Expand all Collapse all | Results 26 - 50 of 217 |
26. CJM 2011 (vol 64 pp. 1248)
Darmon's Points and Quaternionic Shimura Varieties In this paper, we generalize a conjecture due to Darmon and Logan in
an adelic setting. We study the relation between our construction and
Kudla's works on cycles on orthogonal Shimura varieties. This relation
allows us to conjecture a Gross-Kohnen-Zagier theorem for Darmon's
points.
Keywords:elliptic curves, Stark-Heegner points, quaternionic Shimura varieties Categories:11G05, 14G35, 11F67, 11G40 |
27. CJM 2011 (vol 64 pp. 588)
Level Raising and Anticyclotomic Selmer Groups for Hilbert Modular Forms of Weight Two In this article we refine the method of Bertolini and Darmon
and prove several finiteness results for
anticyclotomic Selmer groups of Hilbert modular forms of parallel
weight two.
Keywords:Hilbert modular forms, Selmer groups, Shimura curves Categories:11G40, 11F41, 11G18 |
28. CJM 2011 (vol 64 pp. 935)
The H and K Families of Mock Theta Functions In his last letter to Hardy, Ramanujan
defined 17 functions $F(q)$, $|q|\lt 1$, which he called mock $\theta$-functions.
He observed that as $q$ radially approaches any root of unity $\zeta$ at which
$F(q)$ has an exponential singularity, there is a $\theta$-function
$T_\zeta(q)$ with $F(q)-T_\zeta(q)=O(1)$. Since then, other functions have
been found that possess this property. These functions are related to
a function $H(x,q)$, where $x$ is usually $q^r$ or $e^{2\pi i r}$ for some
rational number $r$. For this reason we refer to $H$ as a ``universal'' mock
$\theta$-function. Modular transformations of $H$ give rise to the functions
$K$, $K_1$, $K_2$. The functions $K$ and $K_1$ appear in Ramanujan's lost
notebook. We prove various linear relations between these functions using
Appell-Lerch sums (also called generalized Lambert series). Some relations
(mock theta ``conjectures'') involving mock $\theta$-functions
of even order and $H$ are listed.
Keywords:mock theta function, $q$-series, Appell-Lerch sum, generalized Lambert series Categories:11B65, 33D15 |
29. CJM 2011 (vol 64 pp. 1122)
$p$-adic $L$-functions and the Rationality of Darmon Cycles Darmon cycles are a higher weight analogue of Stark--Heegner points. They
yield local cohomology classes in the Deligne representation associated with a
cuspidal form on $\Gamma _{0}( N) $ of even weight $k_{0}\geq 2$.
They are conjectured to be the restriction of global cohomology classes in
the Bloch--Kato Selmer group defined over narrow ring class fields attached
to a real quadratic field. We show that suitable linear combinations of them
obtained by genus characters satisfy these conjectures. We also prove $p$-adic Gross--Zagier type formulas, relating the derivatives of $p$-adic $L$-functions of the weight variable attached to imaginary (resp. real)
quadratic fields to Heegner cycles (resp. Darmon cycles). Finally we express
the second derivative of the Mazur--Kitagawa $p$-adic $L$-function of the
weight variable in terms of a global cycle defined over a quadratic
extension of $\mathbb{Q}$.
Categories:11F67, 14G05 |
30. CJM 2011 (vol 64 pp. 282)
Level Lowering Modulo Prime Powers and Twisted Fermat Equations We discuss a clean level lowering theorem modulo prime powers
for weight $2$ cusp forms.
Furthermore, we illustrate how this can be used to completely
solve certain twisted Fermat equations
$ax^n+by^n+cz^n=0$.
Keywords:modular forms, level lowering, Diophantine equations Categories:11D41, 11F33, 11F11, 11F80, 11G05 |
31. CJM 2011 (vol 64 pp. 345)
Salem Numbers and Pisot Numbers via Interlacing We present a general construction of Salem numbers via rational
functions whose zeros and poles mostly lie on the unit circle and
satisfy an interlacing condition. This extends and unifies earlier
work. We then consider the ``obvious'' limit points of the set of Salem
numbers produced by our theorems and show that these are all Pisot
numbers, in support of a conjecture of Boyd. We then show that all
Pisot numbers arise in this way. Combining this with a theorem of
Boyd, we produce all Salem numbers via an interlacing construction.
Keywords:Salem numbers, Pisot numbers Category:11R06 |
32. CJM 2011 (vol 64 pp. 301)
Hermite's Constant for Function Fields We formulate an analog of Hermite's constant for function fields over a finite field and
state a conjectural value for this analog. We prove our conjecture in many cases, and
prove slightly weaker results in all other cases.
Category:11G50 |
33. CJM 2011 (vol 64 pp. 81)
Pseudoprime Reductions of Elliptic Curves
Let $E$ be an elliptic curve over $\mathbb Q$ without complex multiplication,
and for each prime
$p$ of good reduction, let $n_E(p) = | E(\mathbb F_p) |$. For any integer
$b$, we consider elliptic pseudoprimes to the base
$b$. More precisely, let $Q_{E,b}(x)$ be the number of primes $p \leq
x$ such that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$, and let $\pi_{E,
b}^{\operatorname{pseu}}(x)$ be the number of compositive $n_E(p)$ such
that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$ (also called
elliptic curve pseudoprimes). Motivated by cryptography applications,
we address the problem of finding upper bounds for
$Q_{E,b}(x)$ and $\pi_{E, b}^{\operatorname{pseu}}(x)$,
generalising some of the literature for the classical pseudoprimes
to this new setting.
Keywords:Rosser-Iwaniec sieve, group order of elliptic curves over finite fields, pseudoprimes Categories:11N36, 14H52 |
34. CJM 2011 (vol 64 pp. 151)
Moments of the Rank of Elliptic Curves Fix an elliptic curve $E/\mathbb{Q}$ and assume the Riemann Hypothesis
for the $L$-function $L(E_D, s)$ for every quadratic twist $E_D$ of
$E$ by $D\in\mathbb{Z}$. We combine Weil's
explicit formula with techniques of Heath-Brown to derive an asymptotic
upper bound for the weighted moments of the analytic rank of $E_D$. We
derive from this an upper bound for the density of low-lying zeros of
$L(E_D, s)$ that is compatible with the random matrix models of Katz and
Sarnak. We also show that for any unbounded increasing function $f$ on $\mathbb{R}$,
the analytic rank and (assuming in addition the Birch and Swinnerton-Dyer
conjecture)
the number of integral points of $E_D$ are less than $f(D)$
for almost all $D$.
Keywords:elliptic curve, explicit formula, integral point, low-lying zeros, quadratic twist, rank Categories:11G05, 11G40 |
35. CJM 2011 (vol 63 pp. 1328)
On a Conjecture of Chowla and Milnor In this paper, we investigate a conjecture due to S. and P. Chowla and
its generalization by Milnor. These are related to the delicate
question of non-vanishing of $L$-functions associated to periodic
functions at integers greater than $1$. We report on some progress in
relation to these conjectures. In a different vein, we link them to a
conjecture of Zagier on multiple zeta values and also to linear
independence of polylogarithms.
Categories:11F20, 11F11 |
36. CJM 2011 (vol 63 pp. 992)
The Arithmetic of Genus Two Curves with (4,4)-Split Jacobians
In this paper we study genus $2$ curves whose Jacobians admit a
polarized $(4,4)$-isogeny to a product of elliptic curves. We consider
base fields of characteristic different from $2$ and $3$, which we do
not assume to be algebraically closed.
We obtain a full classification of all principally polarized abelian
surfaces that can arise from gluing two elliptic curves along their
$4$-torsion, and we derive the relation their absolute invariants
satisfy.
As an intermediate step, we give a general description of Richelot
isogenies between Jacobians of genus $2$ curves, where previously only
Richelot isogenies with kernels that are pointwise defined over the
base field were considered.
Our main tool is a Galois theoretic characterization of genus $2$
curves admitting multiple Richelot isogenies.
Keywords:Genus 2 curves, isogenies, split Jacobians, elliptic curves Categories:11G30, 14H40 |
37. CJM 2011 (vol 63 pp. 1284)
Non-Existence of Ramanujan Congruences in Modular Forms of Level Four Ramanujan famously found congruences like $p(5n+4)\equiv 0
\operatorname{mod} 5$ for the partition
function. We provide a method to find all simple
congruences of this type in the coefficients of the inverse of a
modular form on $\Gamma_{1}(4)$ that is non-vanishing on the upper
half plane. This is applied to answer open questions about the
(non)-existence of congruences in the generating functions for
overpartitions, crank differences, and 2-colored $F$-partitions.
Keywords:modular form, Ramanujan congruence, generalized Frobenius partition, overpartition, crank Categories:11F33, 11P83 |
38. CJM 2011 (vol 63 pp. 826)
Singular Moduli of Shimura Curves The $j$-function acts as a parametrization of the classical modular
curve. Its values at complex multiplication (CM) points are called
singular moduli and are algebraic integers. A Shimura curve is a
generalization of the modular curve and, if the Shimura curve has
genus~$0$, a rational parameterizing function exists and when
evaluated at a CM point is again algebraic over~$\mathbf{Q}$. This paper shows
that the coordinate maps given by N.~Elkies for the Shimura
curves associated to the quaternion algebras with discriminants $6$
and $10$ are Borcherds lifts of vector-valued modular forms. This
property is then used to explicitly compute the rational norms of
singular moduli on these curves. This method not only verifies
conjectural values for the rational CM points, but also provides a way
of algebraically calculating the norms of CM points with arbitrarily
large negative discriminant.
Categories:11G18, 11F12 |
39. CJM 2011 (vol 63 pp. 1083)
Decomposition of Splitting Invariants in Split Real Groups For a maximal torus in a quasi-split semi-simple simply-connected group over a local field of characteristic $0$,
Langlands and Shelstad constructed a
cohomological invariant called the splitting invariant, which is an important
component of their endoscopic transfer factors. We study this invariant in the
case of a split real group and prove a
decomposition theorem which expresses this invariant for a general torus as a product of the corresponding
invariants for simple tori. We also show how this reduction formula allows for the comparison of splitting invariants
between different tori in the given real group.
Keywords:endoscopy, real lie group, splitting invariant, transfer factor Categories:11F70, 22E47, 11S37, 11F72, 17B22 |
40. CJM 2011 (vol 63 pp. 1107)
Genericity of Representations of p-Adic $Sp_{2n}$ and Local Langlands Parameters Let $G$ be the $F$-rational points of the symplectic group $Sp_{2n}$,
where $F$ is a non-Archimedean local field
of characteristic
$0$. Cogdell, Kim, Piatetski-Shapiro, and Shahidi
constructed local Langlands functorial lifting from irreducible
generic representations of $G$ to irreducible representations of
$GL_{2n+1}(F)$.
Jiang and Soudry constructed the descent map from irreducible
supercuspidal representations of $GL_{2n+1}(F)$ to those of $G$,
showing that the local Langlands functorial lifting from the
irreducible supercuspidal generic representations is surjective. In
this paper, based on above results, using the same descent method of
studying $SO_{2n+1}$ as Jiang and Soudry, we will show the rest
of local Langlands functorial lifting is also surjective, and for any
local Langlands parameter $\phi \in \Phi(G)$, we construct a
representation $\sigma$ such that $\phi$ and $\sigma$ have the same
twisted local factors. As one application, we prove the $G$-case of a
conjecture of
Gross-Prasad and Rallis, that is, a local Langlands parameter $\phi
\in \Phi(G)$ is generic, i.e., the representation attached to
$\phi$ is generic, if and only if the adjoint $L$-function of $\phi$
is holomorphic at $s=1$. As another application, we prove for each
Arthur parameter $\psi$, and the corresponding local Langlands
parameter
$\phi_{\psi}$, the representation attached to $\phi_{\psi}$
is generic if and only if $\phi_{\psi}$ is tempered.
Keywords:generic representations, local Langlands parameters Categories:22E50, 11S37 |
41. CJM 2011 (vol 63 pp. 1220)
Similar Sublattices of Planar Lattices The similar sublattices of a planar lattice can be classified via
its multiplier ring. The latter is the ring of rational integers in
the generic case, and an order in an imaginary quadratic field
otherwise. Several classes of examples are discussed, with special
emphasis on concrete results. In particular, we derive Dirichlet
series generating functions for the number of distinct similar
sublattices of a given index, and relate them to
zeta functions of orders in imaginary quadratic fields.
Categories:11H06, 11R11, 52C05, 82D25 |
42. CJM 2011 (vol 63 pp. 616)
A Modular Quintic Calabi-Yau Threefold of Level 55 In this note we search the parameter space of Horrocks-Mumford quintic
threefolds and locate a Calabi-Yau threefold that is modular, in the
sense that the $L$-function of its middle-dimensional cohomology is
associated with a classical modular form of weight 4 and level 55.
Keywords: Calabi-Yau threefold, non-rigid Calabi-Yau threefold, two-dimensional Galois representation, modular variety, Horrocks-Mumford vector bundle Categories:14J15, 11F23, 14J32, 11G40 |
43. CJM 2011 (vol 63 pp. 591)
Rank One Reducibility for Metaplectic Groups via Theta Correspondence We calculate reducibility for the representations of
metaplectic groups induced from cuspidal representations of
maximal parabolic subgroups via theta correspondence, in terms of the
analogous representations of the odd orthogonal groups. We also
describe the lifts of all relevant subquotients.
Categories:22E50, 11F70 |
44. 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 |
45. 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 |
46. 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 |
47. 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 |
48. 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 |
49. 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 |
50. 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 |