Expand all Collapse all | Results 126 - 150 of 205 |
126. CMB 2004 (vol 47 pp. 373)
On the Diophantine Equation $n(n+d)\cdots(n+(k-1)d)=by^l$ We show that the product of four or five consecutive positive
terms in arithmetic progression can never be a perfect power whenever the
initial term is coprime to the common difference of the arithmetic
progression. This is a generalization of the results of Euler and Obl\'ath
for the case of squares, and an extension of a theorem of Gy\H ory on three
terms in arithmetic progressions. Several other results concerning the
integral solutions of the equation of the title are also obtained. We extend
results of Sander on the rational solutions of the equation in $n,y$ when
$b=d=1$. We show that there are only finitely many solutions in $n,d,b,y$
when $k\geq 3$, $l\geq 2$ are fixed and $k+l>6$.
Category:11D41 |
127. CMB 2004 (vol 47 pp. 271)
Linear Relations Among the Values of Canonical Heights from the Existence of Non-Trivial Endomorphisms |
Linear Relations Among the Values of Canonical Heights from the Existence of Non-Trivial Endomorphisms We study the interplay between canonical heights and endomorphisms of an abelian
variety $A$ over a number field $k$. In particular we show that whenever the ring
of endomorphisms defined over $k$ is strictly larger than $\Z$ there will
be $\Q$-linear relations among the values of a canonical height pairing evaluated
at a basis modulo torsion of $A(k)$.
Categories:11G10, 14K15 |
128. CMB 2004 (vol 47 pp. 237)
Ramification des sÃ©ries formelles Let $p$ be a prime number. Let $k$ be a finite field of characteristic $p$.
The subset $X+X^2 k[[X]]$ of the ring $k[[X]]$ is a group under the substitution
law $\circ $ sometimes called the Nottingham group of $k$; it is denoted by
$\mathcal{R}_k$. The ramification of one series $\gamma\in\mathcal{R}_k$ is
caracterized by its lower ramification numbers: $i_m(\gamma)=\ord_X
\bigl(\gamma^{p^m} (X)/X - 1\bigr)$, as well as its upper ramification numbers:
$$
u_m (\gamma) = i_0 (\gamma) + \frac{i_1 (\gamma) - i_0(\gamma)}{p} +
\cdots + \frac{i_m (\gamma) - i_{m-1} (\gamma)}{p^m} , \quad (m \in
\mathbb{N}).
$$
By Sen's theorem, the $u_m(\gamma)$ are integers. In this paper, we determine
the sequences of integers $(u_m)$ for which there exists $\gamma\in\mathcal{R}_k$
such that $u_m(\gamma)=u_m$ for all integer $m \geq 0$.
Keywords:ramification, Nottingham group Categories:11S15, 20E18 |
129. CMB 2004 (vol 47 pp. 264)
Counting Rational Points on Ruled Varieties In this paper, we prove a general result computing the number of rational points
of bounded height on a projective variety $V$ which is covered by lines. The
main technical result used to achieve this is an upper bound on the number of
rational points of bounded height on a line. This upper bound is such that it
can be easily controlled as the line varies, and hence is used to sum the counting
functions of the lines which cover the original variety $V$.
Categories:11G50, 11D45, 11D04, 14G05 |
130. CMB 2004 (vol 47 pp. 12)
On Newton's Method and Rational Approximations to Quadratic Irrationals In 1988 Rieger exhibited a differentiable function having a zero at
the golden ratio\break
$(-1+\sqrt5)/2$ for which when Newton's method for approximating
roots is applied with an initial value $x_0=0$, all approximates
are so-called ``best rational approximates''---in this case, of the
form $F_{2n}/F_{2n+1}$, where $F_n$ denotes the $n$-th Fibonacci
number. Recently this observation was extended by Komatsu to the
class of all quadratic irrationals whose continued fraction
expansions have period length $2$. Here we generalize these
observations by producing an analogous result for all quadratic
irrationals and thus provide an explanation for these phenomena.
Categories:11A55, 11B37 |
131. CMB 2003 (vol 46 pp. 495)
Canonical Vector Heights on Algebraic K3 Surfaces with Picard Number Two Let $V$ be an algebraic K3 surface defined over a number field $K$.
Suppose $V$ has Picard number two and an infinite group of
automorphisms $\mathcal{A} = \Aut(V/K)$. In this paper, we
introduce the notion of a vector height $\mathbf{h} \colon V \to
\Pic(V) \otimes \mathbb{R}$ and show the existence of a canonical
vector height $\widehat{\mathbf{h}}$ with the following properties:
\begin{gather*}
\widehat{\mathbf{h}} (\sigma P) = \sigma_* \widehat{\mathbf{h}} (P) \\
h_D (P) = \widehat{\mathbf{h}} (P) \cdot D + O(1),
\end{gather*}
where $\sigma \in \mathcal{A}$, $\sigma_*$ is the pushforward of
$\sigma$ (the pullback of $\sigma^{-1}$), and $h_D$ is a Weil
height associated to the divisor $D$. The bounded function implied
by the $O(1)$ does not depend on $P$. This allows us to attack
some arithmetic problems. For example, we show that the number of
rational points with bounded logarithmic height in an
$\mathcal{A}$-orbit satisfies
$$
N_{\mathcal{A}(P)} (t,D) = \# \{Q \in \mathcal{A}(P) : h_D (Q) Categories:11G50, 14J28, 14G40, 14J50, 14G05 |
132. CMB 2003 (vol 46 pp. 546)
$L$-Series of Certain Elliptic Surfaces In this paper, we study the modularity of certain elliptic surfaces
by determining their $L$-series through their monodromy groups.
Categories:14J27, 11M06 |
133. CMB 2003 (vol 46 pp. 473)
A Multiplicative Analogue of Schur's Tauberian Theorem A theorem concerning the asymptotic behaviour of partial sums of the
coefficients of products of Dirichlet series is proved using properties of
regularly varying functions. This theorem is a multiplicative analogue of
Schur's Tauberian theorem for power series.
Category:11N45 |
134. CMB 2003 (vol 46 pp. 344)
Gauss and Eisenstein Sums of Order Twelve Let $q=p^{r}$ with $p$ an odd prime, and $\mathbf{F}_{q}$ denote the finite
field of $q$ elements. Let $\Tr\colon\mathbf{F}_{q} \to\mathbf{F}_{p} $ be
the usual trace map and set $\zeta_{p} =\exp(2\pi i/p)$. For any positive
integer $e$, define the (modified) Gauss sum $g_{r}(e)$ by
$$
g_{r}(e) =\sum_{x\in \mathbf{F}_{q}}\zeta_{p}^{\Tr x^{e}}
$$
Recently, Evans gave an elegant determination of $g_{1}(12)$ in terms of
$g_{1}(3)$, $g_{1}(4)$ and $g_{1}(6)$ which resolved a sign ambiguity
present in a previous evaluation. Here I generalize Evans' result to give
a complete determination of the sum $g_{r}(12)$.
Categories:11L05, 11T24 |
135. CMB 2003 (vol 46 pp. 229)
Counting the Number of Integral Points in General $n$-Dimensional Tetrahedra and Bernoulli Polynomials |
Counting the Number of Integral Points in General $n$-Dimensional Tetrahedra and Bernoulli Polynomials Recently there has been tremendous interest in counting the number of
integral points in $n$-dimen\-sional tetrahedra with non-integral
vertices due to its applications in primality testing and factoring
in number theory and in singularities theory. The purpose of this
note is to formulate a conjecture on sharp upper estimate of the
number of integral points in $n$-dimensional tetrahedra with
non-integral vertices. We show that this conjecture is true for
low dimensional cases as well as in the case of homogeneous
$n$-dimensional tetrahedra. We also show that the Bernoulli
polynomials play a role in this counting.
Categories:11B75, 11H06, 11P21, 11Y99 |
136. CMB 2003 (vol 46 pp. 178)
Sur les invariants d'Iwasawa des tours cyclotomiques We carry out the computation of the Iwasawa invariants $\rho^T_S$,
$\mu^T_S$, $\lambda^T_S$ associated to abelian $T$-ramified
over the finite steps $K_n$ of the cyclotomic
$\mathbb{Z}_\ell$-extension $K_\infty/K$ of a number field of
$\CM$-type.
Nous d\'eterminons explicitement les param\'etres d'Iwasawa
$\rho^T_S$, $\mu^T_S$, $\lambda^T_S$ des $\ell$-groupes de
$S$-classes $T$-infinit\'esimales $\Cl^T_S (K_n)$ attach\'es aux
\'etages finis de la $\mathbb{Z}_\ell$-extension cyclotomique
$K_\infty/K$ d'un corps de nombres \`a conjugaison complexe.
Categories:11R23, 11R37 |
137. CMB 2003 (vol 46 pp. 157)
Torsion Points on Certain Families of Elliptic Curves Fix an elliptic curve $y^2 = x^3+Ax+B$, satisfying $A,B \in \ZZ$,
$A\geq |B| > 0$. We prove that the $\QQ$-torsion subgroup is one of
$(0)$, $\ZZ/3\ZZ$, $\ZZ/9\ZZ$. Related numerical calculations are
discussed.
Category:11G05 |
138. CMB 2003 (vol 46 pp. 39)
Power Residue Criteria for Quadratic Units and the Negative Pell Equation Let $d>1$ be a square-free integer. Power residue criteria for the
fundamental unit $\varepsilon_d$ of the real quadratic fields $\QQ
(\sqrt{d})$ modulo a prime $p$ (for certain $d$ and $p$) are proved by
means of class field theory. These results will then be interpreted
as criteria for the solvability of the negative Pell equation $x^2 -
dp^2 y^2 = -1$. The most important solvability criterion deals with
all $d$ for which $\QQ (\sqrt{-d})$ has an elementary abelian 2-class
group and $p\equiv 5\pmod{8}$ or $p\equiv 9\pmod{16}$.
Categories:11R11, 11R27 |
139. CMB 2003 (vol 46 pp. 71)
The Number of Fields Generated by the Square Root of Values of a Given Polynomial The $abc$-conjecture is applied to various questions involving the
number of distinct fields $\mathbb{Q} \bigl( \sqrt{f(n)} \bigr)$, as we vary
over integers $n$.
Categories:11N32, 11D41 |
140. CMB 2003 (vol 46 pp. 149)
The Ramification Polygon for Curves over a Finite Field A Newton polygon is introduced for a ramified point of a Galois
covering of curves over a finite field. It is shown to be determined
by the sequence of higher ramification groups of the point. It gives
a blowing up of the wildly ramified part which separates the branches
of the curve. There is also a connection with local reciprocity.
Category:11G20 |
141. CMB 2003 (vol 46 pp. 26)
Remarques sur les points rationnels des variÃ©tÃ©s de Fermat Soit $K$ un corps de nombres de degr\'e sur $\mathbb{Q}$ inf\'erieur
ou \'egal \`a $2$. On se propose dans ce travail de faire quelques
remarques sur la question de l'existence de deux \'el\'ements non nuls
$a$ et $b$ de $K$, et d'un entier $n\geq 4$, tels que l'\'equation
$ax^n + by^n = 1$ poss\`ede au moins trois points distincts non
triviaux. Cette \'etude se ram\`ene \`a la recherche de points
rationnels sur $K$ d'une vari\'et\'e projective dans $\mathbb{P}^5$ de
dimension $3$, ou d'une surface de $\mathbb{P}^3$.
Category:11D41 |
142. CMB 2002 (vol 45 pp. 466)
A Note on the Automorphic Langlands Group Langlands has conjectured the existence of a universal group, an
extension of the absolute Galois group, which would play a fundamental
role in the classification of automorphic representations. We shall
describe a possible candidate for this group. We shall also describe
a possible candidate for the complexification of Grothendieck's
motivic Galois group.
Categories:11R39, 22E55 |
143. CMB 2002 (vol 45 pp. 606)
Postcards from the Edge, or Snapshots of the Theory of Generalised Moonshine We begin by reviewing Monstrous Moonshine. The impact of Moonshine on
algebra has been profound, but so far it has had little to teach
number theory. We introduce (using `postcards') a much larger context
in which Monstrous Moonshine naturally sits. This context suggests
Moonshine should indeed have consequences for number theory. We
provide some humble examples of this: new generalisations of Gauss
sums and quadratic reciprocity.
Categories:11F22, 17B67, 81T40 |
144. CMB 2002 (vol 45 pp. 364)
Mellin Transforms of Whittaker Functions In this note we show that for an arbitrary reductive Lie group
and any admissible irreducible Banach representation the Mellin
transforms of Whittaker functions extend to meromorphic functions.
We locate the possible poles and show that they always lie along
translates of walls of Weyl chambers.
Categories:11F30, 22E30, 11F70, 22E45 |
145. CMB 2002 (vol 45 pp. 428)
Criteria for Simultaneous Solutions of $X^2 - DY^2 = c$ and $x^2 - Dy^2 = -c$ The purpose of this article is to provide criteria for the
simultaneous solvability of the Diophantine equations $X^2 - DY^2 =
c$ and $x^2 - Dy^2 = -c$ when $c \in \mathbb{Z}$, and $D \in
\mathbb{N}$ is not a perfect square. This continues work in
\cite{me}--\cite{alfnme}.
Keywords:continued fractions, Diophantine equations, fundamental units, simultaneous solutions Categories:11A55, 11R11, 11D09 |
146. CMB 2002 (vol 45 pp. 337)
Surjectivity of $\mod\ell$ Representations Attached to Elliptic Curves and Congruence Primes For a modular elliptic curve $E/\mathbb{Q}$, we show a number of
links between the primes $\ell$ for which the mod $\ell$
representation of $E/\mathbb{Q}$ has projective dihedral image and
congruence primes for the newform associated to $E/\mathbb{Q}$.
Keywords:torsion points of elliptic curves, Galois representations, congruence primes, Serre tori, grossencharacters, non-split Cartan Categories:11G05, 11F80 |
147. 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 |
148. 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 |
149. 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 |
150. CMB 2002 (vol 45 pp. 231)
Erratum:~~The Lehmer Polynomial and Pretzel Links Erratum to {\it The Lehmer Polynomial and Pretzel Links},
Canad. J. Math. {\bf 44}(2001), 440--451.
Keywords:Alexander polynomial, pretzel knot, Mahler measure, Salem number, Coxeter groups Categories:57M05, 57M25, 11R04, 11R27 |