76. CJM 2011 (vol 63 pp. 298)
 Gun, Sanoli; Murty, V. Kumar

A Variant of Lehmer's Conjecture, II: The CMcase
Let $f$ be a normalized Hecke eigenform with rational integer Fourier
coefficients. It is an interesting question to know how often an
integer $n$ has a factor common with the $n$th Fourier coefficient of
$f$. It has been shown in previous papers that this happens very often. In this
paper, we give an asymptotic formula for the number of integers $n$
for which $(n, a(n)) = 1$, where $a(n)$ is the $n$th Fourier coefficient of
a normalized Hecke eigenform $f$ of weight $2$ with rational integer
Fourier coefficients and having complex multiplication.
Categories:11F11, 11F30 

77. CJM 2010 (vol 63 pp. 241)
 Essouabri, Driss; Matsumoto, Kohji; Tsumura, Hirofumi

Multiple ZetaFunctions Associated with Linear Recurrence Sequences and the Vectorial Sum Formula
We prove the holomorphic continuation of certain multivariable multiple
zetafunctions whose coefficients satisfy a suitable recurrence condition.
In fact, we introduce more general vectorial zetafunctions and prove their
holomorphic continuation. Moreover, we show a vectorial sum formula among
those vectorial zetafunctions from which some generalizations of the
classical sum formula can be deduced.
Keywords:Zetafunctions, holomorphic continuation, recurrence sequences, Fibonacci numbers, sum formulas Categories:11M41, 40B05, 11B39 

78. CJM 2010 (vol 63 pp. 277)
 Ghate, Eknath; Vatsal, Vinayak

Locally Indecomposable Galois Representations
In a previous paper
the authors showed that, under some technical
conditions,
the local Galois representations attached to the members of
a nonCM family of ordinary cusp forms are indecomposable for all
except possibly finitely many
members of the family. In this paper we use deformation theoretic
methods to give examples of nonCM families for
which every classical member of weight at least two has a locally
indecomposable Galois representation.
Category:11F80 

79. CJM 2010 (vol 63 pp. 136)
80. CJM 2010 (vol 63 pp. 38)
 Brüdern, Jörg; Wooley, Trevor D.

Asymptotic Formulae for Pairs of Diagonal Cubic Equations
We investigate the number of integral solutions possessed by a
pair of diagonal cubic equations in a large box. Provided that the number of
variables in the system is at least fourteen, and in addition the number of
variables in any nontrivial linear combination of the underlying forms is at
least eight, we obtain an asymptotic formula for the number of integral
solutions consistent with the product of local densities associated with the
system.
Keywords:exponential sums, Diophantine equations Categories:11D72, 11P55 

81. CJM 2010 (vol 62 pp. 1276)
 El Wassouli, Fouzia

A Generalized Poisson Transform of an $L^{p}$Function over the Shilov Boundary of the $n$Dimensional Lie Ball
Let $\mathcal{D}$ be the $n$dimensional Lie ball and let
$\mathbf{B}(S)$ be the space of hyperfunctions on the Shilov
boundary $S$ of $\mathcal{D}$.
The aim of this paper is to give a
necessary and sufficient condition on the generalized Poisson
transform $P_{l,\lambda}f$ of an element $f$ in the space
$\mathbf{B}(S)$ for $f$ to be in $ L^{p}(S)$, $1 > p > \infty.$
Namely, if $F$ is the Poisson transform of some $f\in
\mathbf{B}(S)$ (i.e., $F=P_{l,\lambda}f$), then for any
$l\in \mathbb{Z}$ and $\lambda\in \mathbb{C}$ such that
$\mathcal{R}e[i\lambda] > \frac{n}{2}1$, we show that $f\in L^{p}(S)$ if and
only if $f$ satisfies the growth condition
$$
\F\_{\lambda,p}=\sup_{0\leq r
< 1}(1r^{2})^{\mathcal{R}e[i\lambda]\frac{n}{2}+l}\Big[\int_{S}F(ru)^{p}du
\Big]^{\frac{1}{p}} < +\infty.
$$
Keywords:Lie ball, Shilov boundary, Fatou's theorem, hyperfuctions, parabolic subgroup, homogeneous line bundle Categories:32A45, 30E20, 33C67, 33C60, 33C55, 32A25, 33C75, 11K70 

82. 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 

83. CJM 2010 (vol 62 pp. 1155)
 Young, Matthew P.

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 

84. CJM 2010 (vol 62 pp. 1099)
 Goldmakher, Leo

Character Sums to Smooth Moduli are Small
Recently, Granville and Soundararajan have made
fundamental breakthroughs in the study of character sums. Building
on their work and using estimates on short character sums developed
by GrahamRingrose and Iwaniec, we improve the
PÃ³lyaVinogradov inequality for characters with smooth conductor.
Categories:11L40, 11M06 

85. 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 

86. CJM 2010 (vol 62 pp. 914)
 Zorn, Christian

Reducibility of the Principal Series for Sp^{~}_{2}(F) over a padic Field
Let $G_n=\mathrm{Sp}_n(F)$ be the rank $n$ symplectic group with
entries in a nondyadic $p$adic field $F$. We further let $\widetilde{G}_n$ be
the metaplectic extension of $G_n$ by $\mathbb{C}^{1}=\{z\in\mathbb{C}^{\times}
\mid z=1\}$ defined using the Leray cocycle. In this paper, we aim to
demonstrate the complete list of reducibility points of the genuine
principal series of ${\widetilde{G}_2}$. In most cases, we will use
some techniques developed by TadiÄ that analyze the Jacquet
modules with respect to all of the parabolics containing a fixed
Borel. The exceptional cases involve representations induced from
unitary characters $\chi$ with $\chi^2=1$. Because such
representations $\pi$ are unitary, to show the irreducibility of
$\pi$, it suffices to show that
$\dim_{\mathbb{C}}\mathrm{Hom}_{{\widetilde{G}}}(\pi,\pi)=1$. We will accomplish this
by examining the poles of certain intertwining operators associated to
simple roots. Then some results of Shahidi and Ban decompose arbitrary
intertwining operators into a composition of operators corresponding
to the simple roots of ${\widetilde{G}_2}$. We will then be able to
show that all such operators have poles at principal series
representations induced from quadratic characters and therefore such
operators do not extend to operators in
$\mathrm{Hom}_{{\widetilde{G}_2}}(\pi,\pi)$ for the $\pi$ in question.
Categories:22E50, 11F70 

87. 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 

88. CJM 2010 (vol 62 pp. 668)
 Vollaard, Inken

The Supersingular Locus of the Shimura Variety for GU(1,s)
In this paper we study the supersingular locus of the reduction modulo $p$ of the Shimura variety for $GU(1,s)$ in the case of an inert prime $p$. Using DieudonnÃ© theory we define a stratification of the corresponding moduli space of $p$divisible groups. We describe the incidence relation of this stratification in terms of the BruhatTits building of a unitary group. In the case of $GU(1,2)$, we show that the supersingular locus is equidimensional of dimension 1 and is of complete intersection. We give an explicit description of the irreducible components and their intersection behaviour.
Categories:14G35, 11G18, 14K10 

89. 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 

90. CJM 2010 (vol 62 pp. 563)
91. CJM 2009 (vol 62 pp. 456)
 Yang, Tonghai

The ChowlaâSelberg Formula and The Colmez Conjecture
In this paper, we reinterpret the Colmez conjecture on the Faltings height of CM abelian varieties in terms of Hilbert (and Siegel) modular forms. We construct an elliptic modular form involving the Faltings height of a CM abelian surface and arithmetic intersection numbers, and prove that the Colmez conjecture for CM abelian surfaces is equivalent to the cuspidality of this modular form.
Categories:11G15, 11F41, 14K22 

92. 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 

93. CJM 2009 (vol 62 pp. 582)
 Konyagin, Sergei V.; Pomerance, Carl; Shparlinski, Igor E.

On the Distribution of Pseudopowers
An xpseudopower to base g is a positive integer that is not a power of g, yet is so modulo p for all primes $ple x$. We improve an upper bound for the least such number, due to E.~Bach, R.~Lukes, J.~Shallit, and H.~C.~Williams. The method is based on a combination of some bounds of exponential sums with new results about the average behaviour of the multiplicative order of g modulo prime numbers.
Categories:11A07, 11L07, 11N36 

94. CJM 2009 (vol 62 pp. 400)
 Prasanna, Kartik

On pAdic Properties of Central LValues of Quadratic Twists of an Elliptic Curve
We study $p$indivisibility of the central values $L(1,E_d)$ of
quadratic twists $E_d$ of a semistable elliptic curve $E$ of
conductor $N$. A consideration of the conjecture of Birch and
SwinnertonDyer shows that the set of quadratic discriminants $d$
splits naturally into several families $\mathcal{F}_S$, indexed by subsets $S$
of the primes dividing $N$. Let $\delta_S= \gcd_{d\in \mathcal{F}_S}
L(1,E_d)^{\operatorname{alg}}$, where $L(1,E_d)^{\operatorname{alg}}$ denotes the algebraic part
of the central $L$value, $L(1,E_d)$. Our main theorem relates the
$p$adic valuations of $\delta_S$ as $S$ varies. As a consequence we
present an application to a refined version of a question of
Kolyvagin. Finally we explain an intriguing (albeit speculative)
relation between Waldspurger packets on $\widetilde{\operatorname{SL}_2}$ and
congruences of modular forms of integral and halfintegral weight. In
this context, we formulate a conjecture on congruences of
halfintegral weight forms and explain its relevance to the problem of
$p$indivisibility of $L$values of quadratic twists.
Categories:11F40, 11F67, 11G05 

95. CJM 2009 (vol 61 pp. 1214)
 Cilleruelo, Javier; Granville, Andrew

Close Lattice Points on Circles
We classify the sets of four lattice points that all lie on a
short arc of a circle that has its center at the origin;
specifically on arcs of length $tR^{1/3}$ on a circle of radius
$R$, for any given $t>0$. In particular we prove that any arc of
length $ (40 + \frac{40}3\sqrt{10} )^{1/3}R^{1/3}$ on a circle of
radius $R$, with $R>\sqrt{65}$, contains at most three lattice
points, whereas we give an explicit infinite family of $4$tuples
of lattice points, $(\nu_{1,n},\nu_{2,n},\nu_{3,n},\nu_{4,n})$,
each of which lies on an arc of length $ (40 +
\frac{40}3\sqrt{10})^{\smash{1/3}}R_n^{\smash{1/3}}+o(1)$ on a circle of
radius $R_n$.
Category:11N36 

96. CJM 2009 (vol 61 pp. 1341)
 Rivoal, Tanguy

Simultaneous Polynomial Approximations of the Lerch Function
We construct bivariate polynomial approximations of the Lerch
function that for certain specialisations of the variables and
parameters turn out to be HermitePad\'e approximants either of
the polylogarithms or of Hurwitz zeta functions. In the former
case, we recover known results, while in the latter the results
are new and generalise some recent works of Beukers and Pr\'evost.
Finally, we make a detailed comparison of our work with Beukers'.
Such constructions are useful in the arithmetical study of the
values of the Riemann zeta function at integer points and of the
KubotaLeopold $p$adic zeta function.
Categories:41A10, 41A21, 11J72 

97. CJM 2009 (vol 61 pp. 1383)
 Wambach, Eric

Integral Representation for $U_{3} \times \GL_{2}$
Gelbart and PiatetskiiShapiro constructed
various integral
representations of RankinSel\berg type for groups $G \times
\GL_{n}$,
where $G$
is of split rank $n$. Here we show that their method
can equally well be applied
to the product $U_{3} \times \GL_{2}$, where $U_{3}$
denotes the quasisplit
unitary group in three variables. As an application, we describe which
cuspidal automorphic representations of $U_{3}$ occur
in the Siegel induced
residual spectrum of the quasisplit $U_{4}$.
Categories:11F70, 11F67 

98. CJM 2009 (vol 61 pp. 1118)
 Pontreau, Corentin

Petits points d'une surface
Pour toute sousvari\'et\'e g\'eom\'etriquement irr\'eductible $V$
du grou\pe multiplicatif
$\mathbb{G}_m^n$, on sait qu'en dehors d'un nombre fini de
translat\'es de tores exceptionnels
inclus dans $V$, tous les points sont de hauteur minor\'ee par une
certaine quantit\'e $q(V)^{1}>0$. On conna\^it de plus une borne
sup\'erieure pour la somme des degr\'es de ces translat\'es de
tores pour des valeurs de $q(V)$ polynomiales en le degr\'e de $V$.
Ceci n'est pas le cas si l'on exige une minoration quasioptimale
pour la hauteur des points de $V$, essentiellement lin\'eaire en l'inverse du degr\'e.
Nous apportons ici une r\'eponse partielle \`a ce probl\`eme\,: nous
donnons une majoration de la somme des degr\'es de ces translat\'es de
soustores de codimension $1$ d'une hypersurface $V$. Les r\'esultats,
obtenus dans le cas de $\mathbb{G}_m^3$, mais compl\`etement
explicites, peuvent toutefois s'\'etendre \`a $\mathbb{G}_m^n$,
moyennant quelques petites complications inh\'erentes \`a la dimension
$n$.
Keywords:Hauteur normalisÃ©e, groupe multiplicatif, problÃ¨me de Lehmer, petits points Categories:11G50, 11J81, 14G40 

99. 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 

100. CJM 2009 (vol 61 pp. 828)
 Howard, Benjamin

Twisted GrossZagier Theorems
The theorems of GrossZagier and Zhang relate the N\'eronTate
heights of complex multiplication points on the modular curve $X_0(N)$
(and on Shimura curve analogues) with the central derivatives of
automorphic $L$function. We extend these results to include certain
CM points on modular curves of the form
$X(\Gamma_0(M)\cap\Gamma_1(S))$ (and on Shimura curve analogues).
These results are motivated by applications to Hida theory
that can be found in the companion article
"Central derivatives of $L$functions in Hida families", Math.\ Ann.\
\textbf{399}(2007), 803818.
Categories:11G18, 14G35 
