions} and the space of \emph{exact functions} and we calculate the codimension of the space of exact functions inside the larger space of closed functions. In particular we provide a new approach for the known cases: the Glauber field and the second-order Ginzburg--Landau field and for the case of the fourth-order Ginzburg--Landau field.

Keywords:Hermite polynomials, Fock space, Fourier coefficients, Fourier transform, group of symmetries
Categories:42B05, 81Q50, 42A16

8. CJM 2005 (vol 57 pp. 1080)

Pritsker, Igor E.
The Gelfond--Schnirelman Method in Prime Number Theory
The original Gelfond--Schnirelman method, proposed in 1936, uses polynomials with integer coefficients and small norms on $[0,1]$ to give a Chebyshev-type lower bound in prime number theory. We study a generalization of this method for polynomials in many variables. Our main result is a lower bound for the integral of Chebyshev's $\psi$-function, expressed in terms of the weighted capacity. This extends previous work of Nair and Chudnovsky, and connects the subject to the potential theory with external fields generated by polynomial-type weights. We also solve the corresponding potential theoretic problem, by finding the extremal measure and its support.

Keywords:distribution of prime numbers, polynomials, integer, coefficients, weighted transfinite diameter, weighted capacity, potentials
Categories:11N05, 31A15, 11C08

9. CJM 2001 (vol 53 pp. 33)

Borwein, Peter; Choi, Kwok-Kwong Stephen
Merit Factors of Polynomials Formed by Jacobi Symbols
We give explicit formulas for the $L_4$ norm (or equivalently for the merit factors) of various sequences of polynomials related to the polynomials $$ f(z) := \sum_{n=0}^{N-1} \leg{n}{N} z^n. $$ and $$ f_t(z) = \sum_{n=0}^{N-1} \leg{n+t}{N} z^n. $$ where $(\frac{\cdot}{N})$ is the Jacobi symbol. Two cases of particular interest are when $N = pq$ is a product of two primes and $p = q+2$ or $p = q+4$. This extends work of H{\o}holdt, Jensen and Jensen and of the authors. This study arises from a number of conjectures of Erd\H{o}s, Littlewood and others that concern the norms of polynomials with $-1,1$ coefficients on the disc. The current best examples are of the above form when $N$ is prime and it is natural to see what happens for composite~$N$.

Keywords:Character polynomial, Class Number, $-1,1$ coefficients, Merit factor, Fekete polynomials, Turyn Polynomials, Littlewood polynomials, Twin Primes, Jacobi Symbols
Categories:11J54, 11B83, 12-04

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