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151. CJM 2003 (vol 55 pp. 292)

Pitman, Jim; Yor, Marc
Infinitely Divisible Laws Associated with Hyperbolic Functions
The infinitely divisible distributions on $\mathbb{R}^+$ of random variables $C_t$, $S_t$ and $T_t$ with Laplace transforms $$ \left( \frac{1}{\cosh \sqrt{2\lambda}} \right)^t, \quad \left( \frac{\sqrt{2\lambda}}{\sinh \sqrt{2\lambda}} \right)^t, \quad \text{and} \quad \left( \frac{\tanh \sqrt{2\lambda}}{\sqrt{2\lambda}} \right)^t $$ respectively are characterized for various $t>0$ in a number of different ways: by simple relations between their moments and cumulants, by corresponding relations between the distributions and their L\'evy measures, by recursions for their Mellin transforms, and by differential equations satisfied by their Laplace transforms. Some of these results are interpreted probabilistically via known appearances of these distributions for $t=1$ or $2$ in the description of the laws of various functionals of Brownian motion and Bessel processes, such as the heights and lengths of excursions of a one-dimensional Brownian motion. The distributions of $C_1$ and $S_2$ are also known to appear in the Mellin representations of two important functions in analytic number theory, the Riemann zeta function and the Dirichlet $L$-function associated with the quadratic character modulo~4. Related families of infinitely divisible laws, including the gamma, logistic and generalized hyperbolic secant distributions, are derived from $S_t$ and $C_t$ by operations such as Brownian subordination, exponential tilting, and weak limits, and characterized in various ways.

Keywords:Riemann zeta function, Mellin transform, characterization of distributions, Brownian motion, Bessel process, Lévy process, gamma process, Meixner process
Categories:11M06, 60J65, 60E07

152. CJM 2003 (vol 55 pp. 225)

Banks, William D.; Harcharras, Asma; Shparlinski, Igor E.
Short Kloosterman Sums for Polynomials over Finite Fields
We extend to the setting of polynomials over a finite field certain estimates for short Kloosterman sums originally due to Karatsuba. Our estimates are then used to establish some uniformity of distribution results in the ring $\mathbb{F}_q[x]/M(x)$ for collections of polynomials either of the form $f^{-1}g^{-1}$ or of the form $f^{-1}g^{-1}+afg$, where $f$ and $g$ are polynomials coprime to $M$ and of very small degree relative to $M$, and $a$ is an arbitrary polynomial. We also give estimates for short Kloosterman sums where the summation runs over products of two irreducible polynomials of small degree. It is likely that this result can be used to give an improvement of the Brun-Titchmarsh theorem for polynomials over finite fields.

Categories:11T23, 11T06

153. CJM 2003 (vol 55 pp. 331)

Savitt, David
The Maximum Number of Points on a Curve of Genus $4$ over $\mathbb{F}_8$ is $25$
We prove that the maximum number of rational points on a smooth, geometrically irreducible genus 4 curve over the field of 8 elements is 25. The body of the paper shows that 27 points is not possible by combining techniques from algebraic geometry with a computer verification. The appendix shows that 26 points is not possible by examining the zeta functions.

Categories:11G20, 14H25

154. CJM 2002 (vol 54 pp. 1305)

Vulakh, L. Ya.
Continued Fractions Associated with $\SL_3 (\mathbf{Z})$ and Units in Complex Cubic Fields
Continued fractions associated with $\GL_3 (\mathbf{Z})$ are introduced and applied to find fundamental units in a two-parameter family of complex cubic fields.

Keywords:fundamental units, continued fractions, diophantine approximation, symmetric space
Categories:11R27, 11J70, 11J13

155. CJM 2002 (vol 54 pp. 1202)

Fernández, J.; Lario, J-C.; Rio, A.
Octahedral Galois Representations Arising From $\mathbf{Q}$-Curves of Degree $2$
Generically, one can attach to a $\mathbf{Q}$-curve $C$ octahedral representations $\rho\colon\Gal(\bar{\mathbf{Q}}/\mathbf{Q})\rightarrow\GL_2(\bar\mathbf{F}_3)$ coming from the Galois action on the $3$-torsion of those abelian varieties of $\GL_2$-type whose building block is $C$. When $C$ is defined over a quadratic field and has an isogeny of degree $2$ to its Galois conjugate, there exist such representations $\rho$ having image into $\GL_2(\mathbf{F}_9)$. Going the other way, we can ask which $\mod 3$ octahedral representations $\rho$ of $\Gal(\bar\mathbf{Q}/\mathbf{Q})$ arise from $\mathbf{Q}$-curves in the above sense. We characterize those arising from quadratic $\mathbf{Q}$-curves of degree $2$. The approach makes use of Galois embedding techniques in $\GL_2(\mathbf{F}_9)$, and the characterization can be given in terms of a quartic polynomial defining the $\mathcal{S}_4$-extension of $\mathbf{Q}$ corresponding to the projective representation $\bar{\rho}$.

Categories:11G05, 11G10, 11R32

156. CJM 2002 (vol 54 pp. 673)

Asgari, Mahdi
Local $L$-Functions for Split Spinor Groups
We study the local $L$-functions for Levi subgroups in split spinor groups defined via the Langlands-Shahidi method and prove a conjecture on their holomorphy in a half plane. These results have been used in the work of Kim and Shahidi on the functorial product for $\GL_2 \times \GL_3$.

Category:11F70

157. CJM 2002 (vol 54 pp. 828)

Moriyama, Tomonori
Spherical Functions for the Semisimple Symmetric Pair $\bigl( \Sp(2,\mathbb{R}), \SL(2,\mathbb{C}) \bigr)$
Let $\pi$ be an irreducible generalized principal series representation of $G = \Sp(2,\mathbb{R})$ induced from its Jacobi parabolic subgroup. We show that the space of algebraic intertwining operators from $\pi$ to the representation induced from an irreducible admissible representation of $\SL(2,\mathbb{C})$ in $G$ is at most one dimensional. Spherical functions in the title are the images of $K$-finite vectors by this intertwining operator. We obtain an integral expression of Mellin-Barnes type for the radial part of our spherical function.

Categories:22E45, 11F70

158. CJM 2002 (vol 54 pp. 468)

Boyd, David W.; Rodriguez-Villegas, Fernando
Mahler's Measure and the Dilogarithm (I)
An explicit formula is derived for the logarithmic Mahler measure $m(P)$ of $P(x,y) = p(x)y - q(x)$, where $p(x)$ and $q(x)$ are cyclotomic. This is used to find many examples of such polynomials for which $m(P)$ is rationally related to the Dedekind zeta value $\zeta_F (2)$ for certain quadratic and quartic fields.

Categories:11G40, 11R06, 11Y35

159. CJM 2002 (vol 54 pp. 449)

Akrout, H.
Théorème de Vorono\"\i\ dans les espaces symétriques
On d\'emontre un th\'eor\`eme de Vorono\"\i\ (caract\'erisation des maxima locaux de l'invariant d'Hermite) pour les familles de r\'eseaux param\'etr\'ees par les espaces sym\'etriques irr\'e\-ductibles non exceptionnels de type non compact. We prove a theorem of Vorono\"\i\ type (characterisation of local maxima of the Hermite invariant) for the lattices parametrized by irreducible nonexceptional symmetric spaces of noncompact type.

Keywords:réseaux, théorème de Vorono\"\i, espaces symétriques
Categories:11H06, 53C35

160. CJM 2002 (vol 54 pp. 352)

Haines, Thomas J.
On Connected Components of Shimura Varieties
We study the cohomology of connected components of Shimura varieties $S_{K^p}$ coming from the group $\GSp_{2g}$, by an approach modeled on the stabilization of the twisted trace formula, due to Kottwitz and Shelstad. More precisely, for each character $\olomega$ on the group of connected components of $S_{K^p}$ we define an operator $L(\omega)$ on the cohomology groups with compact supports $H^i_c (S_{K^p}, \olbbQ_\ell)$, and then we prove that the virtual trace of the composition of $L(\omega)$ with a Hecke operator $f$ away from $p$ and a sufficiently high power of a geometric Frobenius $\Phi^r_p$, can be expressed as a sum of $\omega$-{\em weighted} (twisted) orbital integrals (where $\omega$-{\em weighted} means that the orbital integrals and twisted orbital integrals occuring here each have a weighting factor coming from the character $\olomega$). As the crucial step, we define and study a new invariant $\alpha_1 (\gamma_0; \gamma, \delta)$ which is a refinement of the invariant $\alpha (\gamma_0; \gamma, \delta)$ defined by Kottwitz. This is done by using a theorem of Reimann and Zink.

Categories:14G35, 11F70

161. CJM 2002 (vol 54 pp. 417)

Wooley, Trevor D.
Slim Exceptional Sets for Sums of Cubes
We investigate exceptional sets associated with various additive problems involving sums of cubes. By developing a method wherein an exponential sum over the set of exceptions is employed explicitly within the Hardy-Littlewood method, we are better able to exploit excess variables. By way of illustration, we show that the number of odd integers not divisible by $9$, and not exceeding $X$, that fail to have a representation as the sum of $7$ cubes of prime numbers, is $O(X^{23/36+\eps})$. For sums of eight cubes of prime numbers, the corresponding number of exceptional integers is $O(X^{11/36+\eps})$.

Keywords:Waring's problem, exceptional sets
Categories:11P32, 11P05, 11P55

162. CJM 2002 (vol 54 pp. 263)

Chaudouard, Pierre-Henri
Intégrales orbitales pondérées sur les algèbres de Lie : le cas $p$-adique
Soit $G$ un groupe réductif connexe défini sur un corps $p$-adique $F$ et $\ggo$ son algèbre de Lie. Les intégrales orbitales pondérées sur $\ggo(F)$ sont des distributions $J_M(X,f)$---$f$ est une fonction test---indexées par les sous-groupes de Lévi $M$ de $G$ et les éléments semi-simples réguliers $X \in \mgo(F)\cap \ggo_{\reg}$. Leurs analogues sur $G$ sont les principales composantes du côté géométrique des formules des traces locale et globale d'Arthur. Si $M=G$, on retrouve les intégrales orbitales invariantes qui, vues comme fonction de $X$, sont bornées sur $\mgo(F)\cap \ggo_{\reg}$~: c'est un résultat bien connu de Harish-Chandra. Si $M \subsetneq G$, les intégrales orbitales pondérées explosent au voisinage des éléments singuliers. Nous construisons dans cet article de nouvelles intégrales orbitales pondérées $J_M^b(X,f)$, égales à $J_M(X,f)$ à un terme correctif près, qui tout en conservant les principales propriétés des précédentes (comportement par conjugaison, développement en germes, {\it etc.}) restent bornées quand $X$ parcourt $\mgo(F)\cap\ggo_{\reg}$. Nous montrons également que les intégrales orbitales pondérées globales, associées à des éléments semi-simples réguliers, se décomposent en produits de ces nouvelles intégrales locales.

Categories:22E35, 11F70

163. CJM 2002 (vol 54 pp. 92)

Mezo, Paul
Comparisons of General Linear Groups and their Metaplectic Coverings I
We prepare for a comparison of global trace formulas of general linear groups and their metaplectic coverings. In particular, we generalize the local metaplectic correspondence of Flicker and Kazhdan and describe the terms expected to appear in the invariant trace formulas of the above covering groups. The conjectural trace formulas are then placed into a form suitable for comparison.

Categories:11F70, 11F72, 22E50

164. CJM 2002 (vol 54 pp. 71)

Choi, Kwok-Kwong Stephen; Liu, Jianya
Small Prime Solutions of Quadratic Equations
Let $b_1,\dots,b_5$ be non-zero integers and $n$ any integer. Suppose that $b_1 + \cdots + b_5 \equiv n \pmod{24}$ and $(b_i,b_j) = 1$ for $1 \leq i < j \leq 5$. In this paper we prove that \begin{enumerate}[(ii)] \item[(i)] if $b_j$ are not all of the same sign, then the above quadratic equation has prime solutions satisfying $p_j \ll \sqrt{|n|} + \max \{|b_j|\}^{20+\ve}$; and \item[(ii)] if all $b_j$ are positive and $n \gg \max \{|b_j|\}^{41+ \ve}$, then the quadratic equation $b_1 p_1^2 + \cdots + b_5 p_5^2 = n$ is soluble in primes $p_j$. \end{enumerate}

Categories:11P32, 11P05, 11P55

165. CJM 2001 (vol 53 pp. 1194)

Louboutin, Stéphane
Explicit Upper Bounds for Residues of Dedekind Zeta Functions and Values of $L$-Functions at $s=1$, and Explicit Lower Bounds for Relative Class Numbers of $\CM$-Fields
We provide the reader with a uniform approach for obtaining various useful explicit upper bounds on residues of Dedekind zeta functions of numbers fields and on absolute values of values at $s=1$ of $L$-series associated with primitive characters on ray class groups of number fields. To make it quite clear to the reader how useful such bounds are when dealing with class number problems for $\CM$-fields, we deduce an upper bound for the root discriminants of the normal $\CM$-fields with (relative) class number one.

Keywords:Dedekind zeta functions, $L$-functions, relative class numbers, $\CM$-fields
Categories:11R42, 11R29

166. CJM 2001 (vol 53 pp. 897)

Bennett, Michael A.
On Some Exponential Equations of S.~S.~Pillai
In this paper, we establish a number of theorems on the classic Diophantine equation of S.~S.~Pillai, $a^x-b^y=c$, where $a$, $b$ and $c$ are given nonzero integers with $a,b \geq 2$. In particular, we obtain the sharp result that there are at most two solutions in positive integers $x$ and $y$ and deduce a variety of explicit conditions under which there exists at most a single such solution. These improve or generalize prior work of Le, Leveque, Pillai, Scott and Terai. The main tools used include lower bounds for linear forms in the logarithms of (two) algebraic numbers and various elementary arguments.

Categories:11D61, 11D45, 11J86

167. CJM 2001 (vol 53 pp. 866)

Yang, Yifan
Inverse Problems for Partition Functions
Let $p_w(n)$ be the weighted partition function defined by the generating function $\sum^\infty_{n=0}p_w(n)x^n=\prod^\infty_{m=1} (1-x^m)^{-w(m)}$, where $w(m)$ is a non-negative arithmetic function. Let $P_w(u)=\sum_{n\le u}p_w(n)$ and $N_w(u)=\sum_{n\le u}w(n)$ be the summatory functions for $p_w(n)$ and $w(n)$, respectively. Generalizing results of G.~A.~Freiman and E.~E.~Kohlbecker, we show that, for a large class of functions $\Phi(u)$ and $\lambda(u)$, an estimate for $P_w(u)$ of the form $\log P_w(u)=\Phi(u)\bigl\{1+O(1/\lambda(u)\bigr)\bigr\}$ $(u\to\infty)$ implies an estimate for $N_w(u)$ of the form $N_w(u)=\Phi^\ast(u)\bigl\{1+O\bigl(1/\log\lambda(u)\bigr)\bigr\}$ $(u\to\infty)$ with a suitable function $\Phi^\ast(u)$ defined in terms of $\Phi(u)$. We apply this result and related results to obtain characterizations of the Riemann Hypothesis and the Generalized Riemann Hypothesis in terms of the asymptotic behavior of certain weighted partition functions.

Categories:11P82, 11M26, 40E05

168. CJM 2001 (vol 53 pp. 449)

Akbary, Amir; Murty, V. Kumar
Descending Rational Points on Elliptic Curves to Smaller Fields
In this paper, we study the Mordell-Weil group of an elliptic curve as a Galois module. We consider an elliptic curve $E$ defined over a number field $K$ whose Mordell-Weil rank over a Galois extension $F$ is $1$, $2$ or $3$. We show that $E$ acquires a point (points) of infinite order over a field whose Galois group is one of $C_n \times C_m$ ($n= 1, 2, 3, 4, 6, m= 1, 2$), $D_n \times C_m$ ($n= 2, 3, 4, 6, m= 1, 2$), $A_4 \times C_m$ ($m=1,2$), $S_4 \times C_m$ ($m=1,2$). Next, we consider the case where $E$ has complex multiplication by the ring of integers $\o$ of an imaginary quadratic field $\k$ contained in $K$. Suppose that the $\o$-rank over a Galois extension $F$ is $1$ or $2$. If $\k\neq\Q(\sqrt{-1})$ and $\Q(\sqrt{-3})$ and $h_{\k}$ (class number of $\k$) is odd, we show that $E$ acquires positive $\o$-rank over a cyclic extension of $K$ or over a field whose Galois group is one of $\SL_2(\Z/3\Z)$, an extension of $\SL_2(\Z/3\Z)$ by $\Z/2\Z$, or a central extension by the dihedral group. Finally, we discuss the relation of the above results to the vanishing of $L$-functions.

Categories:11G05, 11G40, 11R32, 11R33

169. CJM 2001 (vol 53 pp. 414)

Rivat, Joël; Sargos, Patrick
Nombres premiers de la forme $\floor{n^c}$
For $c>1$ we denote by $\pi_c(x)$ the number of integers $n \leq x$ such that $\floor{n^c}$ is prime. In 1953, Piatetski-Shapiro has proved that $\pi_c(x) \sim \frac{x}{c\log x}$, $x \rightarrow +\infty$ holds for $c<12/11$. Many authors have extended this range, which measures our progress in exponential sums techniques. In this article we obtain $c < 1.16117\dots\;$.

Categories:11L07, 11L20, 11N05

170. CJM 2001 (vol 53 pp. 310)

Ito, Hiroshi
On a Product Related to the Cubic Gauss Sum, III
We have seen, in the previous works [5], [6], that the argument of a certain product is closely connected to that of the cubic Gauss sum. Here the absolute value of the product will be investigated.

Keywords:Gauss sum, Lagrange resolvent
Categories:11L05, 11R33

171. CJM 2001 (vol 53 pp. 434)

van der Poorten, Alfred J.; Williams, Kenneth S.
Values of the Dedekind Eta Function at Quadratic Irrationalities: Corrigendum
Habib Muzaffar of Carleton University has pointed out to the authors that in their paper [A] only the result \[ \pi_{K,d}(x)+\pi_{K^{-1},d}(x)=\frac{1}{h(d)}\frac{x}{\log x}+O_{K,d}\Bigl(\frac {x}{\log^2x}\Bigr) \] follows from the prime ideal theorem with remainder for ideal classes, and not the stronger result \[ \pi_{K,d}(x)=\frac{1}{2h(d)}\frac{x}{\log x}+O_{K,d}\Bigl(\frac {x}{\log^2x}\Bigr) \] stated in Lemma~5.2. This necessitates changes in Sections~5 and 6 of [A]. The main results of the paper are not affected by these changes. It should also be noted that, starting on page 177 of [A], each and every occurrence of $o(s-1)$ should be replaced by $o(1)$. Sections~5 and 6 of [A] have been rewritten to incorporate the above mentioned correction and are given below. They should replace the original Sections~5 and 6 of [A].

Keywords:Dedekind eta function, quadratic irrationalities, binary quadratic forms, form class group
Categories:11F20, 11E45

172. CJM 2001 (vol 53 pp. 244)

Goldberg, David; Shahidi, Freydoon
On the Tempered Spectrum of Quasi-Split Classical Groups II
We determine the poles of the standard intertwining operators for a maximal parabolic subgroup of the quasi-split unitary group defined by a quadratic extension $E/F$ of $p$-adic fields of characteristic zero. We study the case where the Levi component $M \simeq \GL_n (E) \times U_m (F)$, with $n \equiv m$ $(\mod 2)$. This, along with earlier work, determines the poles of the local Rankin-Selberg product $L$-function $L(s, \tau' \times \tau)$, with $\tau'$ an irreducible unitary supercuspidal representation of $\GL_n (E)$ and $\tau$ a generic irreducible unitary supercuspidal representation of $U_m (F)$. The results are interpreted using the theory of twisted endoscopy.

Categories:22E50, 11S70

173. 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

174. CJM 2001 (vol 53 pp. 122)

Levy, Jason
A Truncated Integral of the Poisson Summation Formula
Let $G$ be a reductive algebraic group defined over $\bQ$, with anisotropic centre. Given a rational action of $G$ on a finite-dimensional vector space $V$, we analyze the truncated integral of the theta series corresponding to a Schwartz-Bruhat function on $V(\bA)$. The Poisson summation formula then yields an identity of distributions on $V(\bA)$. The truncation used is due to Arthur.

Categories:11F99, 11F72

175. CJM 2001 (vol 53 pp. 98)

Khuri-Makdisi, Kamal
On the Curves Associated to Certain Rings of Automorphic Forms
In a 1987 paper, Gross introduced certain curves associated to a definite quaternion algebra $B$ over $\Q$; he then proved an analog of his result with Zagier for these curves. In Gross' paper, the curves were defined in a somewhat {\it ad hoc\/} manner. In this article, we present an interpretation of these curves as projective varieties arising from graded rings of automorphic forms on $B^\times$, analogously to the construction in the Satake compactification. To define such graded rings, one needs to introduce a ``multiplication'' of automorphic forms that arises from the representation ring of $B^\times$. The resulting curves are unions of projective lines equipped with a collection of Hecke correspondences. They parametrize two-dimensional complex tori with quaternionic multiplication. In general, these complex tori are not abelian varieties; they are algebraic precisely when they correspond to $\CM$ points on these curves, and are thus isogenous to a product $E \times E$, where $E$ is an elliptic curve with complex multiplication. For these $\CM$ points one can make a relation between the action of the $p$-th Hecke operator and Frobenius at $p$, similar to the well-known congruence relation of Eichler and Shimura.

Category:11F
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