Expand all Collapse all | Results 126 - 150 of 217 |
126. CJM 2005 (vol 57 pp. 449)
On the Sizes of Gaps in the Fourier Expansion of Modular Forms Let $f= \sum_{n=1}^{\infty} a_f(n)q^n$ be a cusp form with integer
weight $k \geq 2$ that is not a linear combination of forms with
complex multiplication. For $n \geq 1$, let
$$
i_f(n)=\begin{cases}\max\{ i :
a_f(n+j)=0 \text{ for all } 0 \leq j \leq
i\}&\text{if $a_f(n)=0$,}\\
0&\text{otherwise}.\end{cases}
$$
Concerning bounded values
of $i_f(n)$ we prove that for $\epsilon >0$ there exists $M =
M(\epsilon,f)$ such that $\# \{n \leq x : i_f(n) \leq M\} \geq (1
- \epsilon) x$. Using results of Wu, we show that if $f$ is a weight 2
cusp form for an elliptic curve without complex multiplication, then
$i_f(n) \ll_{f, \epsilon} n^{\frac{51}{134} + \epsilon}$. Using a
result of David and Pappalardi, we improve the exponent to
$\frac{1}{3}$ for almost all newforms associated to elliptic curves
without complex multiplication. Inspired by a classical paper of
Selberg, we also investigate $i_f(n)$ on the average using well known
bounds on the Riemann Zeta function.
Category:11F30 |
127. CJM 2005 (vol 57 pp. 616)
Reducibility of Generalized Principal Series In this paper we describe reducibility of non-unitary generalized
principal series for classical $p$-adic groups in terms of the
classification of discrete series due to M\oe glin and Tadi\'c.
Categories:22E35, and, 50, 11F70 |
128. CJM 2005 (vol 57 pp. 267)
Partial Euler Products on the Critical Line The initial version of the Birch and Swinnerton-Dyer conjecture
concerned asymptotics for partial Euler products for an elliptic curve
$L$-function at $s = 1$. Goldfeld later proved that these asymptotics
imply the Riemann hypothesis for the $L$-function and that the
constant in the asymptotics has an unexpected factor of $\sqrt{2}$.
We extend Goldfeld's theorem to an analysis of partial Euler products
for a typical $L$-function along its critical line. The general
$\sqrt{2}$ phenomenon is related to second moments, while the
asymptotic behavior (over number fields) is proved to be equivalent to
a condition that in a precise sense seems much deeper than the Riemann
hypothesis. Over function fields, the Euler product asymptotics can
sometimes be proved unconditionally.
Keywords:Euler product, explicit formula, second moment Categories:11M41, 11S40 |
129. CJM 2005 (vol 57 pp. 298)
On the Waring--Goldbach Problem: Exceptional Sets for Sums of Cubes and Higher Powers We investigate exceptional sets in the Waring--Goldbach problem. For
example, in the cubic case, we show that all but
$O(N^{79/84+\epsilon})$ integers subject to the necessary local
conditions can be represented as the sum of five cubes of primes.
Furthermore, we develop a new device that leads easily to similar
estimates for exceptional sets for sums of fourth and higher powers of
primes.
Categories:11P32, 11L15, 11L20, 11N36, 11P55 |
130. CJM 2005 (vol 57 pp. 328)
On a Conjecture of Birch and Swinnerton-Dyer Let \(E/\mathbb{Q}\) be an elliptic curve defined by the equation
\(y^2=x^3 +ax +b\). For a prime \(p, \linebreak p \nmid\Delta
=-16(4a^3+27b^2)\neq 0\), define \[ N_p = p+1 -a_p =
|E(\mathbb{F}_p)|. \] As a precursor to their celebrated conjecture,
Birch and Swinnerton-Dyer originally conjectured that for some
constant $c$, \[ \prod_{p \leq x, p \nmid\Delta } \frac{N_p}{p} \sim c
(\log x)^r, \quad x \to \infty. \] Let \(\alpha _p\) and \(\beta
_p\) be the eigenvalues of the Frobenius at \(p\). Define \[
\tilde{c}_n = \begin{cases} \frac{\alpha_p^k + \beta_p^k}{k}& n =p^k,
p \textrm{ is a prime, $k$ is a natural number, $p\nmid \Delta$} .
\\ 0 & \text{otherwise}. \end{cases}. \] and \(\tilde{C}(x)=
\sum_{n\leq x} \tilde{c}_n\). In this paper, we establish the
equivalence between the conjecture and the condition
\(\tilde{C}(x)=\mathbf{o}(x)\). The asymptotic condition is indeed
much deeper than what we know so far or what we can know under the
analogue of the Riemann hypothesis. In addition, we provide an
oscillation theorem and an \(\Omega\) theorem which relate to the
constant $c$ in the conjecture.
Categories:11M41, 11M06 |
131. CJM 2005 (vol 57 pp. 338)
Certain Exponential Sums and Random Walks on Elliptic Curves For a given elliptic curve $\E$, we obtain an upper bound
on the discrepancy of sets of
multiples $z_sG$ where $z_s$ runs through a sequence
$\cZ=\(z_1, \dots, z_T\)$
such that $k z_1,\dots, kz_T $ is a permutation of
$z_1, \dots, z_T$, both sequences taken modulo $t$, for
sufficiently many distinct values of $k$ modulo $t$.
We apply this result to studying an analogue of the power generator
over an elliptic curve. These results are elliptic curve analogues
of those obtained for multiplicative groups of finite fields and
residue rings.
Categories:11L07, 11T23, 11T71, 14H52, 94A60 |
132. CJM 2005 (vol 57 pp. 180)
On the Size of the Wild Set To every pair of algebraic number fields with isomorphic Witt rings
one can associate a number, called the {\it minimum number of wild
primes}. Earlier investigations have established lower bounds for this
number. In this paper an analysis is presented that expresses the
minimum number of wild primes in terms of the number of wild dyadic
primes. This formula not only gives immediate upper bounds, but can be
considered to be an exact formula for the minimum number of wild
primes.
Categories:11E12, 11E81, 19F15, 11R29 |
133. CJM 2004 (vol 56 pp. 897)
Finding and Excluding $b$-ary Machin-Type Individual Digit Formulae Constants with formulae of the form treated by D.~Bailey,
P.~Borwein, and S.~Plouffe (\emph{BBP formulae} to a given base $b$) have
interesting computational properties, such as allowing single
digits in their base $b$ expansion to be independently computed,
and there are hints that they
should be \emph{normal} numbers, {\em i.e.,} that their base $b$ digits
are randomly distributed. We study a formally limited subset of BBP
formulae, which we call \emph{Machin-type BBP formulae}, for which it
is relatively easy to determine whether or not a given constant
$\kappa$ has a Machin-type BBP formula. In particular, given $b \in
\mathbb{N}$, $b>2$, $b$ not a proper power, a $b$-ary Machin-type
BBP arctangent formula for $\kappa$ is a formula of the form $\kappa
= \sum_{m} a_m \arctan(-b^{-m})$, $a_m \in \mathbb{Q}$, while when
$b=2$, we also allow terms of the form $a_m \arctan(1/(1-2^m))$. Of
particular interest, we show that $\pi$ has no Machin-type BBP
arctangent formula when $b \neq 2$. To the best of our knowledge,
when there is no Machin-type BBP formula for a constant then no BBP
formula of any form is known for that constant.
Keywords:BBP formulae, Machin-type formulae, arctangents,, logarithms, normality, Mersenne primes, Bang's theorem,, Zsigmondy's theorem, primitive prime factors, $p$-adic analysis Categories:11Y99, 11A51, 11Y50, 11K36, 33B10 |
134. CJM 2004 (vol 56 pp. 673)
DÃ©faut de semi-stabilitÃ© des courbes elliptiques dans le cas non ramifiÃ© Let $\overline {\Q_2}$ be an algebraic closure of $\Q_2$ and $K$ be an unramified
finite extension of $\Q_2$ included in $\overline {\Q_2}$. Let $E$ be an elliptic
curve defined over $K$ with additive reduction over $K$, and having an integral
modular invariant. Let us denote by $K_{nr}$ the maximal unramified extension of
$K$ contained in $\overline {\Q_2}$. There exists a smallest finite extension $L$
of $K_{nr}$ over which $E$ has good reduction. We determine in this paper the
degree of the extension $L/K_{nr}$.
Category:11G07 |
135. CJM 2004 (vol 56 pp. 612)
Solvable Points on Projective Algebraic Curves We examine the problem of finding rational points defined over
solvable extensions on algebraic curves defined over general fields.
We construct non-singular, geometrically irreducible projective curves
without solvable points of genus $g$, when $g$ is at least $40$, over
fields of arbitrary characteristic. We prove that every smooth,
geometrically irreducible projective curve of genus $0$, $2$, $3$ or
$4$ defined over any field has a solvable point. Finally we prove
that every genus $1$ curve defined over a local field of
characteristic zero with residue field of characteristic $p$ has a
divisor of degree prime to $6p$ defined over a solvable extension.
Categories:14H25, 11D88 |
136. CJM 2004 (vol 56 pp. 373)
An Elementary Proof of a Weak Exceptional Zero Conjecture In this paper we extend Darmon's theory of ``integration on $\uh_p\times \uh$''
to cusp forms $f$ of higher even weight. This enables us to prove a ``weak
exceptional zero conjecture'': that when the $p$-adic $L$-function of $f$ has
an exceptional zero at the central point, the $\mathcal{L}$-invariant arising is
independent of a twist by certain Dirichlet characters.
Categories:11F11, 11F67 |
137. CJM 2004 (vol 56 pp. 406)
Theta Series, Eisenstein Series and PoincarÃ© Series over Function Fields We construct analogues of theta series, Eisenstein series and
Poincar\'e series for function fields of one variable over finite
fields, and prove their basic properties.
Category:11F12 |
138. CJM 2004 (vol 56 pp. 356)
Non-Abelian Generalizations of the Erd\H os-Kac Theorem Let $a$ be a natural number greater than $1$.
Let $f_a(n)$ be the order of $a$ mod $n$.
Denote by $\omega(n)$ the number of distinct
prime factors of $n$. Assuming a weak form
of the generalised Riemann hypothesis, we prove
the following conjecture of Erd\"os and Pomerance:
The number of $n\leq x$ coprime to $a$ satisfying
$$\alpha \leq \frac{\omega(f_a(n)) - (\log \log n)^2/2
}{ (\log \log n)^{3/2}/\sqrt{3}} \leq \beta $$
is asymptotic to
$$\left(\frac{ 1 }{ \sqrt{2\pi}} \int_{\alpha}^{\beta}
e^{-t^2/2}dt\right)
\frac{x\phi(a) }{ a}, $$
as $x$ tends to infinity.
Keywords:Tur{\' a}n's theorem, Erd{\H o}s-Kac theorem, Chebotarev density theorem,, Erd{\H o}s-Pomerance conjecture Categories:11K36, 11K99 |
139. CJM 2004 (vol 56 pp. 71)
Euclidean Rings of Algebraic Integers Let $K$ be a finite Galois extension of the field of rational numbers
with unit rank greater than~3. We prove that the ring of integers of
$K$ is a Euclidean domain if and only if it is a principal ideal
domain. This was previously known under the assumption of the
generalized Riemann hypothesis for Dedekind zeta functions. We now
prove this unconditionally.
Categories:11R04, 11R27, 11R32, 11R42, 11N36 |
140. CJM 2004 (vol 56 pp. 55)
$\mathbb{Z}[\sqrt{14}]$ is Euclidean We provide the first unconditional proof that the ring $\mathbb{Z}
[\sqrt{14}]$ is a Euclidean domain. The proof is generalized to
other real quadratic fields and to cyclotomic extensions of
$\mathbb{Q}$. It is proved that if $K$ is a real quadratic field
(modulo the existence of two special primes of $K$) or if $K$ is a
cyclotomic extension of $\mathbb{Q}$ then:
\begin{center}
\emph{%
the ring of integers of $K$ is a Euclidean domain if and only if
it is a principal ideal domain.}
\end{center}
The proof is a modification of the proof of a theorem of Clark and
Murty giving a similar result when $K$ is a totally real extension of
degree at least three. The main changes are a new Motzkin-type lemma
and the addition of the large sieve to the argument. These changes
allow application of a powerful theorem due to Bombieri, Friedlander
and Iwaniec in order to obtain the result in the real quadratic case.
The modification also allows the completion of the classification of
cyclotomic extensions in terms of the Euclidean property.
Categories:11R04, 11R11 |
141. CJM 2004 (vol 56 pp. 168)
On a Certain Residual Spectrum of $\Sp_8$ Let $G=\Sp_{2n}$ be the symplectic group defined over a number
field $F$. Let $\mathbb{A}$ be the ring of adeles. A fundamental
problem in the theory of automorphic forms is to decompose the
right regular representation of $G(\mathbb{A})$ acting on the
Hilbert space $L^2\bigl(G(F)\setminus G(\mathbb{A})\bigr)$. Main
contributions have been made by Langlands. He described, using his
theory of Eisenstein series, an orthogonal decomposition of this
space of the form: $L_{\dis}^2 \bigl( G(F)\setminus G(\mathbb{A})
\bigr)=\bigoplus_{(M,\pi)} L_{\dis}^2(G(F) \setminus G(\mathbb{A})
\bigr)_{(M,\pi)}$, where $(M,\pi)$ is a Levi subgroup with a
cuspidal automorphic representation $\pi$ taken modulo conjugacy
(Here we normalize $\pi$ so that the action of the maximal split
torus in the center of $G$ at the archimedean places is trivial.)
and $L_{\dis}^2\bigl(G(F)\setminus G(\mathbb{A})\bigr)_{(M,\pi)}$
is a space of residues of Eisenstein series associated to
$(M,\pi)$. In this paper, we will completely determine the space
$L_{\dis}^2\bigl(G(F)\setminus G(\mathbb{A})\bigr)_{(M,\pi)}$, when
$M\simeq\GL_2\times\GL_2$. This is the first result on the
residual spectrum for non-maximal, non-Borel parabolic subgroups,
other than $\GL_n$.
Categories:11F70, 22E55 |
142. CJM 2004 (vol 56 pp. 194)
Selmer Groups of Elliptic Curves with Complex Multiplication Suppose $K$ is an imaginary quadratic field and $E$ is an elliptic curve over a
number field $F$ with complex multiplication by the ring of integers in $K$.
Let $p$ be a rational prime that splits as $\mathfrak{p}_{1}\mathfrak{p}_{2}$
in $K$. Let $E_{p^{n}}$ denote the $p^{n}$-division points on $E$. Assume
that $F(E_{p^{n}})$ is abelian over $K$ for all $n\geq 0$. This paper proves
that the Pontrjagin dual of the $\mathfrak{p}_{1}^{\infty}$-Selmer group of
$E$ over $F(E_{p^{\infty}})$ is a finitely generated free $\Lambda$-module,
where $\Lambda$ is the Iwasawa algebra of $\Gal\bigl(F(E_{p^{\infty}})/
F(E_{\mathfrak{p}_{1}^{\infty}\mathfrak{p}_{2}})\bigr)$. It also gives a simple
formula for the rank of the Pontrjagin dual as a $\Lambda$-module.
Categories:11R23, 11G05 |
143. CJM 2004 (vol 56 pp. 23)
Ternary Diophantine Equations via Galois Representations and Modular Forms In this paper, we develop techniques for solving ternary Diophantine
equations of the shape $Ax^n + By^n = Cz^2$, based upon the theory of
Galois representations and modular forms. We subsequently utilize
these methods to completely solve such equations for various choices
of the parameters $A$, $B$ and $C$. We conclude with an application
of our results to certain classical polynomial-exponential equations,
such as those of Ramanujan--Nagell type.
Categories:11D41, 11F11, 11G05 |
144. CJM 2003 (vol 55 pp. 1191)
Decay of Mean Values of Multiplicative Functions For given multiplicative function $f$, with $|f(n)| \leq 1$ for all
$n$, we are interested in how fast its mean value $(1/x) \sum_{n\leq
x} f(n)$ converges. Hal\'asz showed that this depends on the minimum
$M$ (over $y\in \mathbb{R}$) of $\sum_{p\leq x} \bigl( 1 - \Re (f(p)
p^{-iy}) \bigr) / p$, and subsequent authors gave the upper bound $\ll
(1+M) e^{-M}$. For many applications it is necessary to have explicit
constants in this and various related bounds, and we provide these via
our own variant of the Hal\'asz-Montgomery lemma (in fact the constant
we give is best possible up to a factor of 10). We also develop a new
type of hybrid bound in terms of the location of the absolute value of
$y$ that minimizes the sum above. As one application we give bounds
for the least representatives of the cosets of the $k$-th powers
mod~$p$.
Categories:11N60, 11N56, 10K20, 11N37 |
145. CJM 2003 (vol 55 pp. 933)
Renormalized Periods on $\GL(3)$ A theory of renormalization of divergent integrals over torus
periods on $\GL(3)$ is given, based on a relative truncation. It
is shown that the renormalized periods of Eisenstein series have
unexpected functional equations.
Categories:11F12, 11F55 |
146. CJM 2003 (vol 55 pp. 897)
Hypergeometric Abelian Varieties In this paper, we construct abelian varieties associated to Gauss' and
Appell--Lauricella hypergeometric series.
Abelian varieties of this kind and the algebraic curves we define
to construct them were considered by several authors in settings
ranging from monodromy groups (Deligne, Mostow), exceptional sets
(Cohen, Wolfart, W\"ustholz), modular embeddings (Cohen, Wolfart) to
CM-type (Cohen, Shiga, Wolfart) and modularity (Darmon).
Our contribution is to provide a complete, explicit and self-contained
geometric construction.
Categories:11, 14 |
147. CJM 2003 (vol 55 pp. 711)
Adic Topologies for the Rational Integers A topology on $\mathbb{Z}$, which gives a nice proof that the
set of prime integers is infinite, is characterised and examined.
It is found to be homeomorphic to $\mathbb{Q}$, with a compact
completion homeomorphic to the Cantor set. It has a natural place
in a family of topologies on $\mathbb{Z}$, which includes the
$p$-adics, and one in which the set of rational primes $\mathbb{P}$
is dense. Examples from number theory are given, including the
primes and squares, Fermat numbers, Fibonacci numbers and $k$-free
numbers.
Keywords:$p$-adic, metrizable, quasi-valuation, topological ring,, completion, inverse limit, diophantine equation, prime integers,, Fermat numbers, Fibonacci numbers Categories:11B05, 11B25, 11B50, 13J10, 13B35 |
148. CJM 2003 (vol 55 pp. 673)
A Note on Cyclotomic Euler Systems and the Double Complex Method Let $\FF$ be a finite real abelian extension of $\QQ$. Let $M$ be an odd
positive integer. For every squarefree positive integer $r$ the prime
factors of which are congruent to $1$ modulo $M$ and split completely
in $\FF$, the corresponding Kolyvagin class $\kappa_r\in\FF^{\times}/
\FF^{\times M}$ satisfies a remarkable and crucial recursion which
for each prime number $\ell$ dividing $r$ determines the order of
vanishing of $\kappa_r$ at each place of $\FF$ above $\ell$ in terms
of $\kappa_{r/\ell}$. In this note we give the recursion a new and
universal interpretation with the help of the double complex method
introduced by Anderson and further developed by Das and Ouyang. Namely,
we show that the recursion satisfied by Kolyvagin classes is the
specialization of a universal recursion independent of $\FF$ satisfied
by universal Kolyvagin classes in the group cohomology of the universal
ordinary distribution {\it \`a la\/} Kubert tensored with $\ZZ/M\ZZ$.
Further, we show by a method involving a variant of the diagonal shift
operation introduced by Das that certain group cohomology classes belonging
(up to sign) to a basis previously constructed by Ouyang also satisfy the
universal recursion.
Categories:11R18, 11R23, 11R34 |
149. CJM 2003 (vol 55 pp. 432)
Pair Correlation of Squares in $p$-Adic Fields Let $p$ be an odd prime number, $K$ a $p$-adic field of degree $r$
over $\mathbf{Q}_p$, $O$ the ring of integers in $K$, $B = \{\beta_1,\dots,
\beta_r\}$ an integral basis of $K$ over $\mathbf{Q}_p$, $u$ a unit in $O$
and consider sets of the form $\mathcal{N}=\{n_1\beta_1+\cdots+n_r\beta_r:
1\leq n_j\leq N_j, 1\leq j\leq r\}$. We show under certain growth
conditions that the pair correlation of $\{uz^2:z\in\mathcal{N}\}$ becomes
Poissonian.
Categories:11S99, 11K06, 1134 |
150. CJM 2003 (vol 55 pp. 353)
Weak Explicit Matching for Level Zero Discrete Series of Unit Groups of $\mathfrak{p}$-Adic Simple Algebras |
Weak Explicit Matching for Level Zero Discrete Series of Unit Groups of $\mathfrak{p}$-Adic Simple Algebras Let $F$ be a $p$-adic local field and let $A_i^\times$ be the unit
group of a central simple $F$-algebra $A_i$ of reduced degree $n>1$
($i=1,2$). Let $\mathcal{R}^2 (A_i^\times)$ denote the set of
irreducible discrete series representations of $A_i^\times$. The
``Abstract Matching Theorem'' asserts the existence of a bijection,
the ``Jacquet-Langlands'' map, $\mathcal{J} \mathcal{L}_{A_2,A_1}
\colon \mathcal{R}^2 (A_1^\times) \to \mathcal{R}^2 (A_2^\times)$
which, up to known sign, preserves character values for regular
elliptic elements. This paper addresses the question of explicitly
describing the map $\mathcal{J} \mathcal{L}$, but only for ``level
zero'' representations. We prove that the restriction $\mathcal{J}
\mathcal{L}_{A_2,A_1} \colon \mathcal{R}_0^2 (A_1^\times) \to
\mathcal{R}_0^2 (A_2^\times)$ is a bijection of level zero discrete
series (Proposition~3.2) and we give a parameterization of the set of
unramified twist classes of level zero discrete series which does not
depend upon the algebra $A_i$ and is invariant under $\mathcal{J}
\mathcal{L}_{A_2,A_1}$ (Theorem~4.1).
Categories:22E50, 11R39 |