Expand all Collapse all | Results 1 - 25 of 41 |
1. CJM Online first
On Littlewood Polynomials with Prescribed Number of Zeros Inside the Unit Disk We investigate the numbers of complex zeros of Littlewood polynomials
$p(z)$ (polynomials with coefficients $\{-1, 1\}$) inside or
on the unit circle $|z|=1$, denoted by $N(p)$ and $U(p)$, respectively.
Two types of Littlewood polynomials are considered: Littlewood
polynomials with one sign change in the sequence of coefficients
and Littlewood polynomials with one negative coefficient. We
obtain explicit formulas for $N(p)$, $U(p)$ for polynomials $p(z)$
of these types. We show that, if $n+1$ is a prime number, then
for each integer $k$, $0 \leq k \leq n-1$, there exists a Littlewood
polynomial $p(z)$ of degree $n$ with $N(p)=k$ and $U(p)=0$. Furthermore,
we describe some cases when the ratios $N(p)/n$ and $U(p)/n$
have limits as $n \to \infty$ and find the corresponding limit
values.
Keywords:Littlewood polynomials, zeros, complex roots Categories:11R06, 11R09, 11C08 |
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 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 |
4. 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 |
5. 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 |
6. 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 |
7. 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 |
8. 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 |
9. 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 |
10. 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 |
11. CJM 2009 (vol 61 pp. 1073)
On the $2$-Rank of the Hilbert Kernel of Number Fields Let $E/F$ be a quadratic extension of
number fields. In this paper, we show that the genus formula for
Hilbert kernels, proved by M. Kolster and A. Movahhedi, gives the
$2$-rank of the Hilbert kernel of $E$ provided that the $2$-primary
Hilbert kernel of $F$ is trivial. However, since the original genus
formula is not explicit enough in a very particular case, we first
develop a refinement of this formula in order to employ it in the
calculation of the $2$-rank of $E$ whenever $F$ is totally real with
trivial $2$-primary Hilbert kernel. Finally, we apply our results to
quadratic, bi-quadratic, and tri-quadratic fields which include
a complete $2$-rank formula for the family of fields
$\Q(\sqrt{2},\sqrt{\delta})$ where $\delta$ is a squarefree integer.
Categories:11R70, 19F15 |
12. 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 |
13. 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 |
14. 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 |
15. 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 |
16. 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 |
17. 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 |
18. CJM 2007 (vol 59 pp. 553)
Computations of Elliptic Units for Real Quadratic Fields Let $K$ be a real quadratic field, and $p$ a rational prime which is
inert in $K$. Let $\alpha$ be a modular unit on $\Gamma_0(N)$. In an
earlier joint article with Henri Darmon, we presented the definition
of an element $u(\alpha, \tau) \in K_p^\times$ attached to $\alpha$
and each $\tau \in K$. We conjectured that the $p$-adic number
$u(\alpha, \tau)$ lies in a specific ring class extension of $K$
depending on $\tau$, and proposed a ``Shimura reciprocity law"
describing the permutation action of Galois on the set of $u(\alpha,
\tau)$. This article provides computational evidence for these
conjectures. We present an efficient algorithm for computing
$u(\alpha, \tau)$, and implement this algorithm with the modular unit
$\alpha(z) = \Delta(z)^2\Delta(4z)/\Delta(2z)^3.$ Using $p = 3, 5, 7,$
and $11$, and all real quadratic fields $K$ with discriminant $D <
500$ such that $2$ splits in $K$ and $K$ contains no unit of negative
norm, we obtain results supporting our conjectures. One of the
theoretical results in this paper is that a certain measure used to
define $u(\alpha, \tau)$ is shown to be $\mathbf{Z}$-valued rather
than only $\mathbf{Z}_p \cap \mathbf{Q}$-valued; this is an
improvement over our previous result and allows for a precise
definition of $u(\alpha, \tau)$, instead of only up to a root of
unity.
Categories:11R37, 11R11, 11Y40 |
19. CJM 2006 (vol 58 pp. 580)
Annihilators for the Class Group of a Cyclic Field of Prime Power Degree, II We prove, for a field $K$ which is cyclic of odd prime power
degree over the rationals, that the annihilator of the
quotient of the units of $K$ by a suitable large subgroup (constructed
from circular units) annihilates what we call the
non-genus part of the class group.
This leads to stronger annihilation results for the whole
class group than a routine application of the Rubin--Thaine method
would produce, since the
part of the class group determined by genus theory has an obvious
large annihilator which is not detected by
that method; this is our reason for concentrating on
the non-genus part. The present work builds on and strengthens
previous work of the authors; the proofs are more conceptual now,
and we are also able to construct an example which demonstrates
that our results cannot be easily sharpened further.
Categories:11R33, 11R20, 11Y40 |
20. CJM 2006 (vol 58 pp. 419)
Stark's Conjecture and New Stickelberger Phenomena We introduce a new conjecture concerning the construction
of elements in the annihilator ideal
associated to a Galois action on the higher-dimensional algebraic
$K$-groups of rings of integers in number fields. Our conjecture is
motivic in the sense that it involves the (transcendental) Borel
regulator as well as being related to $l$-adic \'{e}tale
cohomology. In addition, the conjecture generalises the well-known
Coates--Sinnott conjecture. For example, for a totally real
extension when $r = -2, -4, -6, \dotsc$ the Coates--Sinnott
conjecture merely predicts that zero annihilates $K_{-2r}$ of the
ring of $S$-integers while our conjecture predicts a non-trivial
annihilator. By way of supporting evidence, we prove the
corresponding (conjecturally equivalent) conjecture for the Galois
action on the \'{e}tale cohomology of the cyclotomic extensions of
the rationals.
Categories:11G55, 11R34, 11R42, 19F27 |
21. CJM 2005 (vol 57 pp. 1155)
The Square Sieve and the Lang--Trotter Conjecture Let $E$ be an elliptic curve defined over $\Q$ and without
complex multiplication. Let $K$ be a fixed imaginary quadratic field.
We find nontrivial upper bounds for the number of ordinary primes $p \leq x$
for which $\Q(\pi_p) = K$, where $\pi_p$ denotes the Frobenius endomorphism
of $E$ at $p$. More precisely, under a generalized Riemann hypothesis
we show that this number is $O_{E}(x^{\slfrac{17}{18}}\log x)$, and unconditionally
we show that this number is $O_{E, K}\bigl(\frac{x(\log \log x)^{\slfrac{13}{12}}}
{(\log x)^{\slfrac{25}{24}}}\bigr)$. We also prove that the number of imaginary quadratic
fields $K$, with $-\disc K \leq x$ and of the form $K = \Q(\pi_p)$, is
$\gg_E\log\log\log x$ for $x\geq x_0(E)$. These results represent progress towards
a 1976 Lang--Trotter conjecture.
Keywords:Elliptic curves modulo $p$; Lang--Trotter conjecture;, applications of sieve methods Categories:11G05, 11N36, 11R45 |
22. CJM 2005 (vol 57 pp. 812)
On the Vanishing of $\mu$-Invariants of Elliptic Curves over $\qq$ Let $E_{/\qq}$ be an elliptic curve with good ordinary reduction at a
prime $p>2$. It has a well-defined Iwasawa $\mu$-invariant $\mu(E)_p$
which encodes part of the information about the growth of the Selmer
group $\sel E{K_n}$ as $K_n$ ranges over the subfields of the
cyclotomic $\zzp$-extension $K_\infty/\qq$. Ralph Greenberg has
conjectured that any such $E$ is isogenous to a curve $E'$ with
$\mu(E')_p=0$. In this paper we prove Greenberg's conjecture for
infinitely many curves $E$ with a rational $p$-torsion point, $p=3$ or
$5$, no two of our examples having isomorphic $p$-torsion. The core
of our strategy is a partial explicit evaluation of the global duality
pairing for finite flat group schemes over rings of integers.
Category:11R23 |
23. CJM 2005 (vol 57 pp. 180)
On the Size of the Wild Set To every pair of algebraic number fields with isomorphic Witt rings
one can associate a number, called the {\it minimum number of wild
primes}. Earlier investigations have established lower bounds for this
number. In this paper an analysis is presented that expresses the
minimum number of wild primes in terms of the number of wild dyadic
primes. This formula not only gives immediate upper bounds, but can be
considered to be an exact formula for the minimum number of wild
primes.
Categories:11E12, 11E81, 19F15, 11R29 |
24. CJM 2004 (vol 56 pp. 194)
Selmer Groups of Elliptic Curves with Complex Multiplication Suppose $K$ is an imaginary quadratic field and $E$ is an elliptic curve over a
number field $F$ with complex multiplication by the ring of integers in $K$.
Let $p$ be a rational prime that splits as $\mathfrak{p}_{1}\mathfrak{p}_{2}$
in $K$. Let $E_{p^{n}}$ denote the $p^{n}$-division points on $E$. Assume
that $F(E_{p^{n}})$ is abelian over $K$ for all $n\geq 0$. This paper proves
that the Pontrjagin dual of the $\mathfrak{p}_{1}^{\infty}$-Selmer group of
$E$ over $F(E_{p^{\infty}})$ is a finitely generated free $\Lambda$-module,
where $\Lambda$ is the Iwasawa algebra of $\Gal\bigl(F(E_{p^{\infty}})/
F(E_{\mathfrak{p}_{1}^{\infty}\mathfrak{p}_{2}})\bigr)$. It also gives a simple
formula for the rank of the Pontrjagin dual as a $\Lambda$-module.
Categories:11R23, 11G05 |
25. CJM 2004 (vol 56 pp. 71)
Euclidean Rings of Algebraic Integers Let $K$ be a finite Galois extension of the field of rational numbers
with unit rank greater than~3. We prove that the ring of integers of
$K$ is a Euclidean domain if and only if it is a principal ideal
domain. This was previously known under the assumption of the
generalized Riemann hypothesis for Dedekind zeta functions. We now
prove this unconditionally.
Categories:11R04, 11R27, 11R32, 11R42, 11N36 |