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176. CMB 2001 (vol 44 pp. 313)

 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, isogenyCategories: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)

Langlands, Robert P.
 The Trace Formula and Its Applications: An Introduction to the Work of James Arthur James Arthur was awarded the Canada Gold Medal of the National Science and Engineering Research Council in 1999. This introduction to his work is an attempt to explain his methods and his goals to the mathematical community at large. Categories:11F70, 11F72, 58G25

179. CMB 2001 (vol 44 pp. 22)

Evans, Ronald
 Gauss Sums of Orders Six and Twelve Precise, elegant evaluations are given for Gauss sums of orders six and twelve. Categories:11L05, 11T24

180. CMB 2001 (vol 44 pp. 19)

Brindza, B.; Pintér, Á.; Schmidt, W. M.
 Multiplicities of Binary Recurrences In this note the multiplicities of binary recurrences over algebraic number fields are investigated under some natural assumptions. Categories:11B37, 11J86

181. CMB 2001 (vol 44 pp. 97)

Ou, Zhiming M.; Williams, Kenneth S.
 On the Density of Cyclic Quartic Fields An asymptotic formula is obtained for the number of cyclic quartic fields over $Q$ with discriminant $\leq x$. Keywords:cyclic quartic fields, density, discriminantCategories:11R16, 11R29

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)

Shahidi, Freydoon
 Twists of a General Class of $L$-Functions by Highly Ramified Characters It is shown that given a local $L$-function defined by Langlands-Shahidi method, there exists a highly ramified character of the group which when is twisted with the original representation leads to a trivial $L$-function. Categories:11F70, 22E35, 22E50

187. CMB 2000 (vol 43 pp. 282)

Boston, Nigel; Ose, David T.
 Characteristic $p$ Galois Representations That Arise from Drinfeld Modules We examine which representations of the absolute Galois group of a field of finite characteristic with image over a finite field of the same characteristic may be constructed by the Galois group's action on the division points of an appropriate Drinfeld module. Categories:11G09, 11R32, 11R58

188. CMB 2000 (vol 43 pp. 304)

Darmon, Henri; Mestre, Jean-François
 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 au-dessus 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 $\{(n^p-n)/p\}_p \in \prod_p \mathbf{F}_p$, 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 sumsCategories:11G05, 14H52

191. CMB 2000 (vol 43 pp. 218)

Mollin, R. A.; van der Poorten, A. J.
 Continued Fractions, Jacobi Symbols, and Quadratic Diophantine Equations The results herein continue observations on norm form equations and continued fractions begun and continued in the works \cite{chows}--\cite{mol}, and \cite{mvdpw}--\cite{schinz}. Categories:11R11, 11D09, 11R29, 11R65

192. CMB 2000 (vol 43 pp. 115)

Schmutz Schaller, Paul
 Perfect Non-Extremal Riemann Surfaces An infinite family of perfect, non-extremal 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 non-Euclidean analogues to the existence of perfect, non-extremal 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 theta-functions, Hilbert class fieldsCategories: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 operatorsCategories: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)

Gittenberger, Bernhard; Thuswaldner, Jörg M.
 The Moments of the Sum-Of-Digits Function in Number Fields We consider the asymptotic behavior of the moments of the sum-of-digits function of canonical number systems in number fields. Using Delange's method we obtain the main term and smaller order terms which contain periodic fluctuations. Categories:11A63, 11N60

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 `2-large'' are given. Several open questions and a conjecture are presented. Categories:11B25, 05D10
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