Expand all Collapse all | Results 126 - 150 of 224 |
126. CJM 2005 (vol 57 pp. 1155)
The Square Sieve and the Lang--Trotter Conjecture Let $E$ be an elliptic curve defined over $\Q$ and without
complex multiplication. Let $K$ be a fixed imaginary quadratic field.
We find nontrivial upper bounds for the number of ordinary primes $p \leq x$
for which $\Q(\pi_p) = K$, where $\pi_p$ denotes the Frobenius endomorphism
of $E$ at $p$. More precisely, under a generalized Riemann hypothesis
we show that this number is $O_{E}(x^{\slfrac{17}{18}}\log x)$, and unconditionally
we show that this number is $O_{E, K}\bigl(\frac{x(\log \log x)^{\slfrac{13}{12}}}
{(\log x)^{\slfrac{25}{24}}}\bigr)$. We also prove that the number of imaginary quadratic
fields $K$, with $-\disc K \leq x$ and of the form $K = \Q(\pi_p)$, is
$\gg_E\log\log\log x$ for $x\geq x_0(E)$. These results represent progress towards
a 1976 Lang--Trotter conjecture.
Keywords:Elliptic curves modulo $p$; Lang--Trotter conjecture;, applications of sieve methods Categories:11G05, 11N36, 11R45 |
127. CJM 2005 (vol 57 pp. 1215)
Reciprocity Law for Compatible Systems of Abelian $\bmod p$ Galois Representations The main result of the paper
is a {\em reciprocity law} which proves that
compatible systems of semisimple, abelian mod $p$ representations
(of arbitrary dimension)
of absolute Galois groups of number fields, arise from Hecke characters.
In the last section analogs for Galois groups of function fields of these
results are explored, and a question is raised whose answer seems to
require developments in transcendence theory in characteristic $p$.
Category:11F80 |
128. CJM 2005 (vol 57 pp. 1102)
Power Residues of Fourier Coefficients of Modular Forms Let $\rho \colon G_{\Q} \to \GL_{n}(\Ql)$ be a motivic $\ell$-adic Galois
representation. For fixed $m > 1$ we initiate an investigation of the
density of the set of primes $p$ such that the trace of the image of an
arithmetic Frobenius at $p$ under $\rho$ is an $m$-th power residue
modulo $p$. Based on numerical investigations with modular forms we
conjecture (with Ramakrishna) that this density equals $1/m$ whenever the
image of $\rho$ is open. We further conjecture that for such $\rho$ the set
of these primes $p$ is independent of any set defined by Cebatorev-style
Galois-theoretic conditions (in an appropriate sense). We then compute these
densities for certain $m$ in the complementary case of modular forms of
CM-type with rational Fourier coefficients; our proofs are a combination of
the Cebatorev density theorem (which does apply in the CM case) and
reciprocity laws applied to Hecke characters. We also discuss a potential
application (suggested by Ramakrishna) to computing inertial degrees at $p$
in abelian extensions of imaginary quadratic fields unramified away from $p$.
Categories:11F30, 11G15, 11A15 |
129. CJM 2005 (vol 57 pp. 1080)
The Gelfond--Schnirelman Method in Prime Number Theory The original Gelfond--Schnirelman method, proposed in 1936, uses
polynomials with integer coefficients and small norms on $[0,1]$
to give a Chebyshev-type lower bound in prime number theory. We
study a generalization of this method for polynomials in many
variables. Our main result is a lower bound for the integral of
Chebyshev's $\psi$-function, expressed in terms of the weighted
capacity. This extends previous work of Nair and Chudnovsky, and
connects the subject to the potential theory with external fields
generated by polynomial-type weights. We also solve the
corresponding potential theoretic problem, by finding the extremal
measure and its support.
Keywords:distribution of prime numbers, polynomials, integer, coefficients, weighted transfinite diameter, weighted capacity, potentials Categories:11N05, 31A15, 11C08 |
130. CJM 2005 (vol 57 pp. 812)
On the Vanishing of $\mu$-Invariants of Elliptic Curves over $\qq$ Let $E_{/\qq}$ be an elliptic curve with good ordinary reduction at a
prime $p>2$. It has a well-defined Iwasawa $\mu$-invariant $\mu(E)_p$
which encodes part of the information about the growth of the Selmer
group $\sel E{K_n}$ as $K_n$ ranges over the subfields of the
cyclotomic $\zzp$-extension $K_\infty/\qq$. Ralph Greenberg has
conjectured that any such $E$ is isogenous to a curve $E'$ with
$\mu(E')_p=0$. In this paper we prove Greenberg's conjecture for
infinitely many curves $E$ with a rational $p$-torsion point, $p=3$ or
$5$, no two of our examples having isomorphic $p$-torsion. The core
of our strategy is a partial explicit evaluation of the global duality
pairing for finite flat group schemes over rings of integers.
Category:11R23 |
131. 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 |
132. CJM 2005 (vol 57 pp. 535)
On Local $L$-Functions and Normalized Intertwining Operators In this paper we make explicit all $L$-functions in the
Langlands--Shahidi method which appear as normalizing factors of
global intertwining operators in the constant term of the
Eisenstein series. We prove, in many cases,
the conjecture of Shahidi regarding the
holomorphy of the local $L$-functions. We also prove
that the normalized local intertwining operators are holomorphic and
non-vaninishing for $\re(s)\geq 1/2$ in many cases. These local
results are essential in global applications such as Langlands
functoriality, residual spectrum and determining poles of
automorphic $L$-functions.
Categories:11F70, 22E55 |
133. 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 |
134. CJM 2005 (vol 57 pp. 494)
Summation Formulae for Coefficients of $L$-functions With applications in mind we establish a summation formula for the
coefficients of a general Dirichlet series satisfying a suitable
functional equation. Among a number of consequences we derive a
generalization of an elegant divisor sum bound due to F.~V. Atkinson.
Categories:11M06, 11M41 |
135. 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 |
136. 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 |
137. 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 |
138. 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 |
139. 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 |
140. 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 |
141. 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 |
142. 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 |
143. 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 |
144. 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 |
145. 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 |
146. 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 |
147. 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 |
148. 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 |
149. 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 |
150. 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:
$$
the~ring~of~integers~of~K~is~a~Euclidean~domain~if~and~only~if~it~is~a~principal~ideal~domain.
$$
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 |