251. 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 

252. 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 

253. 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 

254. 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 

255. 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 

256. 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 

257. 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 

258. 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 

259. 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 

260. 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 

261. 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 

262. 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 

263. 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 

264. 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 
