1. CJM 2016 (vol 68 pp. 571)
 Gras, Georges

Les $\theta$rÃ©gulateurs locaux d'un nombre algÃ©brique : Conjectures $p$adiques
Let $K/\mathbb{Q}$ be Galois and let $\eta\in K^\times$ be such that
$\operatorname{Reg}_\infty (\eta) \ne 0$.
We define the local $\theta$regulators $\Delta_p^\theta(\eta)
\in \mathbb{F}_p$
for the $\mathbb{Q}_p\,$irreducible characters $\theta$ of
$G=\operatorname{Gal}(K/\mathbb{Q})$. A linear representation ${\mathcal L}^\theta\simeq \delta \,
V_\theta$ is associated with
$\Delta_p^\theta (\eta)$ whose nullity is equivalent to $\delta
\geq 1$.
Each $\Delta_p^\theta (\eta)$ yields $\operatorname{Reg}_p^\theta (\eta)$
modulo $p$ in the factorization
$\prod_{\theta}(\operatorname{Reg}_p^\theta (\eta))^{\varphi(1)}$ of
$\operatorname{Reg}_p^G (\eta) := \frac{ \operatorname{Reg}_p(\eta)}{p^{[K : \mathbb{Q}\,]}
}$ (normalized $p$adic regulator).
From $\operatorname{Prob}\big (\Delta_p^\theta(\eta) = 0 \ \& \ {\mathcal
L}^\theta \simeq \delta \, V_\theta\big )
\leq p^{ f \delta^2}$ ($f \geq 1$ is a residue degree) and the
BorelCantelli heuristic,
we conjecture that, for $p$ large enough, $\operatorname{Reg}_p^G (\eta)$
is a $p$adic unit or that
$p^{\varphi(1)} \parallel \operatorname{Reg}_p^G (\eta)$ (a single $\theta$
with $f=\delta=1$); this obstruction may be lifted assuming the
existence of a binomial probability law
confirmed through numerical studies
(groups $C_3$, $C_5$, $D_6$).
This conjecture would imply that, for all $p$ large enough,
Fermat quotients, normalized $p$adic
regulators are $p$adic units and that
number fields are $p$rational.
We recall some deep cohomological results that
may strengthen such conjectures.
Keywords:$p$adic regulators, LeopoldtJaulent conjecture, Frobenius group determinants, characters, Fermat quotient, Abelian $p$ramification, probabilistic number theory Categories:11F85, 11R04, 20C15, 11C20, 11R37, 11R27, 11Y40 

2. CJM 2015 (vol 67 pp. 1326)
 Cojocaru, Alina Carmen; Shulman, Andrew Michael

The Distribution of the First Elementary Divisor of the Reductions of a Generic Drinfeld Module of Arbitrary Rank
Let $\psi$ be a generic Drinfeld module of rank $r \geq 2$. We study
the first elementary divisor
$d_{1, \wp}(\psi)$ of the reduction of $\psi$ modulo a prime $\wp$, as $\wp$ varies.
In particular, we prove the existence of the density of the primes $\wp$ for which $d_{1, \wp} (\psi)$ is fixed. For $r = 2$, we also study the second elementary divisor (the exponent) of the reduction of $\psi$ modulo $\wp$
and prove that, on average, it has a large norm. Our work is motivated by the study of J.P. Serre of an elliptic curve analogue of Artin's Primitive Root Conjecture, and, moreover, by refinements to Serre's study developed by the first author and M.R. Murty.
Keywords:Drinfeld modules, density theorems Categories:11R45, 11G09, 11R58 

3. CJM 2015 (vol 67 pp. 1046)
 Dubickas, Arturas; Sha, Min; Shparlinski, Igor

Explicit Form of Cassels' $p$adic Embedding Theorem for Number Fields
In this paper, we mainly give a general explicit form of Cassels'
$p$adic embedding theorem for number fields. We also give its
refined form in the case of cyclotomic fields. As a byproduct,
given an irreducible polynomial $f$ over $\mathbb{Z}$, we give a general
unconditional upper bound for the smallest prime number $p$ such
that $f$ has a simple root modulo $p$.
Keywords:number field, $p$adic embedding, height, polynomial, cyclotomic field Categories:11R04, 11S85, 11G50, 11R09, 11R18 

4. CJM 2015 (vol 67 pp. 654)
 Lim, Meng Fai; Murty, V. Kumar

Growth of Selmer groups of CM Abelian varieties
Let $p$ be an odd prime. We study the variation of the $p$rank of
the Selmer group of Abelian varieties with complex multiplication in
certain towers of number fields.
Keywords:Selmer group, Abelian variety with complex multiplication, $\mathbb{Z}_p$extension, $p$Hilbert class tower Categories:11G15, 11G10, 11R23, 11R34 

5. CJM 2014 (vol 67 pp. 848)
 Köck, Bernhard; Tait, Joseph

Faithfulness of Actions on RiemannRoch Spaces
Given a faithful action of a finite group $G$ on an algebraic
curve~$X$ of genus $g_X\geq 2$, we give explicit criteria for
the induced action of~$G$ on the RiemannRoch space~$H^0(X,\mathcal{O}_X(D))$
to be faithful, where $D$ is a $G$invariant divisor on $X$ of
degree at least~$2g_X2$. This leads to a concise answer to the
question when the action of~$G$ on the space~$H^0(X, \Omega_X^{\otimes
m})$ of global holomorphic polydifferentials of order $m$ is
faithful. If $X$ is hyperelliptic, we furthermore provide an
explicit basis of~$H^0(X, \Omega_X^{\otimes m})$. Finally, we
give applications in deformation theory and in coding theory
and we discuss the analogous problem for the action of~$G$ on
the first homology $H_1(X, \mathbb{Z}/m\mathbb{Z})$ if $X$ is a Riemann surface.
Keywords:faithful action, RiemannRoch space, polydifferential, hyperelliptic curve, equivariant deformation theory, Goppa code, homology Categories:14H30, 30F30, 14L30, 14D15, 11R32 

6. CJM 2014 (vol 67 pp. 507)
 Borwein, Peter; Choi, Stephen; Ferguson, Ron; Jankauskas, Jonas

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 $k$, $0 \leq k \leq n1$, 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 

7. CJM 2012 (vol 65 pp. 1201)
 Cho, Peter J.; Kim, Henry H.

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 KowalskiMichel, 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 

8. CJM 2012 (vol 64 pp. 254)
9. CJM 2011 (vol 64 pp. 345)
 McKee, James; Smyth, Chris

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 

10. CJM 2011 (vol 63 pp. 1220)
 Baake, Michael; Scharlau, Rudolf; Zeiner, Peter

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 

11. CJM 2010 (vol 62 pp. 1011)
 Buckingham, Paul; Snaith, Victor

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
nontrivial 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
nonabelian) 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 noncommutative Iwasawa algebras.
Categories:11R42, 11R23, 11R70 

12. CJM 2010 (vol 62 pp. 1060)
 Darmon, Henri; Tian, Ye

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 

13. CJM 2010 (vol 62 pp. 543)
 Hare, Kevin G.

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 

14. CJM 2010 (vol 62 pp. 787)
 Landquist, E.; Rozenhart, P.; Scheidler, R.; Webster, J.; Wu, Q.

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 nonsingularity 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 squarefree 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, nonsingularity, integral basis, genus, signature of a place, class number Categories:14H05, 11R58, 14H45, 11G20, 11G30, 11R16, 11R29 

15. CJM 2009 (vol 62 pp. 157)
 Masri, Riad

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
Galoisequivariant $L$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, RubinStark conjecture Categories:11R42, 11F30 

16. CJM 2009 (vol 61 pp. 1073)
 Griffiths, Ross; Lescop, Mikaël

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, biquadratic, and triquadratic 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 

17. CJM 2009 (vol 61 pp. 583)
 Hajir, Farshid

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 $L_n^{(1nr)}(x) = \sum_{j=0}^n
\binom{nj+r}{nj}x^j/j!$ 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 

18. CJM 2009 (vol 61 pp. 518)
 Belliard, JeanRobert

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 

19. CJM 2009 (vol 61 pp. 264)
 Bell, J. P.; Hare, K. G.

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 $d1$ 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 Pisotcyclotomic 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 Pisotcyclotomic numbers of any fixed
order.
\end{compactenum}
Keywords:algebraic integers, Pisot numbers, full rank, discriminant Categories:11R04, 11R06 

20. CJM 2009 (vol 61 pp. 3)
 Behrend, Kai; Dhillon, Ajneet

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 

21. CJM 2008 (vol 60 pp. 1267)
 Blake, Ian F.; Murty, V. Kumar; Xu, Guangwu

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 

22. CJM 2007 (vol 59 pp. 673)
 Ash, Avner; Friedberg, Solomon

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 

23. CJM 2007 (vol 59 pp. 553)
 Dasgupta, Samit

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 

24. CJM 2006 (vol 58 pp. 580)
 Greither, Cornelius; Kučera, Radan

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
nongenus part of the class group.
This leads to stronger annihilation results for the whole
class group than a routine application of the RubinThaine 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 nongenus 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 

25. CJM 2006 (vol 58 pp. 419)
 Snaith, Victor P.

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 higherdimensional 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 wellknown
CoatesSinnott conjecture. For example, for a totally real
extension when $r = 2, 4, 6, \dotsc$ the CoatesSinnott
conjecture merely predicts that zero annihilates $K_{2r}$ of the
ring of $S$integers while our conjecture predicts a nontrivial
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 
