201. 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 

202. 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 

203. CJM 2000 (vol 52 pp. 804)
204. 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 

205. CJM 2000 (vol 52 pp. 369)
206. 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 

207. 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 

208. CJM 2000 (vol 52 pp. 47)
209. 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 

210. 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 

211. CJM 1999 (vol 51 pp. 952)
 Deitmar, Anton; Hoffmann, Werner

On Limit Multiplicities for Spaces of Automorphic Forms
Let $\Gamma$ be a rankone arithmetic subgroup of a
semisimple Lie group~$G$. For fixed $K$Type, the spectral
side of the Selberg trace formula defines a distribution
on the space of infinitesimal characters of~$G$, whose
discrete part encodes the dimensions of the spaces of
squareintegrable $\Gamma$automorphic forms. It is shown
that this distribution converges to the Plancherel measure
of $G$ when $\Ga$ shrinks to the trivial group in a certain
restricted way. The analogous assertion for cocompact
lattices $\Gamma$ follows from results of DeGeorgeWallach
and Delorme.
Keywords:limit multiplicities, automorphic forms, noncompact quotients, Selberg trace formula, functional calculus Categories:11F72, 22E30, 22E40, 43A85, 58G25 

212. CJM 1999 (vol 51 pp. 1020)
 Kozlov, Dmitry N.

On Functions Satisfying Modular Equations for Infinitely Many Primes
In this paper we study properties of the functions which satisfy
modular equations for infinitely many primes. The two main results
are:
\begin{enumerate}
\item[1)] every such function is analytic in the upper half plane;
\item[2)] if such function takes the same value in two different
points $z_1$ and $z_2$ then there exists an $f$preserving analytic
bijection between neighbourhoods of $z_1$ and $z_2$.
\end{enumerate}
Category:11Mxx 

213. CJM 1999 (vol 51 pp. 835)
 Kim, Henry H.

LanglandsShahidi Method and Poles of Automorphic $L$Functions: Application to Exterior Square $L$Functions
In this paper we use LanglandsShahidi method and the result of
Langlands which says that non selfconjugate maximal parabolic
subgroups do not contribute to the residual spectrum, to prove the
holomorphy of several \emph{completed} automorphic $L$functions on the
whole complex plane which appear in constant terms of the Eisenstein
series. They include the exterior square $L$functions of $\GL_n$, $n$
odd, the RankinSelberg $L$functions of $\GL_n\times \GL_m$, $n\ne m$,
and $L$functions $L(s,\sigma,r)$, where $\sigma$ is a generic
cuspidal representation of $\SO_{10}$ and $r$ is the halfspin
representation of $\GSpin(10, \mathbb{C})$. The main part is
proving the holomorphy and nonvanishing of the local normalized
intertwining operators by reducing them to natural conjectures in
harmonic analysis, such as standard module conjecture.
Categories:11F, 22E 

214. CJM 1999 (vol 51 pp. 771)
 Flicker, Yuval Z.

Stable BiPeriod Summation Formula and Transfer Factors
This paper starts by introducing a biperiodic summation formula
for automorphic forms on a group $G(E)$, with periods by a subgroup
$G(F)$, where $E/F$ is a quadratic extension of number fields. The
split case, where $E = F \oplus F$, is that of the standard trace
formula. Then it introduces a notion of stable biconjugacy, and
stabilizes the geometric side of the biperiod summation formula.
Thus weighted sums in the stable biconjugacy class are expressed
in terms of stable biorbital integrals. These stable integrals
are on the same endoscopic groups $H$ which occur in the case of
standard conjugacy.
The spectral side of the biperiod summation formula involves
periods, namely integrals over the group of $F$adele points of
$G$, of cusp forms on the group of $E$adele points on the group
$G$. Our stabilization suggests that such cusp formswith non
vanishing periodsand the resulting biperiod distributions
associated to ``periodic'' automorphic forms, are related to
analogous biperiod distributions associated to ``periodic''
automorphic forms on the endoscopic symmetric spaces $H(E)/H(F)$.
This offers a sharpening of the theory of liftings, where periods
play a key role.
The stabilization depends on the ``fundamental lemma'', which
conjectures that the unit elements of the Hecke algebras on $G$ and
$H$ have matching orbital integrals. Even in stating this
conjecture, one needs to introduce a ``transfer factor''. A
generalization of the standard transfer factor to the biperiodic
case is introduced. The generalization depends on a new definition
of the factors even in the standard case.
Finally, the fundamental lemma is verified for $\SL(2)$.
Categories:11F72, 11F70, 14G27, 14L35 

215. CJM 1999 (vol 51 pp. 266)
 Deitmar, Anton; Hoffman, Werner

Spectral Estimates for Towers of Noncompact Quotients
We prove a uniform upper estimate on the number of cuspidal
eigenvalues of the $\Ga$automorphic Laplacian below a given bound
when $\Ga$ varies in a family of congruence subgroups of a given
reductive linear algebraic group. Each $\Ga$ in the family is assumed
to contain a principal congruence subgroup whose index in $\Ga$ does
not exceed a fixed number. The bound we prove depends linearly on the
covolume of $\Ga$ and is deduced from the analogous result about the
cutoff Laplacian. The proof generalizes the heatkernel method which
has been applied by Donnelly in the case of a fixed lattice~$\Ga$.
Categories:11F72, 58G25, 22E40 

216. CJM 1999 (vol 51 pp. 225)
 Betke, U.; Böröczky, K.

Asymptotic Formulae for the Lattice Point Enumerator
Let $M$ be a convex body such that the boundary has positive
curvature. Then by a well developed theory dating back to Landau and
Hlawka for large $\lambda$ the number of lattice points in $\lambda M$
is given by $G(\lambda M) =V(\lambda M) + O(\lambda^{d1\varepsilon
(d)})$ for some positive $\varepsilon(d)$. Here we give for general
convex bodies the weaker estimate
\[
\left G(\lambda M) V(\lambda M) \right 
\le \frac{1}{2} S_{\Z^d}(M) \lambda^{d1}+o(\lambda^{d1})
\]
where $S_{\Z^d}(M)$ denotes the lattice surface area of $M$. The term
$S_{\Z^d}(M)$ is optimal for all convex bodies and $o(\lambda^{d1})$
cannot be improved in general. We prove that the same estimate even
holds if we allow small deformations of $M$.
Further we deal with families $\{P_\lambda\}$ of convex bodies where
the only condition is that the inradius tends to infinity. Here we have
\[
\left G(P_\lambda)V(P_\lambda) \right
\le dV(P_\lambda,K;1)+o \bigl( S(P_\lambda) \bigr)
\]
where the convex body $K$ satisfies some simple condition,
$V(P_\lambda,K;1)$ is some mixed volume and $S(P_\lambda)$ is the
surface area of $P_\lambda$.
Categories:11P21, 52C07 

217. CJM 1999 (vol 51 pp. 10)
 Chacron, M.; Tignol, J.P.; Wadsworth, A. R.

Tractable Fields
A field $F$ is said to be tractable when a condition
described below on the simultaneous representation of
quaternion algebras holds over $F$. It is shown
that a global field $F$ is tractable i{f}f $F$ has
at most one dyadic place. Several other examples
of tractable and nontractable fields are given.
Categories:12E15, 11R52 

218. CJM 1999 (vol 51 pp. 164)
 Tan, Victor

Poles of Siegel Eisenstein Series on $U(n,n)$
Let $U(n,n)$ be the rank $n$ quasisplit unitary group over a
number field. We show that the normalized Siegel Eisenstein series
of $U(n,n)$ has at most simple poles at the integers or half
integers in certain strip of the complex plane.
Categories:11F70, 11F27, 22E50 

219. CJM 1999 (vol 51 pp. 130)
 Savin, Gordan; Gan, Wee Teck

The Dual Pair $G_2 \times \PU_3 (D)$ ($p$Adic Case)
We study the correspondence of representations arising by
restricting the minimal representation of the linear group of type
$E_7$ and relative rank $4$. The main tool is computations of the
Jacquet modules of the minimal representation with respect to
maximal parabolic subgroups of $G_2$ and $\PU_3(D)$.
Categories:22E35, 22E50, 11F70 

220. CJM 1999 (vol 51 pp. 176)
 van der Poorten, Alfred; Williams, Kenneth S.

Values of the Dedekind Eta Function at Quadratic Irrationalities
Let $d$ be the discriminant of an imaginary quadratic field. Let
$a$, $b$, $c$ be integers such that
$$
b^2  4ac = d, \quad a > 0, \quad \gcd (a,b,c) = 1.
$$
The value of $\bigl\eta \bigl( (b + \sqrt{d})/2a \bigr) \bigr$ is
determined explicitly, where $\eta(z)$ is Dedekind's eta function
$$
\eta (z) = e^{\pi iz/12} \prod^\ty_{m=1} (1  e^{2\pi imz})
\qquad \bigl( \im(z) > 0 \bigr). %\eqno({\rm im}(z)>0).
$$
Keywords:Dedekind eta function, quadratic irrationalities, binary quadratic forms, form class group Categories:11F20, 11E45 

221. CJM 1998 (vol 50 pp. 1253)
 LópezBautista, Pedro Ricardo; VillaSalvador, Gabriel Daniel

Integral representation of $p$class groups in ${\Bbb Z}_p$extensions and the Jacobian variety
For an arbitrary finite Galois $p$extension $L/K$ of
$\zp$cyclotomic number fields of $\CM$type with Galois group $G =
\Gal(L/K)$ such that the Iwasawa invariants $\mu_K^$, $ \mu_L^$
are zero, we obtain unconditionally and explicitly the Galois
module structure of $\clases$, the minus part of the $p$subgroup
of the class group of $L$. For an arbitrary finite Galois
$p$extension $L/K$ of algebraic function fields of one variable
over an algebraically closed field $k$ of characteristic $p$ as its
exact field of constants with Galois group $G = \Gal(L/K)$ we
obtain unconditionally and explicitly the Galois module structure
of the $p$torsion part of the Jacobian variety $J_L(p)$ associated
to $L/k$.
Keywords:${\Bbb Z}_p$extensions, Iwasawa's theory, class group, integral representation, fields of algebraic functions, Jacobian variety, Galois module structure Categories:11R33, 11R23, 11R58, 14H40 

222. CJM 1998 (vol 50 pp. 1323)
 Morales, Jorge

L'invariant de HasseWitt de la forme de Killing
Nous montrons que l'invariant de HasseWitt de la forme de Killing
d'une alg{\`e}bre de Lie semisimple $L$ s'exprime {\`a} l'aide de
l'invariant de Tits de la repr{\'e}sentation irr{\'e}ductible de
$L$ de poids dominant $\rho=\frac{1}{2}$ (somme des racines
positives), et des invariants associ{\'e}s au groupe des
sym{\'e}tries du diagramme de Dynkin de $L$.
Categories:11E04, 11E72, 17B10, 17B20, 11E88, 15A66 

223. CJM 1998 (vol 50 pp. 1007)
 Elder, G. Griffith

Galois module structure of ambiguous ideals in biquadratic extensions
Let $N/K$ be a biquadratic extension of algebraic number fields, and
$G=\Gal (N/K)$. Under a weak restriction on the ramification filtration
associated with each prime of $K$ above $2$, we explicitly describe the
$\bZ[G]$module structure of each ambiguous ideal of $N$. We find under
this restriction that in the representation of each ambiguous ideal as a
$\bZ[G]$module, the exponent (or multiplicity) of each indecomposable
module is determined by the invariants of ramification, alone.
For a given group, $G$, define ${\cal S}_G$ to be the set of
indecomposable $\bZ[G]$modules, ${\cal M}$, such that there
is an extension, $N/K$, for which $G\cong\Gal (N/K)$, and ${\cal M}$
is a $\bZ[G]$module summand of an ambiguous ideal of $N$. Can
${\cal S}_G$ ever be infinite? In this paper we answer this
question of Chinburg in the affirmative.
Keywords:Galois module structure, wild ramification Categories:11R33, 11S15, 20C32 

224. CJM 1998 (vol 50 pp. 1105)
 Roberts, Brooks

Tempered representations and the theta correspondence
Let $V$ be an even dimensional nondegenerate symmetric bilinear
space over a nonarchimedean local field $F$ of characteristic zero,
and let $n$ be a nonnegative integer. Suppose that $\sigma \in
\Irr \bigl(\OO (V)\bigr)$ and $\pi \in \Irr \bigl(\Sp (n,F)\bigr)$
correspond under the theta correspondence. Assuming that $\sigma$
is tempered, we investigate the problem of determining the
Langlands quotient data for $\pi$.
Categories:11F27, 22E50 

225. CJM 1998 (vol 50 pp. 794)
 Louboutin, Stéphane

Upper bounds on $L(1,\chi)$ and applications
We give upper bounds on the modulus of the values at $s=1$ of
Artin $L$functions of abelian extensions unramified at all
the infinite places. We also explain how we can compute better
upper bounds and explain how useful such computed bounds are
when dealing with class number problems for $\CM$fields. For
example, we will reduce the determination of all the
nonabelian normal $\CM$fields of degree $24$ with Galois
group $\SL_2(F_3)$ (the special linear group over the finite
field with three elements) which have class number one to the
computation of the class numbers of $23$ such $\CM$fields.
Keywords:Dedekind zeta function, Dirichlet series, $\CM$field, relative class number Categories:11M20, 11R42, 11Y35, 11R29 
