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176. CMB 2002 (vol 45 pp. 109)

Hall, R. R.; Shiu, P.
The Distribution of Totatives
D.~H.~Lehmer initiated the study of the distribution of totatives, which are numbers coprime with a given integer. This led to various problems considered by P.~Erd\H os, who made a conjecture on such distributions. We prove his conjecture by establishing a theorem on the ordering of residues.

Keywords:Euler's function, totatives
Categories:11A05, 11A07, 11A25

177. CMB 2002 (vol 45 pp. 115)

Luca, Florian
The Number of Non-Zero Digits of $n!$
Let $b$ be an integer with $b>1$. In this note, we prove that the number of non-zero digits in the base $b$ representation of $n!$ grows at least as fast as a constant, depending on $b$, times $\log n$.


178. CMB 2001 (vol 44 pp. 385)

Ballantine, Cristina M.
A Hypergraph with Commuting Partial Laplacians
Let $F$ be a totally real number field and let $\GL_{n}$ be the general linear group of rank $n$ over $F$. Let $\mathfrak{p}$ be a prime ideal of $F$ and $F_{\mathfrak{p}}$ the completion of $F$ with respect to the valuation induced by $\mathfrak{p}$. We will consider a finite quotient of the affine building of the group $\GL_{n}$ over the field $F_{\mathfrak{p}}$. We will view this object as a hypergraph and find a set of commuting operators whose sum will be the usual adjacency operator of the graph underlying the hypergraph.

Keywords:Hecke operators, buildings
Categories:11F25, 20F32

179. CMB 2001 (vol 44 pp. 398)

Cardon, David A.; Ram Murty, M.
Exponents of Class Groups of Quadratic Function Fields over Finite Fields
We find a lower bound on the number of imaginary quadratic extensions of the function field $\F_q(T)$ whose class groups have an element of a fixed order. More precisely, let $q \geq 5$ be a power of an odd prime and let $g$ be a fixed positive integer $\geq 3$. There are $\gg q^{\ell (\frac{1}{2}+\frac{1}{g})}$ polynomials $D \in \F_q[T]$ with $\deg(D) \leq \ell$ such that the class groups of the quadratic extensions $\F_q(T,\sqrt{D})$ have an element of order~$g$.

Keywords:class number, quadratic function field
Categories:11R58, 11R29

180. CMB 2001 (vol 44 pp. 440)

Hironaka, Eriko
The Lehmer Polynomial and Pretzel Links
In this paper we find a formula for the Alexander polynomial $\Delta_{p_1,\dots,p_k} (x)$ of pretzel knots and links with $(p_1,\dots,p_k, \nega 1)$ twists, where $k$ is odd and $p_1,\dots,p_k$ are positive integers. The polynomial $\Delta_{2,3,7} (x)$ is the well-known Lehmer polynomial, which is conjectured to have the smallest Mahler measure among all monic integer polynomials. We confirm that $\Delta_{2,3,7} (x)$ has the smallest Mahler measure among the polynomials arising as $\Delta_{p_1,\dots,p_k} (x)$.

Keywords:Alexander polynomial, pretzel knot, Mahler measure, Salem number, Coxeter groups
Categories:57M05, 57M25, 11R04, 11R27

181. CMB 2001 (vol 44 pp. 282)

Lee, Min Ho; Myung, Hyo Chul
Hecke Operators on Jacobi-like Forms
Jacobi-like forms for a discrete subgroup $\G \subset \SL(2,\mbb R)$ are formal power series with coefficients in the space of functions on the Poincar\'e upper half plane satisfying a certain functional equation, and they correspond to sequences of certain modular forms. We introduce Hecke operators acting on the space of Jacobi-like forms and obtain an explicit formula for such an action in terms of modular forms. We also prove that those Hecke operator actions on Jacobi-like forms are compatible with the usual Hecke operator actions on modular forms.

Categories:11F25, 11F12

182. 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

183. 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

184. 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


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 $\PP^1 (\C_p)$.


186. 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

187. 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, discriminant
Categories:11R16, 11R29

188. CMB 2001 (vol 44 pp. 115)

Roy, Damien
Approximation algébrique simultanée 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.


189. 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

190. 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

191. 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.


192. CMB 2000 (vol 43 pp. 304)

Darmon, Henri; Mestre, Jean-François
Courbes hyperelliptiques à multiplications réelles 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

193. 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

194. 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

195. 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.


196. 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

197. 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

198. 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

199. 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

200. 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 fields
Categories:33C05, 33E05, 11R20, 11R29
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