Expand all Collapse all | Results 151 - 175 of 207 |
151. CMB 2002 (vol 45 pp. 196)
Mahler Measures Close to an Integer We prove that the Mahler measure of an algebraic number cannot be too
close to an integer, unless we have equality. The examples of certain
Pisot numbers show that the respective inequality is sharp up to a
constant. All cases when the measure is equal to the integer are
described in terms of the minimal polynomials.
Keywords:Mahler measure, PV numbers, Salem numbers Categories:11R04, 11R06, 11R09, 11J68 |
152. CMB 2002 (vol 45 pp. 220)
Globalization of Distinguished Supercuspidal Representations of $\GL(n)$ An irreducible supercuspidal representation $\pi$ of $G=
\GL(n,F)$, where $F$ is a nonarchimedean local field of
characteristic zero, is said to be ``distinguished'' by a
subgroup $H$ of $G$ and a quasicharacter $\chi$ of $H$ if
$\Hom_H(\pi,\chi)\noteq 0$. There is a suitable global analogue
of this notion for and irreducible, automorphic, cuspidal
representation associated to $\GL(n)$. Under certain general
hypotheses, it is shown in this paper that every distinguished,
irreducible, supercuspidal representation may be realized as a
local component of a distinguished, irreducible automorphic,
cuspidal representation. Applications to the theory of
distinguished supercuspidal representations are provided.
Categories:22E50, 22E35, 11F70 |
153. CMB 2002 (vol 45 pp. 257)
Modular Forms Associated to Theta Functions We use the theory of Jacobi-like forms to construct modular forms for a
congruence subgroup of $\SL(2,\mathbb{R})$ which can be expressed as linear
combinations of products of certain theta functions.
Categories:11F11, 11F27, 33D10 |
154. CMB 2002 (vol 45 pp. 247)
On a Few Diophantine Equations Related to Fermat's Last Theorem We combine the deep methods of Frey, Ribet, Serre and Wiles with some
results of Darmon, Merel and Poonen to solve certain explicit
diophantine equations. In particular, we prove that the area of a
primitive Pythagorean triangle is never a perfect power, and that each
of the equations $X^4 - 4Y^4 = Z^p$, $X^4 + 4Y^p = Z^2$ has no
non-trivial solution. Proofs are short and rest heavily on results
whose proofs required Wiles' deep machinery.
Keywords:Diophantine equations Category:11D41 |
155. CMB 2002 (vol 45 pp. 123)
Uniform Distribution in Model Sets We give a new measure-theoretical proof of the uniform distribution
property of points in model sets (cut and project sets). Each model
set comes as a member of a family of related model sets, obtained by
joint translation in its ambient (the `physical') space and its
internal space. We prove, assuming only that the window defining the
model set is measurable with compact closure, that almost surely the
distribution of points in any model set from such a family is uniform
in the sense of Weyl, and almost surely the model set is pure point
diffractive.
Categories:52C23, 11K70, 28D05, 37A30 |
156. CMB 2002 (vol 45 pp. 115)
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 $\log
n$.
Category:11A63 |
157. CMB 2002 (vol 45 pp. 86)
On Cyclic Fields of Odd Prime Degree $p$ with Infinite Hilbert $p$-Class Field Towers Let $k$ be a cyclic extension of odd prime degree $p$ of the field of
rational numbers. If $t$ denotes the number of primes that ramify in $k$,
it is known that the Hilbert $p$-class field tower of $k$ is infinite if
$t>3+2\sqrt p$. For each $t>2+\sqrt p$, this paper shows that a positive
proportion of such fields $k$ have infinite Hilbert $p$-class field towers.
Categories:11R29, 11R37, 11R45 |
158. CMB 2002 (vol 45 pp. 97)
Invariant Measures and Natural Extensions We study ergodic properties of a family of interval maps that are
given as the fractional parts of certain real M\"obius
transformations. Included are the maps that are exactly
$n$-to-$1$, the classical Gauss map and the Renyi or backward
continued fraction map. A new approach is presented for deriving
explicit realizations of natural automorphic extensions and their
invariant measures.
Keywords:Continued fractions, interval maps, invariant measures Categories:11J70, 58F11, 58F03 |
159. CMB 2002 (vol 45 pp. 138)
The Discriminant of a Dihedral Quintic Field Defined by a Trinomial $X^5 + aX + b$ Let $X^5 + aX + b \in Z[X]$ have Galois group $D_5$. Let $\theta$ be
a root of $X^5 + aX + b$. An explicit formula is given for the
discriminant of $Q(\theta)$.
Keywords:dihedral quintic field, trinomial, discriminant Categories:11R21, 11R29 |
160. CMB 2002 (vol 45 pp. 109)
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 |
161. CMB 2002 (vol 45 pp. 36)
Modular Equations and Discrete, Genus-Zero Subgroups of $\SL(2,\mathbb{R})$ Containing $\Gamma(N)$ Let $G$ be a discrete subgroup of $\SL(2,\R)$ which contains
$\Gamma(N)$ for some $N$. If the genus of $X(G)$ is zero, then there
is a unique normalised generator of the field of $G$-automorphic
functions which is known as a normalised Hauptmodul. This paper gives
a characterisation of normalised Hauptmoduls as formal $q$ series
using modular polynomials.
Categories:11F03, 11F22, 30F35 |
162. CMB 2001 (vol 44 pp. 398)
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 |
163. CMB 2001 (vol 44 pp. 440)
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 |
164. CMB 2001 (vol 44 pp. 385)
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 |
165. CMB 2001 (vol 44 pp. 282)
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 |
166. 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, isogeny Categories:11R32, 11G05, 12F10, 14K02 |
167. CMB 2001 (vol 44 pp. 242)
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 |
168. CMB 2001 (vol 44 pp. 160)
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 |
169. CMB 2001 (vol 44 pp. 3)
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 |
170. CMB 2001 (vol 44 pp. 12)
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 |
171. CMB 2001 (vol 44 pp. 22)
Gauss Sums of Orders Six and Twelve Precise, elegant evaluations are given for Gauss sums of orders six
and twelve.
Categories:11L05, 11T24 |
172. CMB 2001 (vol 44 pp. 87)
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 |
173. CMB 2001 (vol 44 pp. 115)
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 |
174. CMB 2001 (vol 44 pp. 97)
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 |
175. CMB 2001 (vol 44 pp. 19)
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 |