201. CMB 2000 (vol 43 pp. 304)
 Darmon, Henri; Mestre, JeanFrançois

Courbes hyperelliptiques Ã multiplications rÃ©elles et une construction de Shih
Soient $r$ et $p$ deux nombres premiers distincts, soit $K = \Q(\cos
\frac{2\pi}{r})$, et soit $\F$ le corps r\'esiduel de $K$ en une place
audessus de $p$. Lorsque l'image de $(2  2\cos \frac{2\pi}{r})$
dans $\F$ n'est pas un carr\'e, nous donnons une construction
g\'eom\'etrique d'une extension r\'eguliere de $K(t)$ de groupe de
Galois $\PSL_2 (\F)$. Cette extension correspond \`a un rev\^etement
de $\PP^1_{/K}$ de \og{} signature $(r,p,p)$ \fg{} au sens de [3,
sec.~6.3], et son existence est pr\'edite par le crit\`ere de
rigidit\'e de Belyi, Fried, Thompson et Matzat. Sa construction
s'obtient en tordant la representation galoisienne associ\'ee aux
points d'ordre $p$ d'une famille de vari\'et\'es ab\'eliennes \`a
multiplications r\'eelles par $K$ d\'ecouverte par Tautz, Top et
Verberkmoes [6]. Ces vari\'et\'es ab\'eliennes sont d\'efinies sur un
corps quadratique, et sont isog\`enes \`a leur conjugu\'e galoisien.
Notre construction g\'en\'eralise une m\'ethode de Shih [4], [5], que
l'on retrouve quand $r = 2$ et $r = 3$.
Let $r$ and $p$ be distinct prime numbers, let $K = \Q(\cos
\frac{2\pi}{r})$, and let $\F$ be the residue field of $K$ at a place
above $p$. When the image of $(2  2\cos \frac{2\pi}{r})$ in $\F$ is
not a square, we describe a geometric construction of a regular
extension of $K(t)$ with Galois group $\PSL_2 (\F)$. This extension
corresponds to a covering of $\PP^1_{/K}$ of ``signature $(r,p,p)$''
in the sense of [3, sec.~6.3], and its existence is predicted by the
rigidity criterion of Belyi, Fried, Thompson and Matzat. Its
construction is obtained by twisting the mod $p$ galois representation
attached to a family of abelian varieties with real multiplications by
$K$ discovered by Tautz, Top and Verberkmoes [6]. These abelian
varieties are defined in general over a quadratic field, and are
isogenous to their galois conjugate. Our construction generalises a
method of Shih [4], [5], which one recovers when $r = 2$ and $r = 3$.
Categories:11G30, 14H25 

202. CMB 2000 (vol 43 pp. 239)
 Yu, Gang

On the Number of Divisors of the Quadratic Form $m^2+n^2$
For an integer $n$, let $d(n)$ denote the ordinary divisor function.
This paper studies the asymptotic behavior of the sum
$$
S(x) := \sum_{m\leq x, n\leq x} d(m^2 + n^2).
$$
It is proved in the paper that, as $x \to \infty$,
$$
S(x) := A_1 x^2 \log x + A_2 x^2 + O_\epsilon (x^{\frac32 +
\epsilon}),
$$
where $A_1$ and $A_2$ are certain constants and $\epsilon$ is any
fixed positive real number.
The result corrects a false formula given in a paper of Gafurov
concerning the same problem, and improves the error $O \bigl(
x^{\frac53} (\log x)^9 \bigr)$ claimed by Gafurov.
Keywords:divisor, large sieve, exponential sums Categories:11G05, 14H52 

203. CMB 2000 (vol 43 pp. 236)
 Voloch, José Felipe

On a Question of Buium
We prove that $\{(n^pn)/p\}_p \in \prod_p \mathbf{F}_p$, with $p$
ranging over all primes, is independent of $1$ over the integers,
assuming a conjecture in elementary number theory generalizing
the infinitude of Mersenne primes. This answers a question of
Buium. We also prove a generalization.
Category:11A07 

204. CMB 2000 (vol 43 pp. 218)
205. CMB 2000 (vol 43 pp. 115)
 Schmutz Schaller, Paul

Perfect NonExtremal Riemann Surfaces
An infinite family of perfect, nonextremal Riemann surfaces
is constructed, the first examples of this type of surfaces.
The examples are based on normal subgroups of the modular group
$\PSL(2,{\sf Z})$ of level $6$. They provide nonEuclidean
analogues to the existence of perfect, nonextremal positive
definite quadratic forms. The analogy uses the function {\it syst\/}
which associates to every Riemann surface $M$ the length of a systole,
which is a shortest closed geodesic of $M$.
Categories:11H99, 11F06, 30F45 

206. CMB 1999 (vol 42 pp. 427)
 Berndt, Bruce C.; Chan, Heng Huat

Ramanujan and the Modular $j$Invariant
A new infinite product $t_n$ was introduced by S.~Ramanujan on the
last page of his third notebook. In this paper, we prove
Ramanujan's assertions about $t_n$ by establishing new connections
between the modular $j$invariant and Ramanujan's cubic theory of
elliptic functions to alternative bases. We also show that for
certain integers $n$, $t_n$ generates the Hilbert class field of
$\mathbb{Q} (\sqrt{n})$. This shows that $t_n$ is a new class
invariant according to H.~Weber's definition of class invariants.
Keywords:modular functions, the Borweins' cubic thetafunctions, Hilbert class fields Categories:33C05, 33E05, 11R20, 11R29 

207. CMB 1999 (vol 42 pp. 441)
 Berrizbeitia, P.; Elliott, P. D. T. A.

Product Bases for the Rationals
A sequence of positive rationals generates a subgroup of finite
index in the multiplicative positive rationals, and group product
representations by the sequence need only a bounded number of
terms, if and only if certain related sequences have densities
uniformly bounded from below.
Categories:11N99, 11N05 

208. CMB 1999 (vol 42 pp. 263)
 Choie, Youngju; Lee, Min Ho

Mellin Transforms of Mixed Cusp Forms
We define generalized Mellin transforms of mixed cusp forms, show
their convergence, and prove that the function obtained by such a
Mellin transform of a mixed cusp form satisfies a certain
functional equation. We also prove that a mixed cusp form can be
identified with a holomorphic form of the highest degree on an
elliptic variety.
Categories:11F12, 11F66, 11M06, 14K05 

209. CMB 1999 (vol 42 pp. 393)
 Savin, Gordan

A Class of Supercuspidal Representations of $G_2(k)$
Let $H$ be an exceptional, adjoint group of type $E_6$ and split
rank 2, over a $p$adic field $k$. In this article we discuss the
restriction of the minimal representation of $H$ to a dual pair
$\PD^{\times}\times G_2(k)$, where $D$ is a division algebra of
dimension 9 over $k$. In particular, we discover an interesting
class of supercuspidal representations of $G_2(k)$.
Categories:22E35, 22E50, 11F70 

210. CMB 1999 (vol 42 pp. 129)
 Baker, Andrew

Hecke Operations and the Adams $E_2$Term Based on Elliptic Cohomology
Hecke operators are used to investigate part of the $\E_2$term of
the Adams spectral sequence based on elliptic homology. The main
result is a derivation of $\Ext^1$ which combines use of classical
Hecke operators and $p$adic Hecke operators due to Serre.
Keywords:Adams spectral sequence, elliptic cohomology, Hecke operators Categories:55N20, 55N22, 55T15, 11F11, 11F25 

211. CMB 1999 (vol 42 pp. 68)
212. CMB 1999 (vol 42 pp. 78)
 González, Josep

Fermat Jacobians of Prime Degree over Finite Fields
We study the splitting of Fermat Jacobians of prime
degree $\ell$ over an algebraic closure of a finite field of
characteristic $p$ not equal to $\ell$. We prove that their
decomposition is determined by the residue degree of $p$ in the
cyclotomic field of the $\ell$th roots of unity. We provide a
numerical criterion that allows to compute the absolutely simple
subvarieties and their multiplicity in the Fermat Jacobian.
Categories:11G20, 14H40 

213. CMB 1999 (vol 42 pp. 25)
 Brown, Tom C.; Graham, Ronald L.; Landman, Bruce M.

On the Set of Common Differences in van der Waerden's Theorem on Arithmetic Progressions
Analogues of van der Waerden's theorem on arithmetic progressions
are considered where the family of all arithmetic progressions,
$\AP$, is replaced by some subfamily of $\AP$. Specifically, we
want to know for which sets $A$, of positive integers, the
following statement holds: for all positive integers $r$ and $k$,
there exists a positive integer $n= w'(k,r)$ such that for every
$r$coloring of $[1,n]$ there exists a monochromatic $k$term
arithmetic progression whose common difference belongs to $A$. We
will call any subset of the positive integers that has the above
property {\em large}. A set having this property for a specific
fixed $r$ will be called {\em $r$large}. We give some necessary
conditions for a set to be large, including the fact that every
large set must contain an infinite number of multiples of each
positive integer. Also, no large set $\{a_{n}: n=1,2,\dots\}$ can
have $\liminf\limits_{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}} > 1$.
Sufficient conditions for a set to be large are also given. We
show that any set containing $n$cubes for arbitrarily large $n$,
is a large set. Results involving the connection between the
notions of ``large'' and ``2large'' are given. Several open
questions and a conjecture are presented.
Categories:11B25, 05D10 

214. CMB 1998 (vol 41 pp. 488)
 Sun, Heng

Remarks on certain metaplectic groups
We study metaplectic coverings of the adelized group of a split
connected reductive group $G$ over a number field $F$. Assume its
derived group $G'$ is a simply connected simple Chevalley
group. The purpose is to provide some naturally defined sections
for the coverings with good properties which might be helpful when
we carry some explicit calculations in the theory of automorphic
forms on metaplectic groups. Specifically, we
\begin{enumerate}
\item construct metaplectic coverings of $G({\Bbb A})$ from those
of $G'({\Bbb A})$;
\item for any nonarchimedean place $v$, show the section for a
covering of $G(F_{v})$ constructed from a Steinberg section is an
isomorphism, both algebraically and topologically in an open
subgroup of $G(F_{v})$;
\item define a global section which is a product of local sections
on a maximal torus, a unipotent subgroup and a set of
representatives for the Weyl group.
Categories:20G10, 11F75 

215. CMB 1998 (vol 41 pp. 328)
 Mollin, R. A.

Class number one and primeproducing quadratic polynomials revisited
Over a decade ago, this author produced class number one criteria for
real quadratic fields in terms of primeproducing quadratic
polynomials. The purpose of this article is to revisit the problem
from a new perspective with new criteria. We look at the more general
situation involving arbitrary real quadratic orders rather than the
more restrictive field case, and use the interplay between the various
orders to provide not only more general results, but also simpler proofs.
Categories:11R11, 11R09, 11R29 

216. CMB 1998 (vol 41 pp. 335)
217. CMB 1998 (vol 41 pp. 187)
 Loh, W. K. A.

Exponential sums on reduced residue systems
The aim of this article is to obtain an upper bound for the exponential sums
$\sum e(f(x)/q)$, where the summation runs from $x=1$ to $x=q$ with $(x,q)=1$
and $e(\alpha)$ denotes $\exp(2\pi i\alpha)$.
We shall show that the upper bound depends only on the values of $q$ and $s$, %% where $s$ is the number of terms in the polynomial $f(x)$.
Category:11L07 

218. CMB 1998 (vol 41 pp. 158)
 Gaál, István

Power integral bases in composits of number fields
In the present paper we consider the problem of finding power
integral bases in number fields which are composits of two
subfields with coprime discriminants. Especially, we consider
imaginary quadratic extensions of totally real cyclic number
fields of prime degree. As an example we solve the index form
equation completely in a two parametric family of fields of degree
$10$ of this type.
Categories:11D57, 11R21 

219. CMB 1998 (vol 41 pp. 86)
220. CMB 1998 (vol 41 pp. 15)
 Brown, Tom; Shiue, Peter JauShyong; Yu, X. Y.

Sequences with translates containing many primes
Garrison [3], Forman [2], and Abel and Siebert [1] showed that for all positive integers
$k$ and $N$, there exists a positive integer $\lambda$ such that $n^k+\lambda$ is
prime for at least $N$ positive integers $n$. In other words, there exists $\lambda$
such that $n^k+\lambda$ represents at least $N$ primes.
We give a quantitative version of this result. We show that there exists
$\lambda \leq x^k$ such that $n^k+\lambda$, $1\leq n\leq x$, represents at
least $(\frac 1k+o(1)) \pi(x)$ primes, as $x\rightarrow \infty$. We also give some
related results.
Category:11A48 

221. CMB 1998 (vol 41 pp. 71)
 Hurrelbrink, Jurgen; Rehmann, Ulf

Splitting patterns and trace forms
The splitting pattern of a quadratic form $q$ over
a field $k$ consists of all distinct Witt indices that occur for $q$
over extension fields of $k$. In small dimensions, the complete list
of splitting patterns of quadratic forms is known. We show that
{\it all\/} splitting patterns of quadratic forms of dimension at
most nine can be realized by trace forms.
Keywords:Quadratic forms, Witt indices, generic splitting. Category:11E04 

222. CMB 1998 (vol 41 pp. 125)
 Boyd, David W.

Uniform approximation to Mahler's measure in several variables
If $f(x_1,\dots,x_k)$ is a polynomial with complex coefficients, the Mahler measure
of $f$, $M(f)$ is defined to be the geometric mean of $f$ over the $k$torus
$\Bbb T^k$. We construct a sequence of approximations $M_n(f)$ which satisfy
$d2^{n}\log 2 + \log M_n(f) \le \log M(f) \le \log M_n(f)$. We use these to prove
that $M(f)$ is a continuous function of the coefficients of $f$ for polynomials
of fixed total degree $d$. Since $M_n(f)$ can be computed in a finite number
of arithmetic operations from the coefficients of $f$ this also demonstrates
an effective (but impractical) method for computing $M(f)$ to arbitrary
accuracy.
Categories:11R06, 11K16, 11Y99 

223. CMB 1997 (vol 40 pp. 498)
 Selvaraj, Chikkanna; Selvaraj, Suguna

Matrix transformations based on Dirichlet convolution
This paper is a study of summability methods that are based
on Dirichlet convolution. If $f(n)$ is a function on positive integers
and $x$ is a sequence such that $\lim_{n\to \infty} \sum_{k\le n}
{1\over k}(f\ast x)(k) =L$, then $x$ is said to be {\it $A_f$summable\/}
to $L$. The necessary and sufficient condition for the matrix $A_f$ to
preserve bounded variation of sequences is established. Also, the
matrix $A_f$ is investigated as $\ell  \ell$ and $GG$ mappings. The
strength of the $A_f$matrix is also discussed.
Categories:11A25, 40A05, 40C05, 40D05 

224. CMB 1997 (vol 40 pp. 385)
225. CMB 1997 (vol 40 pp. 402)
 Carpenter, Jenna P.

On the Preservation of Root Numbers and the Behavior of Weil Characters Under Reciprocity Equivalence
This paper studies how the local root numbers and the Weil additive
characters of the Witt ring of a number field behave under
reciprocity equivalence. Given a reciprocity equivalence between
two fields, at each place we define a local square class which
vanishes if and only if the local root numbers are preserved. Thus
this local square class serves as a local obstruction to the
preservation of local root numbers. We establish a set of
necessary and sufficient conditions for a selection of local square
classes (one at each place) to represent a global square class.
Then, given a reciprocity equivalence that has a finite wild set,
we use these conditions to show that the local square classes
combine to give a global square class which serves as a global
obstruction to the preservation of all root numbers. Lastly, we
use these results to study the behavior of Weil characters under
reciprocity equivalence.
Categories:11E12, 11E08 
