76. 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 

77. 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 

78. 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 

79. 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 

80. 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 

81. 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 

82. CJM 2010 (vol 62 pp. 563)
83. 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 

84. 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 

85. 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 

86. 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 

87. 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 

88. 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 

89. 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 

90. 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 

91. 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 

92. CJM 2009 (vol 61 pp. 779)
 Grbac, Neven

Residual Spectra of Split Classical Groups and their Inner Forms
This paper is concerned with the residual spectrum of the
hermitian quaternionic classical groups $G_n'$ and $H_n'$ defined
as algebraic groups for a quaternion algebra over an algebraic
number field. Groups $G_n'$ and
$H_n'$ are not quasisplit. They are inner forms of the split
groups $\SO_{4n}$ and $\Sp_{4n}$. Hence, the parts of the residual
spectrum of $G_n'$ and $H_n'$ obtained in this paper are compared
to the corresponding parts for the split groups $\SO_{4n}$ and
$\Sp_{4n}$.
Categories:11F70, 22E55 

93. 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 

94. CJM 2009 (vol 61 pp. 617)
 Kim, Wook

Square Integrable Representations and the Standard Module Conjecture for General Spin Groups
In this paper we study square integrable representations and
$L$functions for quasisplit general spin groups over a $p$adic
field. In the first part, the holomorphy of $L$functions in a half
plane is proved by using a variant form of Casselman's square
integrability criterion and the LanglandsShahidi method. The
remaining part focuses on the proof of the standard module
conjecture. We generalize Mui\'c's idea via the LanglandsShahidi method
towards a proof of the conjecture. It is used in the work of M. Asgari
and F. Shahidi on generic transfer for general spin groups.
Categories:11F70, 11F85 

95. CJM 2009 (vol 61 pp. 481)
 Banks, William D.; Garaev, Moubariz Z.; Luca, Florian; Shparlinski, Igor E.

Uniform Distribution of Fractional Parts Related to Pseudoprimes
We estimate exponential sums with the Fermatlike quotients
$$
f_g(n) = \frac{g^{n1}  1}{n} \quad\text{and}\quad h_g(n)=\frac{g^{n1}1}{P(n)},
$$
where $g$ and $n$ are positive integers, $n$ is composite, and
$P(n)$ is the largest prime factor of $n$. Clearly, both $f_g(n)$
and $h_g(n)$ are integers if $n$ is a Fermat pseudoprime to base
$g$, and if $n$ is a Carmichael number, this is true for all $g$
coprime to $n$. Nevertheless, our bounds imply that the fractional
parts $\{f_g(n)\}$ and $\{h_g(n)\}$ are uniformly distributed, on
average over~$g$ for $f_g(n)$, and individually for $h_g(n)$. We
also obtain similar results with the functions ${\widetilde f}_g(n)
= gf_g(n)$ and ${\widetilde h}_g(n) = gh_g(n)$.
Categories:11L07, 11N37, 11N60 

96. 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 

97. 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 

98. CJM 2009 (vol 61 pp. 674)
 Pollack, David; Pollack, Robert

A Construction of Rigid Analytic Cohomology Classes for Congruence Subgroups of $\SL_3(\mathbb Z)$
We give a constructive proof, in the special case of ${\rm GL}_3$, of
a theorem of Ash and Stevens which compares overconvergent cohomology
to classical cohomology. Namely, we show that every ordinary
classical Heckeeigenclass can be lifted uniquely to a rigid analytic
eigenclass. Our basic method builds on the ideas of M. Greenberg; we
first form an arbitrary lift of the classical eigenclass to a
distributionvalued cochain. Then, by appropriately iterating the
$U_p$operator, we produce a cocycle whose image in cohomology is the
desired eigenclass. The constructive nature of this proof makes it
possible to perform computer computations to approximate these
interesting overconvergent eigenclasses.
Categories:11F75, 11F85 

99. 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 

100. CJM 2009 (vol 61 pp. 465)
 Woodford, Roger

On Partitions into Powers of Primes and Their Difference Functions
In this paper, we extend the approach first outlined by Hardy and
Ramanujan for calculating the asymptotic formulae for the number of
partitions into $r$th powers of primes, $p_{\mathbb{P}^{(r)}}(n)$,
to include their difference functions. In doing so, we rectify an
oversight of said authors, namely that the first difference function
is perforce positive for all values of $n$, and include the
magnitude of the error term.
Categories:05A17, 11P81 
