126. CJM 2009 (vol 61 pp. 165)
 Laurent, Michel

Exponents of Diophantine Approximation in Dimension Two
Let $\Theta=(\alpha,\beta)$ be a point in $\bR^2$, with $1,\alpha,
\beta$ linearly independent over $\bQ$. We attach to $\Theta$ a
quadruple $\Omega(\Theta)$ of exponents that measure the quality
of approximation to $\Theta$ both by rational points and by
rational lines. The two ``uniform'' components of $\Omega(\Theta)$
are related by an equation due to Jarn\'\i k, and the four
exponents satisfy two inequalities that refine Khintchine's
transference principle. Conversely, we show that for any quadruple
$\Omega$ fulfilling these necessary conditions, there exists
a point $\Theta\in \bR^2$ for which $\Omega(\Theta) =\Omega$.
Categories:11J13, 11J70 

127. 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 

128. 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 

129. CJM 2008 (vol 60 pp. 1306)
 Mui\'c, Goran

Theta Lifts of Tempered Representations for Dual Pairs $(\Sp_{2n}, O(V))$
This paper is the continuation of our previous work on the explicit
determination of the structure of theta lifts for dual pairs
$(\Sp_{2n}, O(V))$ over a nonarchimedean field $F$ of characteristic
different than $2$, where $n$ is the split rank of $\Sp_{2n}$ and the
dimension of the space $V$ (over $F$) is even. We determine the
structure of theta lifts of tempered representations in terms of theta
lifts of representations in discrete series.
Categories:22E35, 22E50, 11F70 

130. CJM 2008 (vol 60 pp. 1406)
 Ricotta, Guillaume; Vidick, Thomas

Hauteur asymptotique des points de Heegner
Geometric intuition suggests that the N\'{e}ronTate 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 GrossZagier formulae, such heights are related to the special
value at the critical point for either the derivative of the
RankinSelberg 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 

131. CJM 2008 (vol 60 pp. 1028)
 Hamblen, Spencer

Lifting $n$Dimensional Galois Representations
We investigate the problem of deforming $n$dimensional mod $p$ Galois
representations to characteristic zero. The existence of 2dimensional
deformations has been proven under certain conditions
by allowing ramification at additional primes in order to
annihilate a dual Selmer group. We use the same general methods to
prove the existence of $n$dimensional deformations.
We then examine under which conditions we may place restrictions on
the shape of our deformations at $p$, with the goal of showing that
under the correct conditions, the deformations may have locally
geometric shape. We also use the existence of these deformations to
prove the existence as Galois groups over $\Q$ of certain infinite
subgroups of $p$adic general linear groups.
Category:11F80 

132. CJM 2008 (vol 60 pp. 975)
 Boca, Florin P.

An AF Algebra Associated with the Farey Tessellation
We associate with the Farey tessellation of the upper
halfplane an
AF algebra $\AA$ encoding the ``cutting sequences'' that define
vertical geodesics.
The EffrosShen AF algebras arise as quotients
of $\AA$. Using the path algebra model for AF algebras we construct, for
each $\tau \in \big(0,\frac{1}{4}\big]$, projections $(E_n)$ in
$\AA$ such that $E_n E_{n\pm 1}E_n \leq \tau E_n$.
Categories:46L05, 11A55, 11B57, 46L55, 37E05, 82B20 

133. CJM 2008 (vol 60 pp. 1149)
 Petersen, Kathleen L.; Sinclair, Christopher D.

Conjugate Reciprocal Polynomials with All Roots on the Unit Circle
We study the geometry, topology and Lebesgue measure of the set of
monic conjugate reciprocal polynomials of fixed degree with all
roots on the unit circle. The set of such polynomials of degree $N$
is naturally associated to a subset of $\R^{N1}$. We calculate
the volume of this set, prove the set is homeomorphic to the $N1$
ball and that its isometry group is isomorphic to the dihedral
group of order $2N$.
Categories:11C08, 28A75, 15A52, 54H10, 58D19 

134. CJM 2008 (vol 60 pp. 790)
 Blasco, Laure

Types, paquets et changement de base : l'exemple de $U(2,1)(F_0)$. I. Types simples maximaux et paquets singletons
Soit $F_0$ un corps local non archim\'edien de caract\'eristique
nulle et de ca\rac\t\'eristique r\'esiduelle impaire.
J. Rogawski a montr\'e l'existence du changement de base entre le
groupe unitaire en trois variables $U(2,1)(F_{0})$, d\'efini
relativement \`a une extension quadratique $F$ de $F_{0}$, et le
groupe lin\'eaire $GL(3,F)$. Par ailleurs, nous
avons d\'ecrit les repr\'esentations supercuspidales irr\'eductibles
de $U(2,1)(F_{0})$ comme induites \`a partir d'un sousgroupe compact
ouvert de $U(2,1)(F_{0})$, description analogue \`a celle des
repr\'esentations admissibles irr\'eductibles de $GL(3,F)$ obtenue
par C. Bushnell et P. Kutzko. A partir de ces
descriptions, nous construisons explicitement le changement de base
des repr\'esentations tr\`es cuspidales de $U(2,1)(F_{0})$.
Categories:22E50, 11F70 

135. CJM 2008 (vol 60 pp. 734)
 Baba, Srinath; Granath, H\aa kan

Genus 2 Curves with Quaternionic Multiplication
We explicitly construct the canonical rational models of Shimura
curves, both analytically in terms of modular forms and
algebraically in terms of coefficients of genus 2 curves, in the
cases of quaternion algebras of discriminant 6 and 10. This emulates
the classical construction in the elliptic curve case. We also give
families of genus 2 QM curves, whose Jacobians are the corresponding
abelian surfaces on the Shimura curve, and with coefficients that
are modular forms of weight 12. We apply these results to show
that our $j$functions are supported exactly at those primes where
the genus 2 curve does not admit potentially good reduction, and
construct fields where this potentially good reduction is attained.
Finally, using $j$, we construct the fields of moduli and definition
for some moduli problems associated to the AtkinLehner group
actions.
Keywords:Shimura curve, canonical model, quaternionic multiplication, modular form, field of moduli Categories:11G18, 14G35 

136. CJM 2008 (vol 60 pp. 532)
 Clark, Pete L.; Xarles, Xavier

Local Bounds for Torsion Points on Abelian Varieties
We say that an abelian variety over a $p$adic field $K$ has
anisotropic reduction (AR) if the special fiber of its N\'eron minimal
model does not contain a nontrivial split torus. This includes all
abelian varieties with potentially good reduction and, in particular,
those with complex or quaternionic multiplication. We give a bound for
the size of the $K$rational torsion subgroup of a $g$dimensional AR
variety depending only on $g$ and the numerical invariants of $K$ (the
absolute ramification index and the cardinality of the residue
field). Applying these bounds to abelian varieties over a number field
with everywhere locally anisotropic reduction, we get bounds which, as
a function of $g$, are close to optimal. In particular, we determine
the possible cardinalities of the torsion subgroup of an AR abelian
surface over the rational numbers, up to a set of 11 values which are
not known to occur. The largest such value is 72.
Categories:11G10, 14K15 

137. CJM 2008 (vol 60 pp. 491)
 Bugeaud, Yann; Mignotte, Maurice; Siksek, Samir

A MultiFrey Approach to Some MultiParameter Families of Diophantine Equations
We solve several multiparameter families of binomial Thue equations of arbitrary
degree; for example, we solve the equation
\[
5^u x^n2^r 3^s y^n= \pm 1,
\]
in nonzero integers $x$, $y$ and positive integers $u$, $r$, $s$ and $n \geq 3$.
Our approach uses several Frey curves simultaneously, Galois representations
and levellowering, new lower bounds for linear
forms in $3$ logarithms due to Mignotte and a famous theorem of Bennett on binomial
Thue equations.
Keywords:Diophantine equations, Frey curves, levellowering, linear forms in logarithms, Thue equation Categories:11F80, 11D61, 11D59, 11J86, 11Y50 

138. CJM 2008 (vol 60 pp. 481)
 Breuer, Florian; Im, BoHae

Heegner Points and the Rank of Elliptic Curves over Large Extensions of Global Fields
Let $k$ be a global field, $\overline{k}$ a separable
closure of $k$, and $G_k$ the absolute Galois group
$\Gal(\overline{k}/k)$ of $\overline{k}$ over $k$. For every
$\sigma\in G_k$, let $\ks$ be the fixed subfield of $\overline{k}$
under $\sigma$. Let $E/k$ be an elliptic curve over $k$. It is known
that the MordellWeil group $E(\ks)$ has infinite rank. We present a
new proof of this fact in the following two cases. First, when $k$
is a global function field of odd characteristic and $E$ is
parametrized by a Drinfeld modular curve, and secondly when $k$ is a
totally real number field and $E/k$ is parametrized by a Shimura
curve. In both cases our approach uses the nontriviality of a
sequence of Heegner points on $E$ defined over ring class fields.
Category:11G05 

139. CJM 2008 (vol 60 pp. 412)
 NguyenChu, G.V.

Quelques calculs de traces compactes et leurs transform{Ã©es de Satake
On calcule les restrictions {\`a} l'alg{\`e}bre de Hecke sph{\'e}rique
des traces tordues compactes d'un ensemble de repr{\'e}sentations
explicitement construites du groupe $\GL(N, F)$, o{\`u} $F$ est
un corps $p$adique. Ces calculs r\'esolve en particulier une
question pos{\'e}e dans un article pr\'ec\'edent du m\^eme auteur.
Categories:22E50, 11F70 

140. CJM 2008 (vol 60 pp. 208)
 Ramakrishna, Ravi

Constructing Galois Representations with Very Large Image
Starting with a 2dimensional mod $p$ Galois representation, we
construct a deformation to a power series ring in infinitely many
variables over the $p$adics. The image of this representation is full
in the sense that it contains $\SL_2$ of this power series
ring. Furthermore, all ${\mathbb Z}_p$ specializations of this
deformation are potentially semistable at $p$.
Keywords:Galois representation, deformation Category:11f80 

141. CJM 2007 (vol 59 pp. 1121)
142. CJM 2007 (vol 59 pp. 1323)
 Ginzburg, David; Lapid, Erez

On a Conjecture of Jacquet, Lai, and Rallis: Some Exceptional Cases
We prove two spectral identities. The first one relates the relative
trace formula for the spherical variety $\GSpin(4,3)/G_2$ with a
weighted trace formula for $\GL_2$. The second relates a spherical
variety pertaining to $F_4$ to one of $\GSp(6)$. These identities are
in accordance with a conjecture made by Jacquet, Lai, and Rallis,
and are obtained without an appeal to a geometric comparison.
Categories:11F70, 11F72, 11F30, 11F67 

143. CJM 2007 (vol 59 pp. 1284)
 Fukshansky, Lenny

On Effective Witt Decomposition and the CartanDieudonn{Ã© Theorem
Let $K$ be a number field, and let $F$ be a symmetric bilinear form in
$2N$ variables over $K$. Let $Z$ be a subspace of $K^N$. A classical
theorem of Witt states that the bilinear space $(Z,F)$ can be
decomposed into an orthogonal sum of hyperbolic planes and singular and
anisotropic components. We prove the existence of such a decomposition
of small height, where all bounds on height are explicit in terms of
heights of $F$ and $Z$. We also prove a special version of Siegel's
lemma for a bilinear space, which provides a smallheight orthogonal
decomposition into onedimensional subspaces. Finally, we prove an
effective version of the CartanDieudonn{\'e} theorem. Namely, we show
that every isometry $\sigma$ of a regular bilinear space $(Z,F)$ can
be represented as a product of reflections of bounded heights with an
explicit bound on heights in terms of heights of $F$, $Z$, and
$\sigma$.
Keywords:quadratic form, heights Categories:11E12, 15A63, 11G50 

144. CJM 2007 (vol 59 pp. 1050)
 Raghuram, A.

On the Restriction to $\D^* \times \D^*$ of Representations of $p$Adic $\GL_2(\D)$
Let $\mathcal{D}$ be a division algebra
over a nonarchimedean local field. Given
an irreducible representation $\pi$ of $\GL_2(\mathcal{D})$, we
describe its restriction to the diagonal subgroup $\mathcal{D}^* \times
\mathcal{D}^*$. The description is in terms of the structure of the
twisted Jacquet module of the representation $\pi$. The proof
involves Kirillov theory that we have developed earlier in joint work
with Dipendra Prasad. The main result on restriction also shows that
$\pi$ is $\mathcal{D}^* \times \mathcal{D}^*$distinguished if and only if
$\pi$ admits a Shalika model. We further prove that if $\mathcal{D}$
is a quaternion division algebra then the twisted Jacquet module
is multiplicityfree by proving an appropriate theorem on invariant
distributions; this then proves a multiplicityone theorem on the
restriction to $\mathcal{D}^* \times \mathcal{D}^*$ in the quaternionic
case.
Categories:22E50, 22E35, 11S37 

145. 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 

146. CJM 2007 (vol 59 pp. 503)
 Chevallier, Nicolas

Cyclic Groups and the Three Distance Theorem
We give a two dimensional extension of the three distance Theorem. Let
$\theta$ be in $\mathbf{R}^{2}$ and let $q$ be in $\mathbf{N}$. There exists a
triangulation of $\mathbf{R}^{2}$ invariant by $\mathbf{Z}^{2}$translations,
whose set of vertices is $\mathbf{Z}^{2}+\{0,\theta,\dots,q\theta\}$, and whose
number of different triangles, up to translations, is bounded above by a
constant which does not depend on $\theta$ and $q$.
Categories:11J70, 11J71, 11J13 

147. 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 

148. CJM 2007 (vol 59 pp. 372)
 Maisner, Daniel; Nart, Enric

Zeta Functions of Supersingular Curves of Genus 2
We determine which isogeny classes of supersingular abelian
surfaces over a finite field $k$ of characteristic $2$ contain
jacobians. We deal with this problem in a direct way by computing
explicitly the zeta function of all supersingular curves of genus
$2$. Our procedure is constructive, so that we are able to exhibit
curves with prescribed zeta function and find formulas for the
number of curves, up to $k$isomorphism, leading to the same zeta
function.
Categories:11G20, 14G15, 11G10 

149. CJM 2007 (vol 59 pp. 85)
 Foster, J. H.; Serbinowska, Monika

On the Convergence of a Class of Nearly Alternating Series
Let $C$ be the class of convex sequences of real numbers. The
quadratic irrational numbers can be partitioned into two types as
follows. If $\alpha$ is of the first type and $(c_k) \in C$, then
$\sum (1)^{\lfloor k\alpha \rfloor} c_k$ converges if and only if
$c_k \log k \rightarrow 0$. If $\alpha$ is of the second type and
$(c_k) \in C$, then $\sum (1)^{\lfloor k\alpha \rfloor} c_k$
converges if and only if $\sum c_k/k$ converges. An example of a
quadratic irrational of the first type is $\sqrt{2}$, and an
example of the second type is $\sqrt{3}$. The analysis of this
problem relies heavily on the representation of $ \alpha$ as a
simple continued fraction and on properties of the sequences of
partial sums $S(n)=\sum_{k=1}^n (1)^{\lfloor k\alpha \rfloor}$
and double partial sums $T(n)=\sum_{k=1}^n S(k)$.
Keywords:Series, convergence, almost alternating, convex, continued fractions Categories:40A05, 11A55, 11B83 

150. CJM 2007 (vol 59 pp. 127)
 Lamzouri, Youness

Smooth Values of the Iterates of the Euler PhiFunction
Let $\phi(n)$ be the Euler phifunction, define
$\phi_0(n) = n$ and $\phi_{k+1}(n)=\phi(\phi_{k}(n))$ for all
$k\geq 0$. We will determine an asymptotic formula for the set of
integers $n$ less than $x$ for which $\phi_k(n)$ is $y$smooth,
conditionally on a weak form of the ElliottHalberstam conjecture.
Categories:11N37, 11B37, 34K05, 45J05 
