176. CJM 2003 (vol 55 pp. 1191)
 Granville, Andrew; Soundararajan, K.

Decay of Mean Values of Multiplicative Functions
For given multiplicative function $f$, with $f(n) \leq 1$ for all
$n$, we are interested in how fast its mean value $(1/x) \sum_{n\leq
x} f(n)$ converges. Hal\'asz showed that this depends on the minimum
$M$ (over $y\in \mathbb{R}$) of $\sum_{p\leq x} \bigl( 1  \Re (f(p)
p^{iy}) \bigr) / p$, and subsequent authors gave the upper bound $\ll
(1+M) e^{M}$. For many applications it is necessary to have explicit
constants in this and various related bounds, and we provide these via
our own variant of the Hal\'aszMontgomery lemma (in fact the constant
we give is best possible up to a factor of 10). We also develop a new
type of hybrid bound in terms of the location of the absolute value of
$y$ that minimizes the sum above. As one application we give bounds
for the least representatives of the cosets of the $k$th powers
mod~$p$.
Categories:11N60, 11N56, 10K20, 11N37 

177. CJM 2003 (vol 55 pp. 897)
 Archinard, Natália

Hypergeometric Abelian Varieties
In this paper, we construct abelian varieties associated to Gauss' and
AppellLauricella hypergeometric series.
Abelian varieties of this kind and the algebraic curves we define
to construct them were considered by several authors in settings
ranging from monodromy groups (Deligne, Mostow), exceptional sets
(Cohen, Wolfart, W\"ustholz), modular embeddings (Cohen, Wolfart) to
CMtype (Cohen, Shiga, Wolfart) and modularity (Darmon).
Our contribution is to provide a complete, explicit and selfcontained
geometric construction.
Categories:11, 14 

178. CJM 2003 (vol 55 pp. 933)
179. CJM 2003 (vol 55 pp. 711)
 Broughan, Kevin A.

Adic Topologies for the Rational Integers
A topology on $\mathbb{Z}$, which gives a nice proof that the
set of prime integers is infinite, is characterised and examined.
It is found to be homeomorphic to $\mathbb{Q}$, with a compact
completion homeomorphic to the Cantor set. It has a natural place
in a family of topologies on $\mathbb{Z}$, which includes the
$p$adics, and one in which the set of rational primes $\mathbb{P}$
is dense. Examples from number theory are given, including the
primes and squares, Fermat numbers, Fibonacci numbers and $k$free
numbers.
Keywords:$p$adic, metrizable, quasivaluation, topological ring,, completion, inverse limit, diophantine equation, prime integers,, Fermat numbers, Fibonacci numbers Categories:11B05, 11B25, 11B50, 13J10, 13B35 

180. CJM 2003 (vol 55 pp. 673)
 Anderson, Greg W.; Ouyang, Yi

A Note on Cyclotomic Euler Systems and the Double Complex Method
Let $\FF$ be a finite real abelian extension of $\QQ$. Let $M$ be an odd
positive integer. For every squarefree positive integer $r$ the prime
factors of which are congruent to $1$ modulo $M$ and split completely
in $\FF$, the corresponding Kolyvagin class $\kappa_r\in\FF^{\times}/
\FF^{\times M}$ satisfies a remarkable and crucial recursion which
for each prime number $\ell$ dividing $r$ determines the order of
vanishing of $\kappa_r$ at each place of $\FF$ above $\ell$ in terms
of $\kappa_{r/\ell}$. In this note we give the recursion a new and
universal interpretation with the help of the double complex method
introduced by Anderson and further developed by Das and Ouyang. Namely,
we show that the recursion satisfied by Kolyvagin classes is the
specialization of a universal recursion independent of $\FF$ satisfied
by universal Kolyvagin classes in the group cohomology of the universal
ordinary distribution {\it \`a la\/} Kubert tensored with $\ZZ/M\ZZ$.
Further, we show by a method involving a variant of the diagonal shift
operation introduced by Das that certain group cohomology classes belonging
(up to sign) to a basis previously constructed by Ouyang also satisfy the
universal recursion.
Categories:11R18, 11R23, 11R34 

181. 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 

182. 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 

183. 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 

184. 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 

185. CJM 2003 (vol 55 pp. 353)
 Silberger, Allan J.; Zink, ErnstWilhelm

Weak Explicit Matching for Level Zero Discrete Series of Unit Groups of $\mathfrak{p}$Adic Simple Algebras
Let $F$ be a $p$adic local field and let $A_i^\times$ be the unit
group of a central simple $F$algebra $A_i$ of reduced degree $n>1$
($i=1,2$). Let $\mathcal{R}^2 (A_i^\times)$ denote the set of
irreducible discrete series representations of $A_i^\times$. The
``Abstract Matching Theorem'' asserts the existence of a bijection,
the ``JacquetLanglands'' map, $\mathcal{J} \mathcal{L}_{A_2,A_1}
\colon \mathcal{R}^2 (A_1^\times) \to \mathcal{R}^2 (A_2^\times)$
which, up to known sign, preserves character values for regular
elliptic elements. This paper addresses the question of explicitly
describing the map $\mathcal{J} \mathcal{L}$, but only for ``level
zero'' representations. We prove that the restriction $\mathcal{J}
\mathcal{L}_{A_2,A_1} \colon \mathcal{R}_0^2 (A_1^\times) \to
\mathcal{R}_0^2 (A_2^\times)$ is a bijection of level zero discrete
series (Proposition~3.2) and we give a parameterization of the set of
unramified twist classes of level zero discrete series which does not
depend upon the algebra $A_i$ and is invariant under $\mathcal{J}
\mathcal{L}_{A_2,A_1}$ (Theorem~4.1).
Categories:22E50, 11R39 

186. 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 

187. CJM 2002 (vol 54 pp. 1305)
188. 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 

189. 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 

190. 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 

191. 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 

192. 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 

193. 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 

194. 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 

195. 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 

196. 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 

197. CJM 2001 (vol 53 pp. 1194)
198. 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 

199. 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 

200. 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 
