Expand all Collapse all | Results 51 - 75 of 224 |
51. CJM 2011 (vol 63 pp. 481)
The Ample Cone for a K3 Surface
In this paper, we give several pictorial fractal
representations of the ample or KÃ¤hler cone for surfaces in a
certain class of $K3$ surfaces. The class includes surfaces
described by smooth $(2,2,2)$ forms in ${\mathbb P^1\times\mathbb P^1\times \mathbb P^1}$ defined over a
sufficiently large number field $K$ that have a line parallel to
one of the axes and have Picard number four. We relate the
Hausdorff dimension of this fractal to the asymptotic growth of
orbits of curves under the action of the surface's group of
automorphisms. We experimentally estimate the Hausdorff dimension
of the fractal to be $1.296 \pm .010$.
Keywords:Fractal, Hausdorff dimension, K3 surface, Kleinian groups, dynamics Categories:14J28, , , , 14J50, 11D41, 11D72, 11H56, 11G10, 37F35, 37D05 |
52. CJM 2011 (vol 63 pp. 634)
On Higher Moments of Fourier Coefficients of Holomorphic Cusp Forms Let $S_{k}(\Gamma)$ be the space of holomorphic cusp forms of even
integral weight $k$ for the full modular group.
Let $\lambda_f(n)$ and $\lambda_g(n)$ be the $n$-th normalized Fourier coefficients of
two holomorphic Hecke eigencuspforms $f(z), g(z) \in S_{k}(\Gamma)$, respectively.
In this paper we are able to show the following results about higher
moments of Fourier coefficients of holomorphic cusp forms.\newline
(i) For any $\varepsilon>0$, we have
\begin{equation*}
\sum_{n\leq x}\lambda_f^5(n) \ll_{f,\varepsilon}x^{\frac{15}{16}+\varepsilon}
\quad\text{and}\quad\sum_{n\leq x}\lambda_f^7(n) \ll_{f,\varepsilon}x^{\frac{63}{64}+\varepsilon}.
\end{equation*}
(ii) If $\operatorname{sym}^3\pi_f \ncong \operatorname{sym}^3\pi_g$, then for any $\varepsilon>0$, we have
\begin{equation*}
\sum_{n \leq x}\lambda_f^3(n)\lambda_g^3(n)\ll_{f,\varepsilon}x^{\frac{31}{32}+\varepsilon};
\end{equation*}
If $\operatorname{sym}^2\pi_f \ncong \operatorname{sym}^2\pi_g$, then for any $\varepsilon>0$, we have
\[
\sum_{n \leq x}\lambda_f^4(n)\lambda_g^2(n)=cx\log x+c'x+O_{f,\varepsilon}\bigl(x^{\frac{31}{32}+\varepsilon}\bigr);
\]
If $\operatorname{sym}^2\pi_f \ncong \operatorname{sym}^2\pi_g$ and $\operatorname{sym}^4\pi_f \ncong \operatorname{sym}^4\pi_g$, then for any $\varepsilon>0$, we have
\[
\sum_{n \leq x}\lambda_f^4(n)\lambda_g^4(n)=xP(\log x)+O_{f,\varepsilon}\bigl(x^{\frac{127}{128}+\varepsilon}\bigr),
\]
where $P(x)$ is a polynomial of degree $3$.
Keywords: Fourier coefficients of cusp forms, symmetric power $L$-function Categories:11F30, , , , 11F11, 11F66 |
53. CJM 2011 (vol 63 pp. 298)
A Variant of Lehmer's Conjecture, II: The CM-case
Let $f$ be a normalized Hecke eigenform with rational integer Fourier
coefficients. It is an interesting question to know how often an
integer $n$ has a factor common with the $n$-th Fourier coefficient of
$f$. It has been shown in previous papers that this happens very often. In this
paper, we give an asymptotic formula for the number of integers $n$
for which $(n, a(n)) = 1$, where $a(n)$ is the $n$-th Fourier coefficient of
a normalized Hecke eigenform $f$ of weight $2$ with rational integer
Fourier coefficients and having complex multiplication.
Categories:11F11, 11F30 |
54. CJM 2010 (vol 63 pp. 241)
Multiple Zeta-Functions Associated with Linear Recurrence Sequences and the Vectorial Sum Formula
We prove the holomorphic continuation of certain multi-variable multiple
zeta-functions whose coefficients satisfy a suitable recurrence condition.
In fact, we introduce more general vectorial zeta-functions and prove their
holomorphic continuation. Moreover, we show a vectorial sum formula among
those vectorial zeta-functions from which some generalizations of the
classical sum formula can be deduced.
Keywords:Zeta-functions, holomorphic continuation, recurrence sequences, Fibonacci numbers, sum formulas Categories:11M41, 40B05, 11B39 |
55. CJM 2010 (vol 63 pp. 277)
Locally Indecomposable Galois Representations
In a previous paper
the authors showed that, under some technical
conditions,
the local Galois representations attached to the members of
a non-CM family of ordinary cusp forms are indecomposable for all
except possibly finitely many
members of the family. In this paper we use deformation theoretic
methods to give examples of non-CM families for
which every classical member of weight at least two has a locally
indecomposable Galois representation.
Category:11F80 |
56. CJM 2010 (vol 63 pp. 136)
Transcendental Nature of Special Values of $L$-Functions
In this paper, we study the non-vanishing and transcendence of
special values of a varying class of $L$-functions and their derivatives.
This allows us to investigate the transcendence of Petersson norms
of certain weight one modular forms.
Categories:11J81, 11J86, 11J91 |
57. CJM 2010 (vol 63 pp. 38)
Asymptotic Formulae for Pairs of Diagonal Cubic Equations
We investigate the number of integral solutions possessed by a
pair of diagonal cubic equations in a large box. Provided that the number of
variables in the system is at least fourteen, and in addition the number of
variables in any non-trivial linear combination of the underlying forms is at
least eight, we obtain an asymptotic formula for the number of integral
solutions consistent with the product of local densities associated with the
system.
Keywords:exponential sums, Diophantine equations Categories:11D72, 11P55 |
58. CJM 2010 (vol 62 pp. 1276)
A Generalized Poisson Transform of an $L^{p}$-Function over the Shilov Boundary of the $n$-Dimensional Lie Ball |
A Generalized Poisson Transform of an $L^{p}$-Function over the Shilov Boundary of the $n$-Dimensional Lie Ball
Let $\mathcal{D}$ be the $n$-dimensional Lie ball and let
$\mathbf{B}(S)$ be the space of hyperfunctions on the Shilov
boundary $S$ of $\mathcal{D}$.
The aim of this paper is to give a
necessary and sufficient condition on the generalized Poisson
transform $P_{l,\lambda}f$ of an element $f$ in the space
$\mathbf{B}(S)$ for $f$ to be in $ L^{p}(S)$, $1 > p > \infty.$
Namely, if $F$ is the Poisson transform of some $f\in
\mathbf{B}(S)$ (i.e., $F=P_{l,\lambda}f$), then for any
$l\in \mathbb{Z}$ and $\lambda\in \mathbb{C}$ such that
$\mathcal{R}e[i\lambda] > \frac{n}{2}-1$, we show that $f\in L^{p}(S)$ if and
only if $f$ satisfies the growth condition
$$
\|F\|_{\lambda,p}=\sup_{0\leq r
< 1}(1-r^{2})^{\mathcal{R}e[i\lambda]-\frac{n}{2}+l}\Big[\int_{S}|F(ru)|^{p}du
\Big]^{\frac{1}{p}} < +\infty.
$$
Keywords:Lie ball, Shilov boundary, Fatou's theorem, hyperfuctions, parabolic subgroup, homogeneous line bundle Categories:32A45, 30E20, 33C67, 33C60, 33C55, 32A25, 33C75, 11K70 |
59. CJM 2010 (vol 62 pp. 1011)
Functoriality of the Canonical Fractional Galois Ideal
The fractional Galois ideal
is a conjectural improvement on the higher Stickelberger
ideals defined at negative integers, and is expected to provide
non-trivial annihilators for higher $K$-groups of rings of integers of
number fields. In this article, we extend the definition of the
fractional Galois ideal to arbitrary (possibly infinite and
non-abelian) Galois extensions of number fields under the assumption
of Stark's conjectures and prove naturality properties under
canonical changes of extension. We discuss applications of this to the
construction of ideals in non-commutative Iwasawa algebras.
Categories:11R42, 11R23, 11R70 |
60. CJM 2010 (vol 62 pp. 1155)
Moments of the Critical Values of Families of Elliptic Curves, with Applications
We make conjectures on the moments of the central values of the family
of all elliptic curves and on the moments of the first derivative of
the central values of a large family of positive rank curves. In both
cases the order of magnitude is the same as that of the moments of the
central values of an orthogonal family of $L$-functions. Notably, we
predict that the critical values of all rank $1$ elliptic curves is
logarithmically larger than the rank $1$ curves in the positive rank
family.
Furthermore, as arithmetical applications, we make a conjecture on the
distribution of $a_p$'s amongst all rank $2$ elliptic curves and
show how the Riemann hypothesis can be deduced from sufficient
knowledge of the first moment of the positive rank family (based on an
idea of Iwaniec)
Categories:11M41, 11G40, 11M26 |
61. CJM 2010 (vol 62 pp. 1099)
Character Sums to Smooth Moduli are Small
Recently, Granville and Soundararajan have made
fundamental breakthroughs in the study of character sums. Building
on their work and using estimates on short character sums developed
by Graham--Ringrose and Iwaniec, we improve the
PÃ³lya--Vinogradov inequality for characters with smooth conductor.
Categories:11L40, 11M06 |
62. CJM 2010 (vol 62 pp. 1060)
Heegner Points over Towers of Kummer Extensions
Let $E$ be an elliptic curve, and let $L_n$ be the Kummer extension
generated by a primitive $p^n$-th root of unity and a $p^n$-th root of
$a$ for a fixed $a\in \mathbb{Q}^\times-\{\pm 1\}$. A detailed case study
by Coates, Fukaya, Kato and Sujatha and V. Dokchitser has led these
authors to predict unbounded and strikingly regular growth for the
rank of $E$ over $L_n$ in certain cases. The aim of this note is to
explain how some of these predictions might be accounted for by
Heegner points arising from a varying collection of Shimura curve
parametrisations.
Categories:11G05, 11R23, 11F46 |
63. CJM 2010 (vol 62 pp. 914)
Reducibility of the Principal Series for Sp^{~}_{2}(F) over a p-adic Field Let $G_n=\mathrm{Sp}_n(F)$ be the rank $n$ symplectic group with
entries in a nondyadic $p$-adic field $F$. We further let $\widetilde{G}_n$ be
the metaplectic extension of $G_n$ by $\mathbb{C}^{1}=\{z\in\mathbb{C}^{\times}
\mid |z|=1\}$ defined using the Leray cocycle. In this paper, we aim to
demonstrate the complete list of reducibility points of the genuine
principal series of ${\widetilde{G}_2}$. In most cases, we will use
some techniques developed by TadiÄ that analyze the Jacquet
modules with respect to all of the parabolics containing a fixed
Borel. The exceptional cases involve representations induced from
unitary characters $\chi$ with $\chi^2=1$. Because such
representations $\pi$ are unitary, to show the irreducibility of
$\pi$, it suffices to show that
$\dim_{\mathbb{C}}\mathrm{Hom}_{{\widetilde{G}}}(\pi,\pi)=1$. We will accomplish this
by examining the poles of certain intertwining operators associated to
simple roots. Then some results of Shahidi and Ban decompose arbitrary
intertwining operators into a composition of operators corresponding
to the simple roots of ${\widetilde{G}_2}$. We will then be able to
show that all such operators have poles at principal series
representations induced from quadratic characters and therefore such
operators do not extend to operators in
$\mathrm{Hom}_{{\widetilde{G}_2}}(\pi,\pi)$ for the $\pi$ in question.
Categories:22E50, 11F70 |
64. CJM 2010 (vol 62 pp. 543)
More Variations on the SierpiÅski Sieve This paper answers a question of Broomhead, Montaldi and Sidorov about the existence of gaskets of a particular type related to the SierpiÅski sieve. These gaskets are given by iterated function systems that do not satisfy the open set condition. We use the methods of Ngai and Wang to compute the dimension of these gaskets.
Categories:28A80, 28A78, 11R06 |
65. CJM 2010 (vol 62 pp. 787)
An Explicit Treatment of Cubic Function Fields with Applications We give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for non-singularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description of the signature of any rational place in a cubic extension that involves only the defining curve and the order of the base field. All these quantities only require simple polynomial arithmetic as well as a few square-free polynomial factorizations and, in some cases, square and cube root extraction modulo an irreducible polynomial. We also illustrate why and how signature computation plays an important role in computing the class number of the function field. This in turn has applications to the study of zeros of zeta functions of function fields.
Keywords:cubic function field, discriminant, non-singularity, integral basis, genus, signature of a place, class number Categories:14H05, 11R58, 14H45, 11G20, 11G30, 11R16, 11R29 |
66. CJM 2010 (vol 62 pp. 668)
The Supersingular Locus of the Shimura Variety for GU(1,s) In this paper we study the supersingular locus of the reduction modulo $p$ of the Shimura variety for $GU(1,s)$ in the case of an inert prime $p$. Using DieudonnÃ© theory we define a stratification of the corresponding moduli space of $p$-divisible groups. We describe the incidence relation of this stratification in terms of the Bruhat--Tits building of a unitary group. In the case of $GU(1,2)$, we show that the supersingular locus is equidimensional of dimension 1 and is of complete intersection. We give an explicit description of the irreducible components and their intersection behaviour.
Categories:14G35, 11G18, 14K10 |
67. CJM 2010 (vol 62 pp. 563)
Whittaker Functions on Real Semisimple Lie Groups of Rank Two We give explicit formulas for Whittaker functions on real semisimple Lie groups of real rank two belonging to the class one principal series representations. By using these formulas we compute certain archimedean zeta integrals.
Categories:11F70, 22E30 |
68. CJM 2009 (vol 62 pp. 582)
On the Distribution of Pseudopowers An x-pseudopower to base g is a positive integer that is not a power of g, yet is so modulo p for all primes $ple x$. We improve an upper bound for the least such number, due to E.~Bach, R.~Lukes, J.~Shallit, and H.~C.~Williams. The method is based on a combination of some bounds of exponential sums with new results about the average behaviour of the multiplicative order of g modulo prime numbers.
Categories:11A07, 11L07, 11N36 |
69. CJM 2009 (vol 62 pp. 400)
On p-Adic Properties of Central L-Values of Quadratic Twists of an Elliptic Curve We study $p$-indivisibility of the central values $L(1,E_d)$ of
quadratic twists $E_d$ of a semi-stable elliptic curve $E$ of
conductor $N$. A consideration of the conjecture of Birch and
Swinnerton-Dyer shows that the set of quadratic discriminants $d$
splits naturally into several families $\mathcal{F}_S$, indexed by subsets $S$
of the primes dividing $N$. Let $\delta_S= \gcd_{d\in \mathcal{F}_S}
L(1,E_d)^{\operatorname{alg}}$, where $L(1,E_d)^{\operatorname{alg}}$ denotes the algebraic part
of the central $L$-value, $L(1,E_d)$. Our main theorem relates the
$p$-adic valuations of $\delta_S$ as $S$ varies. As a consequence we
present an application to a refined version of a question of
Kolyvagin. Finally we explain an intriguing (albeit speculative)
relation between Waldspurger packets on $\widetilde{\operatorname{SL}_2}$ and
congruences of modular forms of integral and half-integral weight. In
this context, we formulate a conjecture on congruences of
half-integral weight forms and explain its relevance to the problem of
$p$-indivisibility of $L$-values of quadratic twists.
Categories:11F40, 11F67, 11G05 |
70. CJM 2009 (vol 62 pp. 157)
Special Values of Class Group $L$-Functions for CM Fields Let $H$ be the Hilbert class field of a CM number field $K$ with
maximal totally real subfield $F$ of degree $n$ over $\mathbb{Q}$. We
evaluate the second term in the Taylor expansion at $s=0$ of the
Galois-equivariant $L$-function $\Theta_{S_{\infty}}(s)$ associated to
the unramified abelian characters of $\operatorname{Gal}(H/K)$. This is an identity
in the group ring $\mathbb{C}[\operatorname{Gal}(H/K)]$ expressing
$\Theta^{(n)}_{S_{\infty}}(0)$ as essentially a linear combination of
logarithms of special values $\{\Psi(z_{\sigma})\}$, where $\Psi\colon
\mathbb{H}^{n} \rightarrow \mathbb{R}$ is a Hilbert modular function for a congruence
subgroup of $SL_{2}(\mathcal{O}_{F})$ and $\{z_{\sigma}: \sigma \in
\operatorname{Gal}(H/K)\}$ are CM points on a universal Hilbert modular variety. We
apply this result to express the relative class number $h_{H}/h_{K}$
as a rational multiple of the determinant of an $(h_{K}-1) \times
(h_{K}-1)$ matrix of logarithms of ratios of special values
$\Psi(z_{\sigma})$, thus giving rise to candidates for higher analogs
of elliptic units. Finally, we obtain a product formula for
$\Psi(z_{\sigma})$ in terms of exponentials of special values of
$L$-functions.
Keywords:Artin $L$-function, CM point, Hilbert modular function, Rubin-Stark conjecture Categories:11R42, 11F30 |
71. CJM 2009 (vol 62 pp. 456)
The ChowlaâSelberg Formula and The Colmez Conjecture In this paper, we reinterpret the Colmez conjecture on the Faltings height of CM abelian varieties in terms of Hilbert (and Siegel) modular forms. We construct an elliptic modular form involving the Faltings height of a CM abelian surface and arithmetic intersection numbers, and prove that the Colmez conjecture for CM abelian surfaces is equivalent to the cuspidality of this modular form.
Categories:11G15, 11F41, 14K22 |
72. CJM 2009 (vol 61 pp. 1383)
Integral Representation for $U_{3} \times \GL_{2}$ Gelbart and Piatetskii-Shapiro constructed
various integral
representations of Rankin--Sel\-berg type for groups $G \times
\GL_{n}$,
where $G$
is of split rank $n$. Here we show that their method
can equally well be applied
to the product $U_{3} \times \GL_{2}$, where $U_{3}$
denotes the quasisplit
unitary group in three variables. As an application, we describe which
cuspidal automorphic representations of $U_{3}$ occur
in the Siegel induced
residual spectrum of the quasisplit $U_{4}$.
Categories:11F70, 11F67 |
73. CJM 2009 (vol 61 pp. 1214)
Close Lattice Points on Circles We classify the sets of four lattice points that all lie on a
short arc of a circle that has its center at the origin;
specifically on arcs of length $tR^{1/3}$ on a circle of radius
$R$, for any given $t>0$. In particular we prove that any arc of
length $ (40 + \frac{40}3\sqrt{10} )^{1/3}R^{1/3}$ on a circle of
radius $R$, with $R>\sqrt{65}$, contains at most three lattice
points, whereas we give an explicit infinite family of $4$-tuples
of lattice points, $(\nu_{1,n},\nu_{2,n},\nu_{3,n},\nu_{4,n})$,
each of which lies on an arc of length $ (40 +
\frac{40}3\sqrt{10})^{\smash{1/3}}R_n^{\smash{1/3}}+o(1)$ on a circle of
radius $R_n$.
Category:11N36 |
74. CJM 2009 (vol 61 pp. 1341)
Simultaneous Polynomial Approximations of the Lerch Function We construct bivariate polynomial approximations of the Lerch
function that for certain specialisations of the variables and
parameters turn out to be Hermite--Pad\'e approximants either of
the polylogarithms or of Hurwitz zeta functions. In the former
case, we recover known results, while in the latter the results
are new and generalise some recent works of Beukers and Pr\'evost.
Finally, we make a detailed comparison of our work with Beukers'.
Such constructions are useful in the arithmetical study of the
values of the Riemann zeta function at integer points and of the
Kubota--Leopold $p$-adic zeta function.
Categories:41A10, 41A21, 11J72 |
75. CJM 2009 (vol 61 pp. 1118)
Petits points d'une surface Pour toute sous-vari\'et\'e g\'eom\'etriquement irr\'eductible $V$
du grou\-pe multiplicatif
$\mathbb{G}_m^n$, on sait qu'en dehors d'un nombre fini de
translat\'es de tores exceptionnels
inclus dans $V$, tous les points sont de hauteur minor\'ee par une
certaine quantit\'e $q(V)^{-1}>0$. On conna\^it de plus une borne
sup\'erieure pour la somme des degr\'es de ces translat\'es de
tores pour des valeurs de $q(V)$ polynomiales en le degr\'e de $V$.
Ceci n'est pas le cas si l'on exige une minoration quasi-optimale
pour la hauteur des points de $V$, essentiellement lin\'eaire en l'inverse du degr\'e.
Nous apportons ici une r\'eponse partielle \`a ce probl\`eme\,: nous
donnons une majoration de la somme des degr\'es de ces translat\'es de
sous-tores de codimension $1$ d'une hypersurface $V$. Les r\'esultats,
obtenus dans le cas de $\mathbb{G}_m^3$, mais compl\`etement
explicites, peuvent toutefois s'\'etendre \`a $\mathbb{G}_m^n$,
moyennant quelques petites complications inh\'erentes \`a la dimension
$n$.
Keywords:Hauteur normalisÃ©e, groupe multiplicatif, problÃ¨me de Lehmer, petits points Categories:11G50, 11J81, 14G40 |