176. CJM 2005 (vol 57 pp. 180)
 Somodi, Marius

On the Size of the Wild Set
To every pair of algebraic number fields with isomorphic Witt rings
one can associate a number, called the {\it minimum number of wild
primes}. Earlier investigations have established lower bounds for this
number. In this paper an analysis is presented that expresses the
minimum number of wild primes in terms of the number of wild dyadic
primes. This formula not only gives immediate upper bounds, but can be
considered to be an exact formula for the minimum number of wild
primes.
Categories:11E12, 11E81, 19F15, 11R29 

177. CJM 2004 (vol 56 pp. 897)
 Borwein, Jonathan M.; Borwein, David; Galway, William F.

Finding and Excluding $b$ary MachinType Individual Digit Formulae
Constants with formulae of the form treated by D.~Bailey,
P.~Borwein, and S.~Plouffe (\emph{BBP formulae} to a given base $b$) have
interesting computational properties, such as allowing single
digits in their base $b$ expansion to be independently computed,
and there are hints that they
should be \emph{normal} numbers, {\em i.e.,} that their base $b$ digits
are randomly distributed. We study a formally limited subset of BBP
formulae, which we call \emph{Machintype BBP formulae}, for which it
is relatively easy to determine whether or not a given constant
$\kappa$ has a Machintype BBP formula. In particular, given $b \in
\mathbb{N}$, $b>2$, $b$ not a proper power, a $b$ary Machintype
BBP arctangent formula for $\kappa$ is a formula of the form $\kappa
= \sum_{m} a_m \arctan(b^{m})$, $a_m \in \mathbb{Q}$, while when
$b=2$, we also allow terms of the form $a_m \arctan(1/(12^m))$. Of
particular interest, we show that $\pi$ has no Machintype BBP
arctangent formula when $b \neq 2$. To the best of our knowledge,
when there is no Machintype BBP formula for a constant then no BBP
formula of any form is known for that constant.
Keywords:BBP formulae, Machintype formulae, arctangents,, logarithms, normality, Mersenne primes, Bang's theorem,, Zsigmondy's theorem, primitive prime factors, $p$adic analysis Categories:11Y99, 11A51, 11Y50, 11K36, 33B10 

178. CJM 2004 (vol 56 pp. 673)
 Cali, Élie

DÃ©faut de semistabilitÃ© des courbes elliptiques dans le cas non ramifiÃ©
Let $\overline {\Q_2}$ be an algebraic closure of $\Q_2$ and $K$ be an unramified
finite extension of $\Q_2$ included in $\overline {\Q_2}$. Let $E$ be an elliptic
curve defined over $K$ with additive reduction over $K$, and having an integral
modular invariant. Let us denote by $K_{nr}$ the maximal unramified extension of
$K$ contained in $\overline {\Q_2}$. There exists a smallest finite extension $L$
of $K_{nr}$ over which $E$ has good reduction. We determine in this paper the
degree of the extension $L/K_{nr}$.
Category:11G07 

179. CJM 2004 (vol 56 pp. 612)
 Pál, Ambrus

Solvable Points on Projective Algebraic Curves
We examine the problem of finding rational points defined over
solvable extensions on algebraic curves defined over general fields.
We construct nonsingular, geometrically irreducible projective curves
without solvable points of genus $g$, when $g$ is at least $40$, over
fields of arbitrary characteristic. We prove that every smooth,
geometrically irreducible projective curve of genus $0$, $2$, $3$ or
$4$ defined over any field has a solvable point. Finally we prove
that every genus $1$ curve defined over a local field of
characteristic zero with residue field of characteristic $p$ has a
divisor of degree prime to $6p$ defined over a solvable extension.
Categories:14H25, 11D88 

180. CJM 2004 (vol 56 pp. 373)
 Orton, Louisa

An Elementary Proof of a Weak Exceptional Zero Conjecture
In this paper we extend Darmon's theory of ``integration on $\uh_p\times \uh$''
to cusp forms $f$ of higher even weight. This enables us to prove a ``weak
exceptional zero conjecture'': that when the $p$adic $L$function of $f$ has
an exceptional zero at the central point, the $\mathcal{L}$invariant arising is
independent of a twist by certain Dirichlet characters.
Categories:11F11, 11F67 

181. CJM 2004 (vol 56 pp. 356)
 Murty, M. Ram; Saidak, Filip

NonAbelian Generalizations of the Erd\H osKac Theorem
Let $a$ be a natural number greater than $1$.
Let $f_a(n)$ be the order of $a$ mod $n$.
Denote by $\omega(n)$ the number of distinct
prime factors of $n$. Assuming a weak form
of the generalised Riemann hypothesis, we prove
the following conjecture of Erd\"os and Pomerance:
The number of $n\leq x$ coprime to $a$ satisfying
$$\alpha \leq \frac{\omega(f_a(n))  (\log \log n)^2/2
}{ (\log \log n)^{3/2}/\sqrt{3}} \leq \beta $$
is asymptotic to
$$\left(\frac{ 1 }{ \sqrt{2\pi}} \int_{\alpha}^{\beta}
e^{t^2/2}dt\right)
\frac{x\phi(a) }{ a}, $$
as $x$ tends to infinity.
Keywords:Tur{\' a}n's theorem, Erd{\H o}sKac theorem, Chebotarev density theorem,, Erd{\H o}sPomerance conjecture Categories:11K36, 11K99 

182. CJM 2004 (vol 56 pp. 406)
183. CJM 2004 (vol 56 pp. 23)
 Bennett, Michael A.; Skinner, Chris M.

Ternary Diophantine Equations via Galois Representations and Modular Forms
In this paper, we develop techniques for solving ternary Diophantine
equations of the shape $Ax^n + By^n = Cz^2$, based upon the theory of
Galois representations and modular forms. We subsequently utilize
these methods to completely solve such equations for various choices
of the parameters $A$, $B$ and $C$. We conclude with an application
of our results to certain classical polynomialexponential equations,
such as those of RamanujanNagell type.
Categories:11D41, 11F11, 11G05 

184. CJM 2004 (vol 56 pp. 55)
 Harper, Malcolm

$\mathbb{Z}[\sqrt{14}]$ is Euclidean
We provide the first unconditional proof that the ring $\mathbb{Z}
[\sqrt{14}]$ is a Euclidean domain. The proof is generalized to
other real quadratic fields and to cyclotomic extensions of
$\mathbb{Q}$. It is proved that if $K$ is a real quadratic field
(modulo the existence of two special primes of $K$) or if $K$ is a
cyclotomic extension of $\mathbb{Q}$ then:
$$
the~ring~of~integers~of~K~is~a~Euclidean~domain~if~and~only~if~it~is~a~principal~ideal~domain.
$$
The proof is a modification of the proof of a theorem of Clark and
Murty giving a similar result when $K$ is a totally real extension of
degree at least three. The main changes are a new Motzkintype lemma
and the addition of the large sieve to the argument. These changes
allow application of a powerful theorem due to Bombieri, Friedlander
and Iwaniec in order to obtain the result in the real quadratic case.
The modification also allows the completion of the classification of
cyclotomic extensions in terms of the Euclidean property.
Categories:11R04, 11R11 

185. CJM 2004 (vol 56 pp. 168)
 Pogge, James Todd

On a Certain Residual Spectrum of $\Sp_8$
Let $G=\Sp_{2n}$ be the symplectic group defined over a number
field $F$. Let $\mathbb{A}$ be the ring of adeles. A fundamental
problem in the theory of automorphic forms is to decompose the
right regular representation of $G(\mathbb{A})$ acting on the
Hilbert space $L^2\bigl(G(F)\setminus G(\mathbb{A})\bigr)$. Main
contributions have been made by Langlands. He described, using his
theory of Eisenstein series, an orthogonal decomposition of this
space of the form: $L_{\dis}^2 \bigl( G(F)\setminus G(\mathbb{A})
\bigr)=\bigoplus_{(M,\pi)} L_{\dis}^2(G(F) \setminus G(\mathbb{A})
\bigr)_{(M,\pi)}$, where $(M,\pi)$ is a Levi subgroup with a
cuspidal automorphic representation $\pi$ taken modulo conjugacy
(Here we normalize $\pi$ so that the action of the maximal split
torus in the center of $G$ at the archimedean places is trivial.)
and $L_{\dis}^2\bigl(G(F)\setminus G(\mathbb{A})\bigr)_{(M,\pi)}$
is a space of residues of Eisenstein series associated to
$(M,\pi)$. In this paper, we will completely determine the space
$L_{\dis}^2\bigl(G(F)\setminus G(\mathbb{A})\bigr)_{(M,\pi)}$, when
$M\simeq\GL_2\times\GL_2$. This is the first result on the
residual spectrum for nonmaximal, nonBorel parabolic subgroups,
other than $\GL_n$.
Categories:11F70, 22E55 

186. CJM 2004 (vol 56 pp. 194)
 Saikia, A.

Selmer Groups of Elliptic Curves with Complex Multiplication
Suppose $K$ is an imaginary quadratic field and $E$ is an elliptic curve over a
number field $F$ with complex multiplication by the ring of integers in $K$.
Let $p$ be a rational prime that splits as $\mathfrak{p}_{1}\mathfrak{p}_{2}$
in $K$. Let $E_{p^{n}}$ denote the $p^{n}$division points on $E$. Assume
that $F(E_{p^{n}})$ is abelian over $K$ for all $n\geq 0$. This paper proves
that the Pontrjagin dual of the $\mathfrak{p}_{1}^{\infty}$Selmer group of
$E$ over $F(E_{p^{\infty}})$ is a finitely generated free $\Lambda$module,
where $\Lambda$ is the Iwasawa algebra of $\Gal\bigl(F(E_{p^{\infty}})/
F(E_{\mathfrak{p}_{1}^{\infty}\mathfrak{p}_{2}})\bigr)$. It also gives a simple
formula for the rank of the Pontrjagin dual as a $\Lambda$module.
Categories:11R23, 11G05 

187. CJM 2004 (vol 56 pp. 71)
 Harper, Malcolm; Murty, M. Ram

Euclidean Rings of Algebraic Integers
Let $K$ be a finite Galois extension of the field of rational numbers
with unit rank greater than~3. We prove that the ring of integers of
$K$ is a Euclidean domain if and only if it is a principal ideal
domain. This was previously known under the assumption of the
generalized Riemann hypothesis for Dedekind zeta functions. We now
prove this unconditionally.
Categories:11R04, 11R27, 11R32, 11R42, 11N36 

188. 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 

189. CJM 2003 (vol 55 pp. 933)
190. 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 

191. 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 

192. 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 

193. 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 

194. 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 

195. 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 

196. 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 

197. 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 

198. 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 

199. CJM 2002 (vol 54 pp. 1305)
200. 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 
