Expand all Collapse all | Results 1 - 13 of 13 |
1. CJM 2013 (vol 65 pp. 1320)
Orbital $L$-functions for the Space of Binary Cubic Forms We introduce the notion of orbital $L$-functions
for the space of binary cubic forms
and investigate their analytic properties.
We study their functional equations and residue formulas in some detail.
Aside from their intrinsic interest,
the results from this paper are used to
prove the existence of secondary terms in counting
functions for cubic fields.
This is worked out in a companion paper.
Keywords:binary cubic forms, prehomogeneous vector spaces, Shintani zeta functions, $L$-functions, cubic rings and fields Categories:11M41, 11E76 |
2. 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 |
3. 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 |
4. 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 |
5. 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 |
6. 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 |
7. CJM 2007 (vol 59 pp. 673)
Hecke $L$-Functions and the Distribution of Totally Positive Integers Let $K$ be a totally real number field of degree $n$. We show that
the number of totally positive integers
(or more generally the number of totally positive elements of a given fractional ideal)
of given trace is evenly distributed around its expected value, which is
obtained from geometric considerations.
This result depends on unfolding an integral over
a compact torus.
Keywords:Eisenstein series, toroidal integral, Fourier series, Hecke $L$-function, totally positive integer, trace Categories:11M41, 11F30, , 11F55, 11H06, 11R47 |
8. CJM 2006 (vol 58 pp. 3)
The Functional Equation of Zeta Distributions Associated With Non-Euclidean Jordan Algebras This paper is devoted to the study of certain zeta distributions
associated with simple non-Euclidean Jordan algebras. An explicit
form of the corresponding functional equation and Bernstein-type
identities is obtained.
Keywords:Zeta distributions, functional equations, Bernstein polynomials, non-Euclidean Jordan algebras Categories:11M41, 17C20, 11S90 |
9. 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 |
10. 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 |
11. 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 |
12. CJM 1997 (vol 49 pp. 641)
Fine spectra and limit laws II First-order 0--1 laws. Using Feferman-Vaught techniques a condition on the fine
spectrum of an admissible class of structures is found
which leads to a first-order 0--1 law.
The condition presented is best possible in the
sense that if it is violated then one can find an admissible
class with the same fine spectrum which does not have
a first-order 0--1 law.
If the condition is satisfied (and hence we have a first-order %% 0--1 law)
Categories:03N45, 11N45, 11N80, 05A15, 05A16, 11M41, 11P81 |
13. CJM 1997 (vol 49 pp. 468)
Fine spectra and limit laws I. First-order laws Using Feferman-Vaught techniques we show a certain property of the fine
spectrum of an admissible class of structures leads to a first-order law.
The condition presented is best possible in the sense that if it is
violated then one can find an admissible class with the same fine
spectrum which does not have a first-order law. We present three
conditions for verifying that the above property actually holds.
The first condition is that the count function of an admissible class
has regular variation with a certain uniformity of convergence. This
applies to a wide range of admissible classes, including those
satisfying Knopfmacher's Axiom A, and those satisfying Bateman
and Diamond's condition.
The second condition is similar to the first condition, but designed
to handle the discrete case, {\it i.e.}, when the sizes of the structures
in an admissible class $K$ are all powers of a single integer. It applies
when either the class of indecomposables or the whole class satisfies
Knopfmacher's Axiom A$^\#$.
The third condition is also for the discrete case, when there is a
uniform bound on the number of $K$-indecomposables of any given size.
Keywords:First order limit laws, generalized number theory Categories:O3C13, 11N45, 11N80, 05A15, 05A16, 11M41, 11P81 |