Expand all Collapse all | Results 176 - 200 of 227 |
176. CJM 2001 (vol 53 pp. 897)
On Some Exponential Equations of S.~S.~Pillai In this paper, we establish a number of theorems on the classic
Diophantine equation of S.~S.~Pillai, $a^x-b^y=c$, where $a$, $b$ and
$c$ are given nonzero integers with $a,b \geq 2$. In particular, we
obtain the sharp result that there are at most two solutions in
positive integers $x$ and $y$ and deduce a variety of explicit
conditions under which there exists at most a single such solution.
These improve or generalize prior work of Le, Leveque, Pillai, Scott
and Terai. The main tools used include lower bounds for linear forms
in the logarithms of (two) algebraic numbers and various elementary
arguments.
Categories:11D61, 11D45, 11J86 |
177. CJM 2001 (vol 53 pp. 866)
Inverse Problems for Partition Functions Let $p_w(n)$ be the weighted partition function defined by the
generating function $\sum^\infty_{n=0}p_w(n)x^n=\prod^\infty_{m=1}
(1-x^m)^{-w(m)}$, where $w(m)$ is a non-negative arithmetic function.
Let $P_w(u)=\sum_{n\le u}p_w(n)$ and $N_w(u)=\sum_{n\le u}w(n)$ be the
summatory functions for $p_w(n)$ and $w(n)$, respectively.
Generalizing results of G.~A.~Freiman and E.~E.~Kohlbecker, we show
that, for a large class of functions $\Phi(u)$ and $\lambda(u)$, an
estimate for $P_w(u)$ of the form
$\log P_w(u)=\Phi(u)\bigl\{1+O(1/\lambda(u)\bigr)\bigr\}$
$(u\to\infty)$ implies an estimate for $N_w(u)$ of the form
$N_w(u)=\Phi^\ast(u)\bigl\{1+O\bigl(1/\log\lambda(u)\bigr)\bigr\}$
$(u\to\infty)$ with a suitable function $\Phi^\ast(u)$ defined in
terms of $\Phi(u)$. We apply this result and related results to
obtain characterizations of the Riemann Hypothesis and the
Generalized Riemann Hypothesis in terms of the asymptotic behavior
of certain weighted partition functions.
Categories:11P82, 11M26, 40E05 |
178. CJM 2001 (vol 53 pp. 449)
Descending Rational Points on Elliptic Curves to Smaller Fields In this paper, we study the Mordell-Weil group of an elliptic curve
as a Galois module. We consider an elliptic curve $E$ defined over a
number field $K$ whose Mordell-Weil rank over a Galois extension $F$ is
$1$, $2$ or $3$. We show that $E$ acquires a point (points) of
infinite order over a field whose Galois group is one of $C_n \times C_m$
($n= 1, 2, 3, 4, 6, m= 1, 2$), $D_n \times C_m$ ($n= 2, 3, 4, 6, m= 1, 2$),
$A_4 \times C_m$ ($m=1,2$), $S_4 \times C_m$ ($m=1,2$). Next, we consider
the case where $E$ has complex multiplication by the ring of integers $\o$
of an imaginary quadratic field $\k$ contained in $K$. Suppose that the
$\o$-rank over a Galois extension $F$ is $1$ or $2$. If $\k\neq\Q(\sqrt{-1})$
and $\Q(\sqrt{-3})$ and $h_{\k}$ (class number of $\k$) is odd, we show that
$E$ acquires positive $\o$-rank over a cyclic extension of $K$ or over a
field whose Galois group is one of $\SL_2(\Z/3\Z)$, an extension of
$\SL_2(\Z/3\Z)$ by $\Z/2\Z$, or a central extension by the dihedral group.
Finally, we discuss the relation of the above results to the vanishing of
$L$-functions.
Categories:11G05, 11G40, 11R32, 11R33 |
179. CJM 2001 (vol 53 pp. 434)
Values of the Dedekind Eta Function at Quadratic Irrationalities: Corrigendum Habib Muzaffar of Carleton University has pointed out to the authors
that in their paper [A] only the result
\[
\pi_{K,d}(x)+\pi_{K^{-1},d}(x)=\frac{1}{h(d)}\frac{x}{\log
x}+O_{K,d}\Bigl(\frac {x}{\log^2x}\Bigr)
\]
follows from the prime ideal theorem with remainder for ideal classes,
and not the stronger result
\[
\pi_{K,d}(x)=\frac{1}{2h(d)}\frac{x}{\log
x}+O_{K,d}\Bigl(\frac {x}{\log^2x}\Bigr)
\]
stated in Lemma~5.2. This necessitates changes in Sections~5 and 6 of
[A]. The main results of the paper are not affected by these changes.
It should also be noted that, starting on page 177 of [A], each and
every occurrence of $o(s-1)$ should be replaced by $o(1)$.
Sections~5 and 6 of [A] have been rewritten to incorporate the above
mentioned correction and are given below. They should replace the
original Sections~5 and 6 of [A].
Keywords:Dedekind eta function, quadratic irrationalities, binary quadratic forms, form class group Categories:11F20, 11E45 |
180. CJM 2001 (vol 53 pp. 310)
On a Product Related to the Cubic Gauss Sum, III We have seen, in the previous works [5], [6], that the argument of a
certain product is closely connected to that of the cubic Gauss sum.
Here the absolute value of the product will be investigated.
Keywords:Gauss sum, Lagrange resolvent Categories:11L05, 11R33 |
181. CJM 2001 (vol 53 pp. 244)
On the Tempered Spectrum of Quasi-Split Classical Groups II We determine the poles of the standard intertwining operators for a
maximal parabolic subgroup of the quasi-split unitary group defined by
a quadratic extension $E/F$ of $p$-adic fields of characteristic
zero. We study the case where the Levi component $M \simeq \GL_n (E)
\times U_m (F)$, with $n \equiv m$ $(\mod 2)$. This, along with
earlier work, determines the poles of the local Rankin-Selberg product
$L$-function $L(s, \tau' \times \tau)$, with $\tau'$ an irreducible
unitary supercuspidal representation of $\GL_n (E)$ and $\tau$ a
generic irreducible unitary supercuspidal representation of $U_m
(F)$. The results are interpreted using the theory of twisted
endoscopy.
Categories:22E50, 11S70 |
182. CJM 2001 (vol 53 pp. 414)
Nombres premiers de la forme $\floor{n^c}$ For $c>1$ we denote by $\pi_c(x)$ the number of integers $n \leq x$
such that $\floor{n^c}$ is prime. In 1953, Piatetski-Shapiro has
proved that $\pi_c(x) \sim \frac{x}{c\log x}$, $x \rightarrow +\infty$
holds for $c<12/11$. Many authors have extended this range, which
measures our progress in exponential sums techniques.
In this article we obtain $c < 1.16117\dots\;$.
Categories:11L07, 11L20, 11N05 |
183. CJM 2001 (vol 53 pp. 33)
Merit Factors of Polynomials Formed by Jacobi Symbols We give explicit formulas for the $L_4$ norm (or equivalently for the
merit factors) of various sequences of polynomials related to the
polynomials
$$
f(z) := \sum_{n=0}^{N-1} \leg{n}{N} z^n.
$$
and
$$
f_t(z) = \sum_{n=0}^{N-1} \leg{n+t}{N} z^n.
$$
where $(\frac{\cdot}{N})$ is the Jacobi symbol.
Two cases of particular interest are when $N = pq$ is a product of two
primes and $p = q+2$ or $p = q+4$. This extends work of H{\o}holdt,
Jensen and Jensen and of the authors.
This study arises from a number of conjectures of Erd\H{o}s,
Littlewood and others that concern the norms of polynomials with
$-1,1$ coefficients on the disc. The current best examples are of the
above form when $N$ is prime and it is natural to see what happens for
composite~$N$.
Keywords:Character polynomial, Class Number, $-1,1$ coefficients, Merit factor, Fekete polynomials, Turyn Polynomials, Littlewood polynomials, Twin Primes, Jacobi Symbols Categories:11J54, 11B83, 12-04 |
184. CJM 2001 (vol 53 pp. 122)
A Truncated Integral of the Poisson Summation Formula Let $G$ be a reductive algebraic group defined over $\bQ$, with
anisotropic centre. Given a rational action of $G$ on a finite-dimensional
vector space $V$, we analyze the truncated integral of the theta series
corresponding to a Schwartz-Bruhat function on $V(\bA)$. The Poisson
summation formula then yields an identity of distributions on $V(\bA)$.
The truncation used is due to Arthur.
Categories:11F99, 11F72 |
185. CJM 2001 (vol 53 pp. 98)
On the Curves Associated to Certain Rings of Automorphic Forms In a 1987 paper, Gross introduced certain curves associated to a
definite quaternion algebra $B$ over $\Q$; he then proved an analog of
his result with Zagier for these curves. In Gross' paper, the curves
were defined in a somewhat {\it ad hoc\/} manner. In this article, we
present an interpretation of these curves as projective varieties
arising from graded rings of automorphic forms on $B^\times$,
analogously to the construction in the Satake compactification. To
define such graded rings, one needs to introduce a ``multiplication''
of automorphic forms that arises from the representation ring of
$B^\times$. The resulting curves are unions of projective lines
equipped with a collection of Hecke correspondences. They parametrize
two-dimensional complex tori with quaternionic multiplication. In
general, these complex tori are not abelian varieties; they are
algebraic precisely when they correspond to $\CM$ points on these curves,
and are thus isogenous to a product $E \times E$, where $E$ is an
elliptic curve with complex multiplication. For these $\CM$ points one
can make a relation between the action of the $p$-th Hecke operator
and Frobenius at $p$, similar to the well-known congruence relation of
Eichler and Shimura.
Category:11F |
186. CJM 2000 (vol 52 pp. 1121)
Ramanujan Type Buildings We will construct a finite union of finite quotients of the affine
building of the group $\GL_3$ over the field of $p$-adic numbers
$\mathbb{Q}_p$. We will view this object as a hypergraph and estimate
the spectrum of its underlying graph.
Keywords:automorphic representations, buildings Category:11F70 |
187. CJM 2000 (vol 52 pp. 1269)
Well Ramified Extensions of Complete Discrete Valuation Fields with Applications to the Kato Conductor |
Well Ramified Extensions of Complete Discrete Valuation Fields with Applications to the Kato Conductor We study extensions $L/K$ of complete discrete valuation fields $K$
with residue field $\oK$ of characteristic $p > 0$, which we do not
assume to be perfect. Our work concerns ramification theory for such
extensions, in particular we show that all classical properties which
are true under the hypothesis {\it ``the residue field extension
$\oL/\oK$ is separable''} are still valid under the more general
hypothesis that the valuation ring extension is monogenic. We also
show that conversely, if classical ramification properties hold true
for an extension $L/K$, then the extension of valuation rings is
monogenic. These are the ``{\it well ramified}'' extensions. We show
that there are only three possible types of well ramified extensions
and we give examples. In the last part of the paper we consider, for
the three types, Kato's generalization of the conductor, which we show
how to bound in certain cases.
Categories:11S, 11S15, 11S20 |
188. CJM 2000 (vol 52 pp. 833)
W-Groups under Quadratic Extensions of Fields To each field $F$ of characteristic not $2$, one can associate a
certain Galois group $\G_F$, the so-called W-group of $F$, which
carries essentially the same information as the Witt ring $W(F)$ of
$F$. In this paper we investigate the connection between $\wg$ and
$\G_{F(\sqrt{a})}$, where $F(\sqrt{a})$ is a proper quadratic
extension of $F$. We obtain a precise description in the case when
$F$ is a pythagorean formally real field and $a = -1$, and show that
the W-group of a proper field extension $K/F$ is a subgroup of the
W-group of $F$ if and only if $F$ is a formally real pythagorean field
and $K = F(\sqrt{-1})$. This theorem can be viewed as an analogue of
the classical Artin-Schreier's theorem describing fields fixed by
finite subgroups of absolute Galois groups. We also obtain precise
results in the case when $a$ is a double-rigid element in $F$. Some
of these results carry over to the general setting.
Categories:11E81, 12D15 |
189. CJM 2000 (vol 52 pp. 804)
The Distributions in the Invariant Trace Formula Are Supported on Characters J.~Arthur put the trace formula in invariant form for all connected
reductive groups and certain disconnected ones. However his work was
written so as to apply to the general disconnected case, modulo two
missing ingredients. This paper supplies one of those missing
ingredients, namely an argument in Galois cohomology of a kind first
used by D.~Kazhdan in the connected case.
Categories:22E50, 11S37, 10D40 |
190. CJM 2000 (vol 52 pp. 673)
Sums of Two Squares in Short Intervals Let $\calS$ denote the set of integers representable as a sum of two
squares. Since $\calS$ can be described as the unsifted elements of a
sieving process of positive dimension, it is to be expected that
$\calS$ has many properties in common with the set of prime numbers.
In this paper we exhibit ``unexpected irregularities'' in the
distribution of sums of two squares in short intervals, a phenomenon
analogous to that discovered by Maier, over a decade ago, in the
distribution of prime numbers. To be precise, we show that there are
infinitely many short intervals containing considerably more elements
of $\calS$ than expected, and infinitely many intervals containing
considerably fewer than expected.
Keywords:sums of two squares, sieves, short intervals, smooth numbers Categories:11N36, 11N37, 11N25 |
191. CJM 2000 (vol 52 pp. 737)
An Automorphic Theta Module for Quaternionic Exceptional Groups We construct an automorphic realization of the global minimal
representation of quaternionic exceptional groups, using the theory
of Eisenstein series, and use this for the study of theta
correspondences.
Categories:11F27, 11F70 |
192. CJM 2000 (vol 52 pp. 613)
Small Solutions of $\phi_1 x_1^2 + \cdots + \phi_n x_n^2 = 0$ Let $\phi_1,\dots,\phi_n$ $(n\geq 2)$ be nonzero integers such that
the equation
$$
\sum_{i=1}^n \phi_i x_i^2 = 0
$$
is solvable in integers $x_1,\dots,x_n$ not all zero. It is shown
that there exists a solution satisfying
$$
0 < \sum_{i=1}^n |\phi_i| x_i^2 \leq 2 |\phi_1 \cdots \phi_n|,
$$
and that the constant 2 is best possible.
Keywords:small solutions, diagonal quadratic forms Category:11E25 |
193. CJM 2000 (vol 52 pp. 369)
An Upper Bound on the Least Inert Prime in a Real Quadratic Field It is shown by a combination of analytic and computational
techniques that for any positive fundamental discriminant $D >
3705$, there is always at least one prime $p < \sqrt{D}/2$ such
that the Kronecker symbol $\left(D/p\right) = -1$.
Categories:11R11, 11Y40 |
194. CJM 2000 (vol 52 pp. 31)
On Russell-Type Modular Equations In this paper, we revisit Russell-type modular equations, a
collection of modular equations first studied systematically by
R.~Russell in 1887. We give a proof of Russell's main theorem and
indicate the relations between such equations and the constructions
of Hilbert class fields of imaginary quadratic fields. Motivated by
Russell's theorem, we state and prove its cubic analogue which
allows us to construct Russell-type modular equations in the theory
of signature~$3$.
Categories:33D10, 33C05, 11F11 |
195. CJM 2000 (vol 52 pp. 47)
Comparison of $K$-Theory Galois Module Structure Invariants We prove that two, apparently different, class-group valued Galois
module structure invariants associated to the algebraic $K$-groups
of rings of algebraic integers coincide. This comparison result is
particularly important in making explicit calculations.
Categories:11S99, 19F15, 19F27 |
196. CJM 2000 (vol 52 pp. 172)
Cubic Base Change for $\GL(2)$ We prove a relative trace formula that establishes the cubic base
change for $\GL(2)$. One also gets a classification of the image
of base change. The case when the field extension is nonnormal
gives an example where a trace formula is used to prove lifting
which is not endoscopic.
Categories:11F70, 11F72 |
197. CJM 1999 (vol 51 pp. 1307)
Quadratic Integers and Coxeter Groups Matrices whose entries belong to certain rings of algebraic
integers can be associated with discrete groups of transformations
of inversive $n$-space or hyperbolic $(n+1)$-space
$\mbox{H}^{n+1}$. For small $n$, these may be Coxeter groups,
generated by reflections, or certain subgroups whose generators
include direct isometries of $\mbox{H}^{n+1}$. We show how linear
fractional transformations over rings of rational and (real or
imaginary) quadratic integers are related to the symmetry groups of
regular tilings of the hyperbolic plane or 3-space. New light is
shed on the properties of the rational modular group $\PSL_2
(\bbZ)$, the Gaussian modular (Picard) group $\PSL_2 (\bbZ[{\it
i}])$, and the Eisenstein modular group $\PSL_2 (\bbZ[\omega ])$.
Categories:11F06, 20F55, 20G20, 20H10, 22E40 |
198. CJM 1999 (vol 51 pp. 1258)
Similarity Submodules and Root Systems in Four Dimensions Lattices and $\ZZ$-modules in Euclidean space possess an infinitude
of subsets that are images of the original set under similarity
transformation. We classify such self-similar images according to
their indices for certain 4D examples that are related to 4D root
systems, both crystallographic and non-crystallographic. We
encapsulate their statistics in terms of Dirichlet series
generating functions and derive some of their asymptotic properties.
Categories:11S45, 11H05, 52C07 |
199. CJM 1999 (vol 51 pp. 1020)
On Functions Satisfying Modular Equations for Infinitely Many Primes In this paper we study properties of the functions which satisfy
modular equations for infinitely many primes. The two main results
are:
\begin{enumerate}
\item[1)] every such function is analytic in the upper half plane;
\item[2)] if such function takes the same value in two different
points $z_1$ and $z_2$ then there exists an $f$-preserving analytic
bijection between neighbourhoods of $z_1$ and $z_2$.
\end{enumerate}
Category:11Mxx |
200. CJM 1999 (vol 51 pp. 952)
On Limit Multiplicities for Spaces of Automorphic Forms Let $\Gamma$ be a rank-one arithmetic subgroup of a
semisimple Lie group~$G$. For fixed $K$-Type, the spectral
side of the Selberg trace formula defines a distribution
on the space of infinitesimal characters of~$G$, whose
discrete part encodes the dimensions of the spaces of
square-integrable $\Gamma$-automorphic forms. It is shown
that this distribution converges to the Plancherel measure
of $G$ when $\Ga$ shrinks to the trivial group in a certain
restricted way. The analogous assertion for cocompact
lattices $\Gamma$ follows from results of DeGeorge-Wallach
and Delorme.
Keywords:limit multiplicities, automorphic forms, noncompact quotients, Selberg trace formula, functional calculus Categories:11F72, 22E30, 22E40, 43A85, 58G25 |