Expand all Collapse all | Results 26 - 50 of 227 |
26. CJM 2012 (vol 65 pp. 1201)
Application of the Strong Artin Conjecture to the Class Number Problem We construct unconditionally several families of number fields with
the largest possible class numbers. They are number fields of degree 4
and 5 whose Galois closures have the Galois group $A_4, S_4$ and
$S_5$. We first construct families of number fields with smallest
regulators, and by using the strong Artin conjecture and applying zero
density result of Kowalski-Michel, we choose subfamilies of
$L$-functions which are zero free close to 1.
For these subfamilies, the $L$-functions have the extremal value at
$s=1$, and by the class number formula, we obtain the extreme class
numbers.
Keywords:class number, strong Artin conjecture Categories:11R29, 11M41 |
27. CJM 2012 (vol 65 pp. 544)
Iterated Integrals and Higher Order Invariants We show that higher order invariants of smooth functions can be
written as linear combinations of full invariants times iterated
integrals.
The non-uniqueness of such a presentation is captured in the kernel of
the ensuing map from the tensor product. This kernel is computed
explicitly.
As a consequence, it turns out that higher order invariants are a free
module of the algebra of full invariants.
Keywords:higher order forms, iterated integrals Categories:14F35, 11F12, 55D35, 58A10 |
28. CJM 2012 (vol 65 pp. 403)
On the Dihedral Main Conjectures of Iwasawa Theory for Hilbert Modular Eigenforms We construct a bipartite Euler system in the sense of Howard for Hilbert modular eigenforms of parallel
weight two over totally real fields, generalizing works of Bertolini-Darmon, Longo, Nekovar, Pollack-Weston
and others. The construction has direct applications to Iwasawa main conjectures. For instance, it implies
in many cases one divisibility of the associated dihedral or anticyclotomic main conjecture, at the same
time reducing the other divisibility to a certain nonvanishing criterion for the associated $p$-adic $L$-functions.
It also has applications to cyclotomic main conjectures for Hilbert modular forms over CM fields via the technique
of Skinner and Urban.
Keywords:Iwasawa theory, Hilbert modular forms, abelian varieties Categories:11G10, 11G18, 11G40 |
29. CJM 2012 (vol 65 pp. 171)
Optimal Polynomial Recurrence Let $P\in\mathbb Z[n]$ with $P(0)=0$ and $\varepsilon\gt 0$.
We show, using Fourier analytic techniques, that if $N\geq
\exp\exp(C\varepsilon^{-1}\log\varepsilon^{-1})$ and
$A\subseteq\{1,\dots,N\}$, then there must exist $n\in\mathbb N$ such that
\[\frac{|A\cap (A+P(n))|}{N}\gt \left(\frac{|A|}{N}\right)^2-\varepsilon.\]
In addition to this we also show, using the same Fourier analytic
methods, that if $A\subseteq\mathbb N$, then the set of
$\varepsilon$-optimal return times
\[R(A,P,\varepsilon)=\left\{n\in \mathbb N
\,:\,\delta(A\cap(A+P(n)))\gt \delta(A)^2-\varepsilon\right\}\]
is syndetic for every $\varepsilon\gt 0$. Moreover, we show that
$R(A,P,\varepsilon)$ is dense in every sufficiently long interval, in particular we show that
there exists an $L=L(\varepsilon,P,A)$ such that
\[\left|R(A,P,\varepsilon)\cap I\right|
\geq c(\varepsilon,P)|I|\]
for all intervals $I$ of natural numbers with $|I|\geq L$ and
$c(\varepsilon,P)=\exp\exp(-C\,\varepsilon^{-1}\log\varepsilon^{-1})$.
Keywords:Sarkozy, syndetic, polynomial return times Category:11B30 |
30. CJM 2012 (vol 64 pp. 1019)
On a Theorem of Bombieri, Friedlander, and Iwaniec In this article, we show to which extent one can improve a theorem of Bombieri, Friedlander and Iwaniec by using Hooley's variant of the divisor switching technique. We also give an application of the theorem in question, which is a Bombieri-Vinogradov type theorem for the Tichmarsh divisor problem in arithmetic progressions.
Keywords:primes in arithmetic progressions, Titchmarsh divisor problem Category:11N13 |
31. CJM 2012 (vol 64 pp. 254)
Corrigendum to ``On $\mathbb{Z}$-modules of Algebraic Integers'' We fix a mistake in the proof of Theorem 1.6 in the paper in the title.
Keywords:Pisot numbers, algebraic integers, number rings, Schmidt subspace theorem Categories:11R04, 11R06 |
32. CJM 2012 (vol 64 pp. 497)
Le lemme fondamental pondÃ©rÃ© pour le groupe mÃ©taplectique Dans cet article, on Ã©nonce une variante du lemme fondamental
pondÃ©rÃ© d'Arthur pour le groupe mÃ©taplectique de Weil, qui sera un
ingrÃ©dient indispensable de la stabilisation de la formule des
traces. Pour un corps de caractÃ©ristique rÃ©siduelle suffisamment
grande, on en donne une dÃ©monstration Ã l'aide de la mÃ©thode de
descente, qui est conditionnelle: on admet le lemme fondamental
pondÃ©rÃ© non standard sur les algÃ¨bres de Lie. Vu les travaux de
Chaudouard et Laumon, on s'attend Ã ce que cette condition soit
ultÃ©rieurement vÃ©rifiÃ©e.
Keywords:fundamental lemma, metaplectic group, endoscopy, trace formula Categories:11F70, 11F27, 22E50 |
33. CJM 2011 (vol 64 pp. 1201)
The Central Limit Theorem for Subsequences in Probabilistic Number Theory Let $(n_k)_{k \geq 1}$ be an increasing sequence of positive integers, and let $f(x)$ be a real function satisfying
\begin{equation}
\tag{1}
f(x+1)=f(x), \qquad \int_0^1 f(x) ~dx=0,\qquad
\operatorname{Var_{[0,1]}}
f \lt \infty.
\end{equation}
If $\lim_{k \to \infty} \frac{n_{k+1}}{n_k} = \infty$
the distribution of
\begin{equation}
\tag{2}
\frac{\sum_{k=1}^N f(n_k x)}{\sqrt{N}}
\end{equation}
converges to a Gaussian distribution. In the case
$$
1 \lt \liminf_{k \to \infty} \frac{n_{k+1}}{n_k}, \qquad \limsup_{k \to \infty} \frac{n_{k+1}}{n_k} \lt \infty
$$
there is a complex interplay between the analytic properties of the
function $f$, the number-theoretic properties of $(n_k)_{k \geq 1}$,
and the limit distribution of (2).
In this paper we prove that any sequence $(n_k)_{k \geq 1}$ satisfying
$\limsup_{k \to \infty} \frac{n_{k+1}}{n_k} = 1$ contains a nontrivial
subsequence $(m_k)_{k \geq 1}$ such that for any function satisfying
(1) the distribution of
$$
\frac{\sum_{k=1}^N f(m_k x)}{\sqrt{N}}
$$
converges to a Gaussian distribution. This result is best possible: for any
$\varepsilon\gt 0$ there exists a sequence $(n_k)_{k \geq 1}$ satisfying $\limsup_{k \to
\infty} \frac{n_{k+1}}{n_k} \leq 1 + \varepsilon$ such that for every nontrivial
subsequence $(m_k)_{k \geq 1}$ of $(n_k)_{k \geq 1}$ the distribution
of (2) does not converge to a Gaussian distribution for some $f$.
Our result can be viewed as a Ramsey type result: a sufficiently dense
increasing integer sequence contains a subsequence having a certain
requested number-theoretic property.
Keywords:central limit theorem, lacunary sequences, linear Diophantine equations, Ramsey type theorem Categories:60F05, 42A55, 11D04, 05C55, 11K06 |
34. CJM 2011 (vol 65 pp. 22)
Non-vanishing of $L$-functions, the Ramanujan Conjecture, and Families of Hecke Characters We prove a non-vanishing result for families of
$\operatorname{GL}_n\times\operatorname{GL}_n$ Rankin-Selberg $L$-functions in the critical strip,
as one factor runs over twists by Hecke characters. As an
application, we simplify the proof, due to Luo, Rudnick, and Sarnak,
of the best known bounds towards the Generalized Ramanujan Conjecture
at the infinite places for cusp forms on $\operatorname{GL}_n$. A key ingredient is
the regularization of the units in residue classes by the use of an
Arakelov ray class group.
Keywords:non-vanishing, automorphic forms, Hecke characters, Ramanujan conjecture Categories:11F70, 11M41 |
35. CJM 2011 (vol 64 pp. 1036)
Harmonic Analysis Related to Homogeneous Varieties in Three Dimensional Vector Spaces over Finite Fields |
Harmonic Analysis Related to Homogeneous Varieties in Three Dimensional Vector Spaces over Finite Fields In this paper we study the extension problem, the
averaging problem, and the generalized ErdÅs-Falconer distance
problem associated with arbitrary homogeneous varieties in three
dimensional vector spaces over finite fields. In the case when the
varieties do not contain any plane passing through the origin, we
obtain the best possible results on the aforementioned three problems. In
particular, our result on the extension problem modestly generalizes
the result by Mockenhaupt and Tao who studied the particular conical
extension problem. In addition, investigating the Fourier decay on
homogeneous varieties enables us to give complete mapping properties
of averaging operators. Moreover, we improve the size condition on a
set such that the cardinality of its distance set is nontrivial.
Keywords:extension problems, averaging operator, finite fields, ErdÅs-Falconer distance problems, homogeneous polynomial Categories:42B05, 11T24, 52C17 |
36. 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 |
37. 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 |
38. 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 |
39. 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 |
40. 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 |
41. 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 |
42. 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 |
43. 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 |
44. 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 |
45. 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 |
46. 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 |
47. 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 |
48. 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 |
49. 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 |
50. 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 |