201. 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 

202. CJM 2001 (vol 53 pp. 1194)
203. 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 

204. 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 

205. 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 

206. 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 

207. 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 

208. 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 

209. 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 

210. 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 

211. CJM 2001 (vol 53 pp. 98)
 KhuriMakdisi, 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
twodimensional 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 wellknown congruence relation of
Eichler and Shimura.
Category:11F 

212. 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 

213. CJM 2000 (vol 52 pp. 1121)
 Ballantine, Cristina M.

Ramanujan Type Buildings
We will construct a finite union of finite quotients of the affine
building of the group $\GL_3$ over the field of $p$adic numbers
$\mathbb{Q}_p$. We will view this object as a hypergraph and estimate
the spectrum of its underlying graph.
Keywords:automorphic representations, buildings Category:11F70 

214. CJM 2000 (vol 52 pp. 1269)
 Spriano, Luca

Well Ramified Extensions of Complete Discrete Valuation Fields with Applications to the Kato Conductor
We study extensions $L/K$ of complete discrete valuation fields $K$
with residue field $\oK$ of characteristic $p > 0$, which we do not
assume to be perfect. Our work concerns ramification theory for such
extensions, in particular we show that all classical properties which
are true under the hypothesis {\it ``the residue field extension
$\oL/\oK$ is separable''} are still valid under the more general
hypothesis that the valuation ring extension is monogenic. We also
show that conversely, if classical ramification properties hold true
for an extension $L/K$, then the extension of valuation rings is
monogenic. These are the ``{\it well ramified}'' extensions. We show
that there are only three possible types of well ramified extensions
and we give examples. In the last part of the paper we consider, for
the three types, Kato's generalization of the conductor, which we show
how to bound in certain cases.
Categories:11S, 11S15, 11S20 

215. CJM 2000 (vol 52 pp. 804)
216. CJM 2000 (vol 52 pp. 673)
 Balog, Antal; Wooley, Trevor D.

Sums of Two Squares in Short Intervals
Let $\calS$ denote the set of integers representable as a sum of two
squares. Since $\calS$ can be described as the unsifted elements of a
sieving process of positive dimension, it is to be expected that
$\calS$ has many properties in common with the set of prime numbers.
In this paper we exhibit ``unexpected irregularities'' in the
distribution of sums of two squares in short intervals, a phenomenon
analogous to that discovered by Maier, over a decade ago, in the
distribution of prime numbers. To be precise, we show that there are
infinitely many short intervals containing considerably more elements
of $\calS$ than expected, and infinitely many intervals containing
considerably fewer than expected.
Keywords:sums of two squares, sieves, short intervals, smooth numbers Categories:11N36, 11N37, 11N25 

217. CJM 2000 (vol 52 pp. 737)
218. CJM 2000 (vol 52 pp. 833)
 Mináč, Ján; Smith, Tara L.

WGroups under Quadratic Extensions of Fields
To each field $F$ of characteristic not $2$, one can associate a
certain Galois group $\G_F$, the socalled Wgroup of $F$, which
carries essentially the same information as the Witt ring $W(F)$ of
$F$. In this paper we investigate the connection between $\wg$ and
$\G_{F(\sqrt{a})}$, where $F(\sqrt{a})$ is a proper quadratic
extension of $F$. We obtain a precise description in the case when
$F$ is a pythagorean formally real field and $a = 1$, and show that
the Wgroup of a proper field extension $K/F$ is a subgroup of the
Wgroup of $F$ if and only if $F$ is a formally real pythagorean field
and $K = F(\sqrt{1})$. This theorem can be viewed as an analogue of
the classical ArtinSchreier's theorem describing fields fixed by
finite subgroups of absolute Galois groups. We also obtain precise
results in the case when $a$ is a doublerigid element in $F$. Some
of these results carry over to the general setting.
Categories:11E81, 12D15 

219. CJM 2000 (vol 52 pp. 613)
 Ou, Zhiming M.; Williams, Kenneth S.

Small Solutions of $\phi_1 x_1^2 + \cdots + \phi_n x_n^2 = 0$
Let $\phi_1,\dots,\phi_n$ $(n\geq 2)$ be nonzero integers such that
the equation
$$
\sum_{i=1}^n \phi_i x_i^2 = 0
$$
is solvable in integers $x_1,\dots,x_n$ not all zero. It is shown
that there exists a solution satisfying
$$
0 < \sum_{i=1}^n \phi_i x_i^2 \leq 2 \phi_1 \cdots \phi_n,
$$
and that the constant 2 is best possible.
Keywords:small solutions, diagonal quadratic forms Category:11E25 

220. CJM 2000 (vol 52 pp. 369)
221. CJM 2000 (vol 52 pp. 47)
222. CJM 2000 (vol 52 pp. 172)
 Mao, Zhengyu; Rallis, Stephen

Cubic Base Change for $\GL(2)$
We prove a relative trace formula that establishes the cubic base
change for $\GL(2)$. One also gets a classification of the image
of base change. The case when the field extension is nonnormal
gives an example where a trace formula is used to prove lifting
which is not endoscopic.
Categories:11F70, 11F72 

223. CJM 2000 (vol 52 pp. 31)
 Chan, Heng Huat; Liaw, WenChin

On RussellType Modular Equations
In this paper, we revisit Russelltype modular equations, a
collection of modular equations first studied systematically by
R.~Russell in 1887. We give a proof of Russell's main theorem and
indicate the relations between such equations and the constructions
of Hilbert class fields of imaginary quadratic fields. Motivated by
Russell's theorem, we state and prove its cubic analogue which
allows us to construct Russelltype modular equations in the theory
of signature~$3$.
Categories:33D10, 33C05, 11F11 

224. CJM 1999 (vol 51 pp. 1258)
 Baake, Michael; Moody, Robert V.

Similarity Submodules and Root Systems in Four Dimensions
Lattices and $\ZZ$modules in Euclidean space possess an infinitude
of subsets that are images of the original set under similarity
transformation. We classify such selfsimilar images according to
their indices for certain 4D examples that are related to 4D root
systems, both crystallographic and noncrystallographic. We
encapsulate their statistics in terms of Dirichlet series
generating functions and derive some of their asymptotic properties.
Categories:11S45, 11H05, 52C07 

225. CJM 1999 (vol 51 pp. 1307)
 Johnson, Norman W.; Weiss, Asia Ivić

Quadratic Integers and Coxeter Groups
Matrices whose entries belong to certain rings of algebraic
integers can be associated with discrete groups of transformations
of inversive $n$space or hyperbolic $(n+1)$space
$\mbox{H}^{n+1}$. For small $n$, these may be Coxeter groups,
generated by reflections, or certain subgroups whose generators
include direct isometries of $\mbox{H}^{n+1}$. We show how linear
fractional transformations over rings of rational and (real or
imaginary) quadratic integers are related to the symmetry groups of
regular tilings of the hyperbolic plane or 3space. New light is
shed on the properties of the rational modular group $\PSL_2
(\bbZ)$, the Gaussian modular (Picard) group $\PSL_2 (\bbZ[{\it
i}])$, and the Eisenstein modular group $\PSL_2 (\bbZ[\omega ])$.
Categories:11F06, 20F55, 20G20, 20H10, 22E40 
