Expand all Collapse all | Results 76 - 100 of 222 |
76. CJM 2009 (vol 61 pp. 828)
Twisted Gross--Zagier Theorems The theorems of Gross--Zagier and Zhang relate the N\'eron--Tate
heights of complex multiplication points on the modular curve $X_0(N)$
(and on Shimura curve analogues) with the central derivatives of
automorphic $L$-function. We extend these results to include certain
CM points on modular curves of the form
$X(\Gamma_0(M)\cap\Gamma_1(S))$ (and on Shimura curve analogues).
These results are motivated by applications to Hida theory
that can be found in the companion article
"Central derivatives of $L$-functions in Hida families", Math.\ Ann.\
\textbf{399}(2007), 803--818.
Categories:11G18, 14G35 |
77. CJM 2009 (vol 61 pp. 674)
A Construction of Rigid Analytic Cohomology Classes for Congruence Subgroups of $\SL_3(\mathbb Z)$ We give a constructive proof, in the special case of ${\rm GL}_3$, of
a theorem of Ash and Stevens which compares overconvergent cohomology
to classical cohomology. Namely, we show that every ordinary
classical Hecke-eigenclass can be lifted uniquely to a rigid analytic
eigenclass. Our basic method builds on the ideas of M. Greenberg; we
first form an arbitrary lift of the classical eigenclass to a
distribution-valued cochain. Then, by appropriately iterating the
$U_p$-operator, we produce a cocycle whose image in cohomology is the
desired eigenclass. The constructive nature of this proof makes it
possible to perform computer computations to approximate these
interesting overconvergent eigenclasses.
Categories:11F75, 11F85 |
78. CJM 2009 (vol 61 pp. 617)
Square Integrable Representations and the Standard Module Conjecture for General Spin Groups In this paper we study square integrable representations and
$L$-functions for quasisplit general spin groups over a $p$-adic
field. In the first part, the holomorphy of $L$-functions in a half
plane is proved by using a variant form of Casselman's square
integrability criterion and the Langlands--Shahidi method. The
remaining part focuses on the proof of the standard module
conjecture. We generalize Mui\'c's idea via the Langlands--Shahidi method
towards a proof of the conjecture. It is used in the work of M. Asgari
and F. Shahidi on generic transfer for general spin groups.
Categories:11F70, 11F85 |
79. CJM 2009 (vol 61 pp. 583)
Algebraic Properties of a Family of Generalized Laguerre Polynomials We study the algebraic properties of Generalized Laguerre Polynomials
for negative integral values of the parameter. For integers $r,n\geq
0$, we conjecture that $L_n^{(-1-n-r)}(x) = \sum_{j=0}^n
\binom{n-j+r}{n-j}x^j/j!$ is a $\Q$-irreducible polynomial whose
Galois group contains the alternating group on $n$ letters. That this
is so for $r=n$ was conjectured in the 1950's by Grosswald and proven
recently by Filaseta and Trifonov. It follows from recent work of
Hajir and Wong that the conjecture is true when $r$ is large with
respect to $n\geq 5$. Here we verify it in three situations: i) when
$n$ is large with respect to $r$, ii) when $r \leq 8$, and iii) when
$n\leq 4$. The main tool is the theory of $p$-adic Newton Polygons.
Categories:11R09, 05E35 |
80. CJM 2009 (vol 61 pp. 481)
Uniform Distribution of Fractional Parts Related to Pseudoprimes We estimate exponential sums with the Fermat-like quotients
$$
f_g(n) = \frac{g^{n-1} - 1}{n} \quad\text{and}\quad h_g(n)=\frac{g^{n-1}-1}{P(n)},
$$
where $g$ and $n$ are positive integers, $n$ is composite, and
$P(n)$ is the largest prime factor of $n$. Clearly, both $f_g(n)$
and $h_g(n)$ are integers if $n$ is a Fermat pseudoprime to base
$g$, and if $n$ is a Carmichael number, this is true for all $g$
coprime to $n$. Nevertheless, our bounds imply that the fractional
parts $\{f_g(n)\}$ and $\{h_g(n)\}$ are uniformly distributed, on
average over~$g$ for $f_g(n)$, and individually for $h_g(n)$. We
also obtain similar results with the functions ${\widetilde f}_g(n)
= gf_g(n)$ and ${\widetilde h}_g(n) = gh_g(n)$.
Categories:11L07, 11N37, 11N60 |
81. CJM 2009 (vol 61 pp. 518)
Global Units Modulo Circular Units: Descent Without Iwasawa's Main Conjecture Iwasawa's classical asymptotical formula relates the orders of the $p$-parts $X_n$ of the ideal
class groups along a $\mathbb{Z}_p$-extension $F_\infty/F$ of a number
field $F$ to Iwasawa structural invariants $\la$ and $\mu$
attached to the inverse limit $X_\infty=\varprojlim X_n$.
It relies on ``good" descent properties satisfied by
$X_n$. If $F$ is abelian and $F_\infty$ is cyclotomic, it is known
that the $p$-parts of the orders of the global units modulo
circular units $U_n/C_n$ are asymptotically equivalent to the
$p$-parts of the ideal class numbers. This suggests that these
quotients $U_n/C_n$, so to speak unit class groups, also satisfy
good descent properties. We show this directly, \emph{i.e.,} without using Iwasawa's Main Conjecture.
Category:11R23 |
82. CJM 2009 (vol 61 pp. 373)
An Infinite Order Whittaker Function In this paper we construct a flat smooth section of an induced space
$I(s,\eta)$ of $SL_2(\mathbb{R})$ so that the attached Whittaker function
is not of finite order.
An asymptotic method of classical analysis is used.
Categories:11F70, 22E45, 41A60, 11M99, 30D15, 33C15 |
83. CJM 2009 (vol 61 pp. 465)
On Partitions into Powers of Primes and Their Difference Functions In this paper, we extend the approach first outlined by Hardy and
Ramanujan for calculating the asymptotic formulae for the number of
partitions into $r$-th powers of primes, $p_{\mathbb{P}^{(r)}}(n)$,
to include their difference functions. In doing so, we rectify an
oversight of said authors, namely that the first difference function
is perforce positive for all values of $n$, and include the
magnitude of the error term.
Categories:05A17, 11P81 |
84. CJM 2009 (vol 61 pp. 395)
$L$-Functions for $\GSp(2)\times \GL(2)$: Archimedean Theory and Applications Let $\Pi$ be a generic cuspidal automorphic representation of
$\GSp(2)$ defined over a totally real algebraic number field $\gfk$
whose archimedean type is either a (limit of) large discrete series
representation or a certain principal series representation. Through
explicit computation of archimedean local zeta integrals, we prove the
functional equation of tensor product $L$-functions $L(s,\Pi \times
\sigma)$ for an arbitrary cuspidal automorphic representation $\sigma$
of $\GL(2)$. We also give an application to the spinor $L$-function
of $\Pi$.
Categories:11F70, 11F41, 11F46 |
85. CJM 2009 (vol 61 pp. 336)
The Large Sieve Inequality for the Exponential Sequence $\lambda^{[O(n^{15/14+o(1)})]}$ Modulo Primes |
The Large Sieve Inequality for the Exponential Sequence $\lambda^{[O(n^{15/14+o(1)})]}$ Modulo Primes Let $\lambda$ be a fixed integer exceeding $1$ and $s_n$ any
strictly increasing sequence of positive integers satisfying $s_n\le
n^{15/14+o(1)}.$ In this paper we give a version of the large sieve
inequality for the sequence $\lambda^{s_n}.$ In particular, we
obtain nontrivial estimates of the associated trigonometric sums
``on average" and establish equidistribution properties of the
numbers $\lambda^{s_n} , n\le p(\log p)^{2+\varepsilon}$,
modulo $p$ for most primes $p.$
Keywords:Large sieve, exponential sums Categories:11L07, 11N36 |
86. CJM 2009 (vol 61 pp. 264)
On $\BbZ$-Modules of Algebraic Integers Let $q$ be an algebraic integer of degree $d \geq 2$.
Consider the rank of the multiplicative subgroup of $\BbC^*$ generated
by the conjugates of $q$.
We say $q$ is of {\em full rank} if either the rank is $d-1$ and $q$
has norm $\pm 1$, or the rank is $d$.
In this paper we study some properties of $\BbZ[q]$ where $q$ is an
algebraic integer of full rank.
The special cases of when $q$ is a Pisot number and when $q$ is a Pisot-cyclotomic number
are also studied.
There are four main results.
\begin{compactenum}[\rm(1)]
\item If $q$ is an algebraic integer of full rank and $n$ is a fixed positive
integer,
then there are only finitely many $m$ such that
$\disc\left(\BbZ[q^m]\right)=\disc\left(\BbZ[q^n]\right)$.
\item If $q$ and $r$ are algebraic integers of degree $d$ of full rank
and $\BbZ[q^n] = \BbZ[r^n]$ for
infinitely many $n$, then either $q = \omega r'$ or $q={\rm Norm}(r)^{2/d}\omega/r'$,
where
$r'$ is some conjugate of $r$ and $\omega$ is some root of unity.
\item Let $r$ be an algebraic integer of degree at most $3$.
Then there are at most $40$ Pisot numbers $q$ such that
$\BbZ[q] = \BbZ[r]$.
\item There are only finitely many Pisot-cyclotomic numbers of any fixed
order.
\end{compactenum}
Keywords:algebraic integers, Pisot numbers, full rank, discriminant Categories:11R04, 11R06 |
87. CJM 2009 (vol 61 pp. 165)
Exponents of Diophantine Approximation in Dimension Two Let $\Theta=(\alpha,\beta)$ be a point in $\bR^2$, with $1,\alpha,
\beta$ linearly independent over $\bQ$. We attach to $\Theta$ a
quadruple $\Omega(\Theta)$ of exponents that measure the quality
of approximation to $\Theta$ both by rational points and by
rational lines. The two ``uniform'' components of $\Omega(\Theta)$
are related by an equation due to Jarn\'\i k, and the four
exponents satisfy two inequalities that refine Khintchine's
transference principle. Conversely, we show that for any quadruple
$\Omega$ fulfilling these necessary conditions, there exists
a point $\Theta\in \bR^2$ for which $\Omega(\Theta) =\Omega$.
Categories:11J13, 11J70 |
88. CJM 2009 (vol 61 pp. 3)
Connected Components of Moduli Stacks of Torsors via Tamagawa Numbers Let $X$ be a smooth projective geometrically connected curve over
a finite field with function field $K$. Let $\G$ be a connected semisimple group
scheme over $X$. Under certain hypotheses we prove the equality of
two numbers associated with $\G$.
The first is an arithmetic invariant, its Tamagawa number. The second
is a geometric invariant, the number of connected components of the moduli
stack of $\G$-torsors on $X$. Our results are most useful for studying
connected components as much is known about Tamagawa numbers.
Categories:11E, 11R, 14D, 14H |
89. CJM 2009 (vol 61 pp. 141)
On the Littlewood Problem Modulo a Prime Let $p$ be a prime, and let $f \from \mathbb{Z}/p\mathbb{Z} \rightarrow
\mathbb{R}$ be a function with $\E f = 0$ and $\Vert \widehat{f}
\Vert_1 \leq 1$. Then
$\min_{x \in \Zp} |f(x)| = O(\log p)^{-1/3 + \epsilon}$.
One should think of $f$ as being ``approximately continuous''; our
result is then an ``approximate intermediate value theorem''.
As an immediate consequence we show that if $A \subseteq \Zp$ is a
set of cardinality $\lfloor p/2\rfloor$, then
$\sum_r |\widehat{1_A}(r)| \gg (\log p)^{1/3 - \epsilon}$. This
gives a result on a ``mod $p$'' analogue of Littlewood's well-known
problem concerning the smallest possible $L^1$-norm of the Fourier
transform of a set of $n$ integers.
Another application is to answer a question of Gowers. If $A
\subseteq \Zp$ is a set of size $\lfloor p/2 \rfloor$, then there is
some $x \in \Zp$ such that
\[ | |A \cap (A + x)| - p/4 | = o(p).\]
Categories:42A99, 11B99 |
90. CJM 2008 (vol 60 pp. 1306)
Theta Lifts of Tempered Representations for Dual Pairs $(\Sp_{2n}, O(V))$ This paper is the continuation of our previous work on the explicit
determination of the structure of theta lifts for dual pairs
$(\Sp_{2n}, O(V))$ over a non-archimedean field $F$ of characteristic
different than $2$, where $n$ is the split rank of $\Sp_{2n}$ and the
dimension of the space $V$ (over $F$) is even. We determine the
structure of theta lifts of tempered representations in terms of theta
lifts of representations in discrete series.
Categories:22E35, 22E50, 11F70 |
91. CJM 2008 (vol 60 pp. 1267)
Nonadjacent Radix-$\tau$ Expansions of Integers in Euclidean Imaginary Quadratic Number Fields In his seminal papers, Koblitz proposed curves
for cryptographic use. For fast operations on these curves,
these papers also
initiated a study of the radix-$\tau$ expansion of integers in the number
fields $\Q(\sqrt{-3})$ and $\Q(\sqrt{-7})$. The (window)
nonadjacent form of $\tau$-expansion of integers in
$\Q(\sqrt{-7})$ was first investigated by Solinas.
For integers in $\Q(\sqrt{-3})$, the nonadjacent form
and the window nonadjacent form of the $\tau$-expansion were
studied. These are used for efficient
point multiplications on Koblitz curves.
In this paper, we complete
the picture by producing the (window)
nonadjacent radix-$\tau$ expansions
for integers in all Euclidean imaginary quadratic number fields.
Keywords:algebraic integer, radix expression, window nonadjacent expansion, algorithm, point multiplication of elliptic curves, cryptography Categories:11A63, 11R04, 11Y16, 11Y40, 14G50 |
92. CJM 2008 (vol 60 pp. 1406)
Hauteur asymptotique des points de Heegner Geometric intuition suggests that the N\'{e}ron--Tate height of Heegner
points on a rational elliptic curve $E$ should be asymptotically
governed by the degree of its modular parametrisation. In this paper,
we show that this geometric intuition asymptotically holds on average
over a subset of discriminants. We also study the asymptotic behaviour
of traces of Heegner points on average over a subset of discriminants
and find a difference according to the rank of the elliptic curve. By
the Gross--Zagier formulae, such heights are related to the special
value at the critical point for either the derivative of the
Rankin--Selberg convolution of $E$ with a certain weight one theta
series attached to the principal ideal class of an imaginary quadratic
field or the twisted $L$-function of $E$ by a quadratic Dirichlet
character. Asymptotic formulae for the first moments associated with
these $L$-series and $L$-functions are proved, and experimental results
are discussed. The appendix contains some conjectural applications of
our results to the problem of the discretisation of odd quadratic
twists of elliptic curves.
Categories:11G50, 11M41 |
93. CJM 2008 (vol 60 pp. 1149)
Conjugate Reciprocal Polynomials with All Roots on the Unit Circle We study the geometry, topology and Lebesgue measure of the set of
monic conjugate reciprocal polynomials of fixed degree with all
roots on the unit circle. The set of such polynomials of degree $N$
is naturally associated to a subset of $\R^{N-1}$. We calculate
the volume of this set, prove the set is homeomorphic to the $N-1$
ball and that its isometry group is isomorphic to the dihedral
group of order $2N$.
Categories:11C08, 28A75, 15A52, 54H10, 58D19 |
94. CJM 2008 (vol 60 pp. 975)
An AF Algebra Associated with the Farey Tessellation We associate with the Farey tessellation of the upper
half-plane an
AF algebra $\AA$ encoding the ``cutting sequences'' that define
vertical geodesics.
The Effros--Shen AF algebras arise as quotients
of $\AA$. Using the path algebra model for AF algebras we construct, for
each $\tau \in \big(0,\frac{1}{4}\big]$, projections $(E_n)$ in
$\AA$ such that $E_n E_{n\pm 1}E_n \leq \tau E_n$.
Categories:46L05, 11A55, 11B57, 46L55, 37E05, 82B20 |
95. CJM 2008 (vol 60 pp. 1028)
Lifting $n$-Dimensional Galois Representations We investigate the problem of deforming $n$-dimensional mod $p$ Galois
representations to characteristic zero. The existence of 2-dimensional
deformations has been proven under certain conditions
by allowing ramification at additional primes in order to
annihilate a dual Selmer group. We use the same general methods to
prove the existence of $n$-dimensional deformations.
We then examine under which conditions we may place restrictions on
the shape of our deformations at $p$, with the goal of showing that
under the correct conditions, the deformations may have locally
geometric shape. We also use the existence of these deformations to
prove the existence as Galois groups over $\Q$ of certain infinite
subgroups of $p$-adic general linear groups.
Category:11F80 |
96. CJM 2008 (vol 60 pp. 790)
Types, paquets et changement de base : l'exemple de $U(2,1)(F_0)$. I. Types simples maximaux et paquets singletons |
Types, paquets et changement de base : l'exemple de $U(2,1)(F_0)$. I. Types simples maximaux et paquets singletons Soit $F_0$ un corps local non archim\'edien de caract\'eristique
nulle et de ca\-rac\-t\'eristique r\'esiduelle impaire.
J. Rogawski a montr\'e l'existence du changement de base entre le
groupe unitaire en trois variables $U(2,1)(F_{0})$, d\'efini
relativement \`a une extension quadratique $F$ de $F_{0}$, et le
groupe lin\'eaire $GL(3,F)$. Par ailleurs, nous
avons d\'ecrit les repr\'esentations supercuspidales irr\'eductibles
de $U(2,1)(F_{0})$ comme induites \`a partir d'un sous-groupe compact
ouvert de $U(2,1)(F_{0})$, description analogue \`a celle des
repr\'esentations admissibles irr\'eductibles de $GL(3,F)$ obtenue
par C. Bushnell et P. Kutzko. A partir de ces
descriptions, nous construisons explicitement le changement de base
des repr\'esentations tr\`es cuspidales de $U(2,1)(F_{0})$.
Categories:22E50, 11F70 |
97. CJM 2008 (vol 60 pp. 734)
Genus 2 Curves with Quaternionic Multiplication We explicitly construct the canonical rational models of Shimura
curves, both analytically in terms of modular forms and
algebraically in terms of coefficients of genus 2 curves, in the
cases of quaternion algebras of discriminant 6 and 10. This emulates
the classical construction in the elliptic curve case. We also give
families of genus 2 QM curves, whose Jacobians are the corresponding
abelian surfaces on the Shimura curve, and with coefficients that
are modular forms of weight 12. We apply these results to show
that our $j$-functions are supported exactly at those primes where
the genus 2 curve does not admit potentially good reduction, and
construct fields where this potentially good reduction is attained.
Finally, using $j$, we construct the fields of moduli and definition
for some moduli problems associated to the Atkin--Lehner group
actions.
Keywords:Shimura curve, canonical model, quaternionic multiplication, modular form, field of moduli Categories:11G18, 14G35 |
98. CJM 2008 (vol 60 pp. 532)
Local Bounds for Torsion Points on Abelian Varieties We say that an abelian variety over a $p$-adic field $K$ has
anisotropic reduction (AR) if the special fiber of its N\'eron minimal
model does not contain a nontrivial split torus. This includes all
abelian varieties with potentially good reduction and, in particular,
those with complex or quaternionic multiplication. We give a bound for
the size of the $K$-rational torsion subgroup of a $g$-dimensional AR
variety depending only on $g$ and the numerical invariants of $K$ (the
absolute ramification index and the cardinality of the residue
field). Applying these bounds to abelian varieties over a number field
with everywhere locally anisotropic reduction, we get bounds which, as
a function of $g$, are close to optimal. In particular, we determine
the possible cardinalities of the torsion subgroup of an AR abelian
surface over the rational numbers, up to a set of 11 values which are
not known to occur. The largest such value is 72.
Categories:11G10, 14K15 |
99. CJM 2008 (vol 60 pp. 481)
Heegner Points and the Rank of Elliptic Curves over Large Extensions of Global Fields Let $k$ be a global field, $\overline{k}$ a separable
closure of $k$, and $G_k$ the absolute Galois group
$\Gal(\overline{k}/k)$ of $\overline{k}$ over $k$. For every
$\sigma\in G_k$, let $\ks$ be the fixed subfield of $\overline{k}$
under $\sigma$. Let $E/k$ be an elliptic curve over $k$. It is known
that the Mordell--Weil group $E(\ks)$ has infinite rank. We present a
new proof of this fact in the following two cases. First, when $k$
is a global function field of odd characteristic and $E$ is
parametrized by a Drinfeld modular curve, and secondly when $k$ is a
totally real number field and $E/k$ is parametrized by a Shimura
curve. In both cases our approach uses the non-triviality of a
sequence of Heegner points on $E$ defined over ring class fields.
Category:11G05 |
100. CJM 2008 (vol 60 pp. 491)
A Multi-Frey Approach to Some Multi-Parameter Families of Diophantine Equations We solve several multi-parameter families of binomial Thue equations of arbitrary
degree; for example, we solve the equation
\[
5^u x^n-2^r 3^s y^n= \pm 1,
\]
in non-zero integers $x$, $y$ and positive integers $u$, $r$, $s$ and $n \geq 3$.
Our approach uses several Frey curves simultaneously, Galois representations
and level-lowering, new lower bounds for linear
forms in $3$ logarithms due to Mignotte and a famous theorem of Bennett on binomial
Thue equations.
Keywords:Diophantine equations, Frey curves, level-lowering, linear forms in logarithms, Thue equation Categories:11F80, 11D61, 11D59, 11J86, 11Y50 |