251. 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 

252. 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 

253. 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 

254. 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 

255. 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 

256. 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 

257. 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 

258. 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 

259. 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 

260. 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 

261. 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 

262. CJM 1998 (vol 50 pp. 465)
 Balog, Antal

Six primes and an almost prime in four linear equations
There are infinitely many triplets of primes $p,q,r$ such that the
arithmetic means of any two of them, ${p+q\over2}$, ${p+r\over2}$,
${q+r\over2}$ are also primes. We give an asymptotic formula for
the number of such triplets up to a limit. The more involved
problem of asking that in addition to the above the arithmetic mean
of all three of them, ${p+q+r\over3}$ is also prime seems to be out
of reach. We show by combining the HardyLittlewood method with the
sieve method that there are quite a few triplets for which six of
the seven entries are primes and the last is almost prime.}
Categories:11P32, 11N36 

263. CJM 1998 (vol 50 pp. 563)
 Goldston, D. A.; Yildirim, C. Y.

Primes in short segments of arithmetic progressions
Consider the variance for the number of primes that are both in the
interval $[y,y+h]$ for $y \in [x,2x]$ and in an arithmetic
progression of modulus $q$. We study the total variance
obtained by adding these variances over all the reduced residue
classes modulo $q$. Assuming a strong form of the twin prime
conjecture and the Riemann Hypothesis one can obtain an asymptotic
formula for the total variance in the range when $1 \leq h/q \leq
x^{1/2\epsilon}$, for any $\epsilon >0$. We show that one can still
obtain some weaker asymptotic results assuming the Generalized Riemann
Hypothesis (GRH) in place of the twin prime conjecture. In their
simplest form, our results are that on GRH the same asymptotic formula
obtained with the twin prime conjecture is true for ``almost all'' $q$
in the range $1 \leq h/q \leq h^{1/4\epsilon}$, that on averaging
over $q$ one obtains an asymptotic formula in the extended range $1
\leq h/q \leq h^{1/2\epsilon}$, and that there are lower bounds with
the correct order of magnitude for all $q$ in the range $1 \leq h/q
\leq x^{1/3\epsilon}$.
Category:11M26 

264. CJM 1998 (vol 50 pp. 412)
 McIntosh, Richard J.

Asymptotic transformations of $q$series
For the $q$series $\sum_{n=0}^\infty a^nq^{bn^2+cn}/(q)_n$
we construct a companion $q$series such that the asymptotic
expansions of their logarithms as $q\to 1^{\scriptscriptstyle }$
differ only in the dominant few terms. The asymptotic expansion
of their quotient then has a simple closed form; this gives rise
to a new $q$hypergeometric identity. We give an asymptotic
expansion of a general class of $q$series containing some of
Ramanujan's mock theta functions and Selberg's identities.
Categories:11B65, 33D10, 34E05, 41A60 

265. CJM 1998 (vol 50 pp. 74)
 Flicker, Yuval Z.

Elementary proof of the fundamental lemma for a unitary group
The fundamental lemma in the theory of automorphic forms is proven
for the (quasisplit) unitary group $U(3)$ in three variables
associated with a quadratic extension of $p$adic fields, and its
endoscopic group $U(2)$, by means of a new, elementary technique.
This lemma is a prerequisite for an application of the trace
formula to classify the automorphic and admissible representations
of $U(3)$ in terms of those of $U(2)$ and base change to $\GL(3)$.
It compares the (unstable) orbital integral of the characteristic
function of the standard maximal compact subgroup $K$ of $U(3)$ at
a regular element (whose centralizer $T$ is a torus), with an
analogous (stable) orbital integral on the endoscopic group $U(2)$.
The technique is based on computing the sum over the double coset
space $T\bs G/K$ which describes the integral, by means of an
intermediate double coset space $H\bs G/K$ for a subgroup $H$ of
$G=U(3)$ containing $T$. Such an argument originates from
Weissauer's work on the symplectic group. The lemma is proven for
both ramified and unramified regular elements, for which endoscopy
occurs (the stable conjugacy class is not a single orbit).
Categories:22E35, 11F70, 11F85, 11S37 

266. CJM 1997 (vol 49 pp. 1265)
 Snaith, V. P.

Hecke algebras and classgroup invariant
Let $G$ be a finite group. To a set of subgroups of order two we associate
a $\mod 2$ Hecke algebra and construct a homomorphism, $\psi$, from its
units to the classgroup of ${\bf Z}[G]$. We show that this homomorphism
takes values in the subgroup, $D({\bf Z}[G])$. Alternative constructions of
Chinburg invariants arising from the Galois module structure of
higherdimensional algebraic $K$groups of rings of algebraic integers
often differ by elements in the image of $\psi$. As an application we show
that two such constructions coincide.
Categories:16S34, 19A99, 11R65 

267. CJM 1997 (vol 49 pp. 1139)
 Kraus, Alain

Majorations effectives pour l'Ã©quation de Fermat gÃ©nÃ©ralisÃ©e
Soient $A$, $B$ et $C$ trois entiers
non nuls premiers entre eux deux \`a deux, et $p$ un nombre premier.
Comme cons\'equence des travaux de A. Wiles et F. Diamond sur la
conjecture de TaniyamaWeil, on explicite une constante $f(A,B,C)$
telle que, sous certaines conditions portant sur $A$, $B$ et $C$,
l'\'equation $Ax^p+By^p+Cz^p=0$ n'a aucune solution non triviale
dans $\Z$, si $p$ est $>f(A,B,C)$. On d\'emontre par ailleurs,
sans condition suppl\'ementaire portant sur $A$, $B$ et $C$, que
cette \'equation n'a aucune solution non triviale dans $\Z$, si
$p$ divise $xyz$, et si $p$ est $>f(A,B,C)$.
Category:11G 

268. CJM 1997 (vol 49 pp. 887)
 Borwein, Peter; Pinner, Christopher

Polynomials with $\{ 0, +1, 1\}$ coefficients and a root close to a given point
For a fixed algebraic number $\alpha$ we
discuss how closely $\alpha$ can be approximated by
a root of a $\{0,+1,1\}$ polynomial of given degree.
We show that the worst rate of approximation tends to
occur for roots of unity, particularly those of small degree.
For roots of unity these bounds depend on
the order of vanishing, $k$, of the polynomial at $\alpha$.
In particular we obtain the following. Let
${\cal B}_{N}$ denote the set of roots of all
$\{0,+1,1\}$ polynomials of degree at most $N$ and
${\cal B}_{N}(\alpha,k)$ the roots of those
polynomials that have a root of order at most $k$
at $\alpha$. For a Pisot number $\alpha$ in $(1,2]$
we show that
\[
\min_{\beta \in {\cal B}_{N}\setminus \{ \alpha \}} \alpha
\beta \asymp \frac{1}{\alpha^{N}},
\]
and for a root of unity $\alpha$ that
\[
\min_{\beta \in {\cal B}_{N}(\alpha,k)\setminus \{\alpha\}}
\alpha \beta\asymp \frac{1}{N^{(k+1) \left\lceil
\frac{1}{2}\phi (d)\right\rceil +1}}.
\]
We study in detail the case of $\alpha=1$, where, by far, the
best approximations are real.
We give fairly precise bounds on the closest real root to 1.
When $k=0$ or 1 we
can describe the extremal polynomials explicitly.
Keywords:Mahler measure, zero one polynomials, Pisot numbers, root separation Categories:11J68, 30C10 

269. CJM 1997 (vol 49 pp. 749)
 Howe, Lawrence

Twisted HasseWeil $L$functions and the rank of MordellWeil groups
Following a method outlined by Greenberg, root
number computations give a conjectural lower bound for the ranks of
certain MordellWeil groups of elliptic curves. More specifically,
for $\PQ_{n}$ a \pgl{{\bf Z}/p^{n}{\bf Z}}extension of ${\bf Q}$ and
$E$ an elliptic curve over {\bf Q}, define the motive $E \otimes
\rho$, where $\rho$ is any irreducible representation of
$\Gal (\PQ_{n}/{\bf Q})$. Under some restrictions, the root number in
the conjectural functional equation for the $L$function of $E
\otimes \rho$ is easily computible, and a `$1$' implies, by the
Birch and SwinnertonDyer conjecture, that $\rho$ is found in
$E(\PQ_{n}) \otimes {\bf C}$. Summing the dimensions of such $\rho$
gives a conjectural lower bound of
$$
p^{2n}  p^{2n  1}  p  1
$$
for the rank of $E(\PQ_{n})$.
Categories:11G05, 14G10 

270. CJM 1997 (vol 49 pp. 722)
 Elder, G. Griffith; Madan, Manohar L.

Galois module structure of the integers in wildly ramified $C_p\times C_p$ extensions
Let $L/K$ be a finite Galois extension of local fields which are finite
extensions of $\bQ_p$, the field of $p$adic numbers. Let $\Gal (L/K)=G$,
and $\euO_L$ and $\bZ_p$ be the rings of integers in $L$ and $\bQ_p$,
respectively. And let $\euP_L$ denote the maximal ideal of $\euO_L$. We
determine, explicitly in terms of specific indecomposable $\bZ_p[G]$modules,
the $\bZ_p[G]$module structure of $\euO_L$ and $\euP_L$, for $L$, a
composite of two arithmetically disjoint, ramified cyclic extensions of
$K$, one of which is only weakly ramified in the sense of Erez \cite{erez}.
Keywords:Galois module structureintegral representation. Categories:11S15, 20C32 

271. CJM 1997 (vol 49 pp. 641)
 Burris, Stanley; Compton, Kevin; Odlyzko, Andrew; Richmond, Bruce

Fine spectra and limit laws II Firstorder 01 laws.
Using FefermanVaught techniques a condition on the fine
spectrum of an admissible class of structures is found
which leads to a firstorder 01 law.
The condition presented is best possible in the
sense that if it is violated then one can find an admissible
class with the same fine spectrum which does not have
a firstorder 01 law.
If the condition is satisfied (and hence we have a firstorder %% 01 law)
Categories:03N45, 11N45, 11N80, 05A15, 05A16, 11M41, 11P81 

272. CJM 1997 (vol 49 pp. 468)
 Burris, Stanley; Sárközy, András

Fine spectra and limit laws I. Firstorder laws
Using FefermanVaught techniques we show a certain property of the fine
spectrum of an admissible class of structures leads to a firstorder law.
The condition presented is best possible in the sense that if it is
violated then one can find an admissible class with the same fine
spectrum which does not have a firstorder law. We present three
conditions for verifying that the above property actually holds.
The first condition is that the count function of an admissible class
has regular variation with a certain uniformity of convergence. This
applies to a wide range of admissible classes, including those
satisfying Knopfmacher's Axiom A, and those satisfying Bateman
and Diamond's condition.
The second condition is similar to the first condition, but designed
to handle the discrete case, {\it i.e.}, when the sizes of the structures
in an admissible class $K$ are all powers of a single integer. It applies
when either the class of indecomposables or the whole class satisfies
Knopfmacher's Axiom A$^\#$.
The third condition is also for the discrete case, when there is a
uniform bound on the number of $K$indecomposables of any given size.
Keywords:First order limit laws, generalized number theory Categories:O3C13, 11N45, 11N80, 05A15, 05A16, 11M41, 11P81 

273. CJM 1997 (vol 49 pp. 499)
 Fitzgerald, Robert W.

Gorenstein Witt rings II
The abstract Witt rings which are Gorenstein have been classified
when the dimension is one and the classification problem for those of
dimension zero has been reduced to the case of socle degree three. Here we
classifiy the Gorenstein Witt rings of fields with dimension zero and
socle degree three. They are of elementary type.
Categories:11E81, 13H10 

274. CJM 1997 (vol 49 pp. 405)
 Vulakh, L. Ya.

On Hurwitz constants for Fuchsian groups
Explicit bounds for the Hurwitz constants for general cofinite
Fuchsian groups have been found. It is shown that the bounds
obtained are exact for the Hecke groups and triangular groups with
signature $(0:2,p,q)$.
Categories:11J04, 20H10 

275. CJM 1997 (vol 49 pp. 283)
 McCall, Thomas M.; Parry, Charles J.; Ranalli, Ramona R.

The $2$rank of the class group of imaginary bicyclic biquadratic fields
A formula is obtained for the rank of the $2$Sylow subgroup of the
ideal class group of imaginary bicyclic biquadratic fields. This
formula involves the number of primes that ramify in the field, the
ranks of the $2$Sylow subgroups of the ideal class groups of the
quadratic subfields and the rank of a $Z_2$matrix determined by
Legendre symbols involving pairs of ramified primes. As
applications, all subfields with both $2$class and class group
$Z_2 \times Z_2$ are determined. The final results assume the
completeness of D.~A.~Buell's list of imaginary fields with small
class numbers.
Categories:11R16, 11R29, 11R20 
