176. 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 BrunTitchmarsh theorem for
polynomials over finite fields.
Categories:11T23, 11T06 

177. 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 

178. CJM 2003 (vol 55 pp. 432)
 Zaharescu, Alexandru

Pair Correlation of Squares in $p$Adic Fields
Let $p$ be an odd prime number, $K$ a $p$adic field of degree $r$
over $\mathbf{Q}_p$, $O$ the ring of integers in $K$, $B = \{\beta_1,\dots,
\beta_r\}$ an integral basis of $K$ over $\mathbf{Q}_p$, $u$ a unit in $O$
and consider sets of the form $\mathcal{N}=\{n_1\beta_1+\cdots+n_r\beta_r:
1\leq n_j\leq N_j, 1\leq j\leq r\}$. We show under certain growth
conditions that the pair correlation of $\{uz^2:z\in\mathcal{N}\}$ becomes
Poissonian.
Categories:11S99, 11K06, 1134 

179. 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
onedimensional 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, LÃ©vy process, gamma process, Meixner process Categories:11M06, 60J65, 60E07 

180. CJM 2002 (vol 54 pp. 1305)
181. CJM 2002 (vol 54 pp. 1202)
 Fernández, J.; Lario, JC.; 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 

182. 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 MellinBarnes type for the radial part of our
spherical function.
Categories:22E45, 11F70 

183. 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 LanglandsShahidi 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 

184. CJM 2002 (vol 54 pp. 468)
 Boyd, David W.; RodriguezVillegas, 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 

185. CJM 2002 (vol 54 pp. 449)
 Akrout, H.

ThÃ©orÃ¨me de Vorono\"\i\ dans les espaces symÃ©triques
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:rÃ©seaux, thÃ©orÃ¨me de Vorono\"\i, espaces symÃ©triques Categories:11H06, 53C35 

186. CJM 2002 (vol 54 pp. 263)
 Chaudouard, PierreHenri

IntÃ©grales orbitales pondÃ©rÃ©es sur les algÃ¨bres de Lie : le cas $p$adique
Soit $G$ un groupe rÃ©ductif connexe dÃ©fini sur un corps $p$adique $F$ et $\ggo$
son algÃ¨bre de Lie. Les intÃ©grales orbitales pondÃ©rÃ©es sur $\ggo(F)$ sont des
distributions $J_M(X,f)$$f$ est une fonction testindexÃ©es par les
sousgroupes de LÃ©vi $M$ de $G$ et les Ã©lÃ©ments semisimples rÃ©guliers
$X \in \mgo(F)\cap \ggo_{\reg}$. Leurs analogues sur $G$ sont les principales
composantes du cÃ´tÃ© gÃ©omÃ©trique des formules des traces locale et globale d'Arthur.
Si $M=G$, on retrouve les intÃ©grales orbitales invariantes qui, vues comme fonction
de $X$, sont bornÃ©es sur $\mgo(F)\cap \ggo_{\reg}$~: c'est un rÃ©sultat bien connu
de HarishChandra. Si $M \subsetneq G$, les intÃ©grales orbitales pondÃ©rÃ©es
explosent au voisinage des Ã©lÃ©ments singuliers. Nous construisons dans cet article
de nouvelles intÃ©grales orbitales pondÃ©rÃ©es $J_M^b(X,f)$, Ã©gales Ã $J_M(X,f)$ Ã
un terme correctif prÃ¨s, qui tout en conservant les principales propriÃ©tÃ©s des
prÃ©cÃ©dentes (comportement par conjugaison, dÃ©veloppement en germes, {\it etc.})
restent bornÃ©es quand $X$ parcourt $\mgo(F)\cap\ggo_{\reg}$. Nous montrons
Ã©galement que les intÃ©grales orbitales pondÃ©rÃ©es globales, associÃ©es Ã des
Ã©lÃ©ments semisimples rÃ©guliers, se dÃ©composent en produits de ces nouvelles
intÃ©grales locales.
Categories:22E35, 11F70 

187. 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 HardyLittlewood 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 

188. 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 

189. 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 

190. CJM 2002 (vol 54 pp. 71)
 Choi, KwokKwong Stephen; Liu, Jianya

Small Prime Solutions of Quadratic Equations
Let $b_1,\dots,b_5$ be nonzero 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 

191. CJM 2001 (vol 53 pp. 1194)
192. 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, $a^xb^y=c$, 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 

193. 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 $\sum^\infty_{n=0}p_w(n)x^n=\prod^\infty_{m=1}
(1x^m)^{w(m)}$, where $w(m)$ is a nonnegative 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 

194. 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 MordellWeil group of an elliptic curve
as a Galois module. We consider an elliptic curve $E$ defined over a
number field $K$ whose MordellWeil 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 

195. 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 $o(s1)$ 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 

196. 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, PiatetskiShapiro 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 

197. 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 

198. CJM 2001 (vol 53 pp. 244)
 Goldberg, David; Shahidi, Freydoon

On the Tempered Spectrum of QuasiSplit Classical Groups II
We determine the poles of the standard intertwining operators for a
maximal parabolic subgroup of the quasisplit 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 RankinSelberg 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 

199. 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 finitedimensional
vector space $V$, we analyze the truncated integral of the theta series
corresponding to a SchwartzBruhat 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 

200. CJM 2001 (vol 53 pp. 33)
 Borwein, Peter; Choi, KwokKwong 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
$$
f(z) := \sum_{n=0}^{N1} \leg{n}{N} z^n.
$$
and
$$
f_t(z) = \sum_{n=0}^{N1} \leg{n+t}{N} z^n.
$$
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, 1204 
