176. CMB 2001 (vol 44 pp. 313)
 Reverter, Amadeu; Vila, Núria

Images of mod $p$ Galois Representations Associated to Elliptic Curves
We give an explicit recipe for the determination of the images
associated to the Galois action on $p$torsion points of elliptic
curves. We present a table listing the image for all the elliptic
curves defined over $\QQ$ without complex multiplication with
conductor less than 200 and for each prime number~$p$.
Keywords:Galois groups, elliptic curves, Galois representation, isogeny Categories:11R32, 11G05, 12F10, 14K02 

177. CMB 2001 (vol 44 pp. 242)
 Schueller, Laura Mann

The Zeta Function of a Pair of Quadratic Forms
The zeta function of a nonsingular pair of quadratic forms defined over a
finite field, $k$, of arbitrary characteristic is calculated. A.~Weil made
this computation when $\rmchar k \neq 2$. When the pair has even order, a
relationship between the number of zeros of the pair and the number of
places of degree one in an appropriate hyperelliptic function field is
Category:11G25 

178. CMB 2001 (vol 44 pp. 160)
179. CMB 2001 (vol 44 pp. 22)
180. CMB 2001 (vol 44 pp. 19)
181. CMB 2001 (vol 44 pp. 97)
182. CMB 2001 (vol 44 pp. 115)
 Roy, Damien

Approximation algÃ©brique simultanÃ©e de nombres de Liouville
The purpose of this paper is to show the limitations of the
conjectures of algebraic approximation. For this, we construct
points of $\bC^m$ which do not admit good algebraic approximations
of bounded degree and height, when the bounds on the degree and the
height are taken from specific sequences. The coordinates of these
points are Liouville numbers.
Category:11J82 

183. CMB 2001 (vol 44 pp. 87)
 Lieman, Daniel; Shparlinski, Igor

On a New Exponential Sum
Let $p$ be prime and let $\vartheta\in\Z^*_p$ be of
multiplicative order $t$ modulo $p$. We consider exponential
sums of the form
$$
S(a) = \sum_{x =1}^{t} \exp(2\pi i a \vartheta^{x^2}/p)
$$
and prove that for any $\varepsilon > 0$
$$
\max_{\gcd(a,p) = 1} S(a) = O( t^{5/6 + \varepsilon}p^{1/8}) .
$$
Categories:11L07, 11T23, 11B50, 11K31, 11K38 

184. CMB 2001 (vol 44 pp. 12)
 Anisca, Razvan; Ilie, Monica

A Technique of Studying Sums of Central Cantor Sets
This paper is concerned with the structure of the arithmetic sum of a
finite number of central Cantor sets. The technique used to study this
consists of a duality between central Cantor sets and sets of subsums
of certain infinite series. One consequence is that the sum of a finite
number of central Cantor sets is one of the following: a finite union
of closed intervals, homeomorphic to the Cantor ternary set or an
$M$Cantorval.
Category:11B05 

185. CMB 2001 (vol 44 pp. 3)
 Alexandru, Victor; Popescu, Nicolae; Zaharescu, Alexandru

The Generating Degree of $\C_p$
The generating degree $\gdeg (A)$ of a topological commutative ring
$A$ with $\Char A = 0$ is the cardinality of the smallest subset $M$
of $A$ for which the subring $\Z[M]$ is dense in $A$. For a prime
number $p$, $\C_p$ denotes the topological completion of an algebraic
closure of the field $\Q_p$ of $p$adic numbers. We prove that $\gdeg
(\C_p) = 1$, \ie, there exists $t$ in $\C_p$ such that $\Z[t]$ is
dense in $\C_p$. We also compute $\gdeg \bigl( A(U) \bigr)$ where
$A(U)$ is the ring of rigid analytic functions defined on a ball $U$
in $\C_p$. If $U$ is a closed ball then $\gdeg \bigl( A(U) \bigr) =
2$ while if $U$ is an open ball then $\gdeg \bigl( A(U) \bigr)$ is
infinite. We show more generally that $\gdeg \bigl( A(U) \bigr)$ is
finite for any {\it affinoid} $U$ in $\PP^1 (\C_p)$ and $\gdeg \bigl(
A(U) \bigr)$ is infinite for any {\it wide open} subset $U$ of $\PP^1
(\C_p)$.
Category:11S99 

186. CMB 2000 (vol 43 pp. 380)
187. CMB 2000 (vol 43 pp. 282)
188. 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 

189. 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 

190. 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 

191. CMB 2000 (vol 43 pp. 218)
192. 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 

193. 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 

194. 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 

195. 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 

196. 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 

197. 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 

198. 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 

199. CMB 1999 (vol 42 pp. 68)
200. 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 
