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126. 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 sous-groupe 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

127. 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 Atkin--Lehner group actions.

Keywords:Shimura curve, canonical model, quaternionic multiplication, modular form, field of moduli
Categories:11G18, 14G35

128. CJM 2008 (vol 60 pp. 481)

Breuer, Florian; Im, Bo-Hae
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 Mordell--Weil 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 non-triviality of a sequence of Heegner points on $E$ defined over ring class fields.


129. 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

130. CJM 2008 (vol 60 pp. 491)

Bugeaud, Yann; Mignotte, Maurice; Siksek, Samir
A Multi-Frey Approach to Some Multi-Parameter Families of Diophantine Equations
We solve several multi-parameter families of binomial Thue equations of arbitrary degree; for example, we solve the equation \[ 5^u x^n-2^r 3^s y^n= \pm 1, \] in non-zero integers $x$, $y$ and positive integers $u$, $r$, $s$ and $n \geq 3$. Our approach uses several Frey curves simultaneously, Galois representations and level-lowering, 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, level-lowering, linear forms in logarithms, Thue equation
Categories:11F80, 11D61, 11D59, 11J86, 11Y50

131. CJM 2008 (vol 60 pp. 412)

Nguyen-Chu, G.-V.
Quelques calculs de traces compactes et leurs transform{ées 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

132. CJM 2008 (vol 60 pp. 208)

Ramakrishna, Ravi
Constructing Galois Representations with Very Large Image
Starting with a 2-dimensional 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

133. 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

134. CJM 2007 (vol 59 pp. 1284)

Fukshansky, Lenny
On Effective Witt Decomposition and the Cartan--Dieudonn{é 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 small-height orthogonal decomposition into one-dimensional subspaces. Finally, we prove an effective version of the Cartan--Dieudonn{\'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

135. CJM 2007 (vol 59 pp. 1121)

Alayont, Feryâl
Meromorphic Continuation of Spherical Cuspidal Data Eisenstein Series
Meromorphic continuation of the Eisenstein series induced from spherical, cuspidal data on parabolic subgroups is achieved via reworking Bernstein's adaptation of Selberg's third proof of meromorphic continuation.

Categories:11F72, 32N10, 32D15

136. 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 multiplicity-free by proving an appropriate theorem on invariant distributions; this then proves a multiplicity-one theorem on the restriction to $\mathcal{D}^* \times \mathcal{D}^*$ in the quaternionic case.

Categories:22E50, 22E35, 11S37

137. 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

138. 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

139. 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

140. 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

141. CJM 2007 (vol 59 pp. 148)

Muić, Goran
On Certain Classes of Unitary Representations for Split Classical Groups
In this paper we prove the unitarity of duals of tempered representations supported on minimal parabolic subgroups for split classical $p$-adic groups. We also construct a family of unitary spherical representations for real and complex classical groups

Categories:22E35, 22E50, 11F70

142. 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

143. CJM 2007 (vol 59 pp. 127)

Lamzouri, Youness
Smooth Values of the Iterates of the Euler Phi-Function
Let $\phi(n)$ be the Euler phi-function, 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 Elliott--Halberstam conjecture.

Categories:11N37, 11B37, 34K05, 45J05

144. CJM 2007 (vol 59 pp. 211)

Roy, Damien
On Two Exponents of Approximation Related to a Real Number and Its Square
For each real number $\xi$, let $\lambdahat_2(\xi)$ denote the supremum of all real numbers $\lambda$ such that, for each sufficiently large $X$, the inequalities $|x_0| \le X$, $|x_0\xi-x_1| \le X^{-\lambda}$ and $|x_0\xi^2-x_2| \le X^{-\lambda}$ admit a solution in integers $x_0$, $x_1$ and $x_2$ not all zero, and let $\omegahat_2(\xi)$ denote the supremum of all real numbers $\omega$ such that, for each sufficiently large $X$, the dual inequalities $|x_0+x_1\xi+x_2\xi^2| \le X^{-\omega}$, $|x_1| \le X$ and $|x_2| \le X$ admit a solution in integers $x_0$, $x_1$ and $x_2$ not all zero. Answering a question of Y.~Bugeaud and M.~Laurent, we show that the exponents $\lambdahat_2(\xi)$ where $\xi$ ranges through all real numbers with $[\bQ(\xi)\wcol\bQ]>2$ form a dense subset of the interval $[1/2, (\sqrt{5}-1)/2]$ while, for the same values of $\xi$, the dual exponents $\omegahat_2(\xi)$ form a dense subset of $[2, (\sqrt{5}+3)/2]$. Part of the proof rests on a result of V.~Jarn\'{\i}k showing that $\lambdahat_2(\xi) = 1-\omegahat_2(\xi)^{-1}$ for any real number $\xi$ with $[\bQ(\xi)\wcol\bQ]>2$.

Categories:11J13, 11J82

145. CJM 2006 (vol 58 pp. 1203)

Heiermann, Volker
Orbites unipotentes et pôles d'ordre maximal de la fonction $\mu $ de Harish-Chandra
Dans un travail ant\'erieur, nous avions montr\'e que l'induite parabolique (normalis\'ee) d'une repr\'esentation irr\'eductible cuspidale $\sigma $ d'un sous-groupe de Levi $M$ d'un groupe $p$-adique contient un sous-quotient de carr\'e int\'egrable, si et seulement si la fonction $\mu $ de Harish-Chandra a un p\^ole en $\sigma $ d'ordre \'egal au rang parabolique de $M$. L'objet de cet article est d'interpr\'eter ce r\'esultat en termes de fonctorialit\'e de Langlands.

Categories:11F70, 11F80, 22E50

146. CJM 2006 (vol 58 pp. 1095)

Sakellaridis, Yiannis
A Casselman--Shalika Formula for the Shalika Model of $\operatorname{GL}_n$
The Casselman--Shalika method is a way to compute explicit formulas for periods of irreducible unramified representations of $p$-adic groups that are associated to unique models (i.e., multiplicity-free induced representations). We apply this method to the case of the Shalika model of $GL_n$, which is known to distinguish lifts from odd orthogonal groups. In the course of our proof, we further develop a variant of the method, that was introduced by Y. Hironaka, and in effect reduce many such problems to straightforward calculations on the group.

Keywords:Casselman--Shalika, periods, Shalika model, spherical functions, Gelfand pairs
Categories:22E50, 11F70, 11F85

147. CJM 2006 (vol 58 pp. 843)

Õzlük, A. E.; Snyder, C.
On the One-Level Density Conjecture for Quadratic Dirichlet L-Functions
In a previous article, we studied the distribution of ``low-lying" zeros of the family of quad\-ratic Dirichlet $L$-functions assuming the Generalized Riemann Hypothesis for all Dirichlet $L$-functions. Even with this very strong assumption, we were limited to using weight functions whose Fourier transforms are supported in the interval $(-2,2)$. However, it is widely believed that this restriction may be removed, and this leads to what has become known as the One-Level Density Conjecture for the zeros of this family of quadratic $L$-functions. In this note, we make use of Weil's explicit formula as modified by Besenfelder to prove an analogue of this conjecture.


148. CJM 2006 (vol 58 pp. 796)

Im, Bo-Hae
Mordell--Weil Groups and the Rank of Elliptic Curves over Large Fields
Let $K$ be a number field, $\overline{K}$ an algebraic closure of $K$ and $E/K$ an elliptic curve defined over $K$. In this paper, we prove that if $E/K$ has a $K$-rational point $P$ such that $2P\neq O$ and $3P\neq O$, then for each $\sigma\in \Gal(\overline{K}/K)$, the Mordell--Weil group $E(\overline{K}^{\sigma})$ of $E$ over the fixed subfield of $\overline{K}$ under $\sigma$ has infinite rank.


149. CJM 2006 (vol 58 pp. 643)

Yu, Xiaoxiang
Centralizers and Twisted Centralizers: Application to Intertwining Operators
ABSTRACT The equality of the centralizer and twisted centralizer is proved based on a case-by-case analysis when the unipotent radical of a maximal parabolic subgroup is abelian. Then this result is used to determine the poles of intertwining operators.


150. 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 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
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