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51. 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 non-CM 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 non-CM families for which every classical member of weight at least two has a locally indecomposable Galois representation.

Category:11F80

52. CJM 2010 (vol 63 pp. 241)

Essouabri, Driss; Matsumoto, Kohji; Tsumura, Hirofumi
Multiple Zeta-Functions Associated with Linear Recurrence Sequences and the Vectorial Sum Formula
We prove the holomorphic continuation of certain multi-variable multiple zeta-functions whose coefficients satisfy a suitable recurrence condition. In fact, we introduce more general vectorial zeta-functions and prove their holomorphic continuation. Moreover, we show a vectorial sum formula among those vectorial zeta-functions from which some generalizations of the classical sum formula can be deduced.

Keywords:Zeta-functions, holomorphic continuation, recurrence sequences, Fibonacci numbers, sum formulas
Categories:11M41, 40B05, 11B39

53. CJM 2010 (vol 63 pp. 136)

Gun, Sanoli; Murty, M. Ram; Rath, Purusottam
Transcendental Nature of Special Values of $L$-Functions
In this paper, we study the non-vanishing and transcendence of special values of a varying class of $L$-functions and their derivatives. This allows us to investigate the transcendence of Petersson norms of certain weight one modular forms.

Categories:11J81, 11J86, 11J91

54. 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 $$ \|F\|_{\lambda,p}=\sup_{0\leq r < 1}(1-r^{2})^{\mathcal{R}e[i\lambda]-\frac{n}{2}+l}\Big[\int_{S}|F(ru)|^{p}du \Big]^{\frac{1}{p}} < +\infty. $$

Keywords:Lie ball, Shilov boundary, Fatou's theorem, hyperfuctions, parabolic subgroup, homogeneous line bundle
Categories:32A45, 30E20, 33C67, 33C60, 33C55, 32A25, 33C75, 11K70

55. 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 non-trivial 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

56. 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 non-trivial 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 non-abelian) 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 non-commutative Iwasawa algebras.

Categories:11R42, 11R23, 11R70

57. 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

58. 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 Graham--Ringrose and Iwaniec, we improve the Pólya--Vinogradov inequality for characters with smooth conductor.

Categories:11L40, 11M06

59. CJM 2010 (vol 62 pp. 914)

Zorn, Christian
Reducibility of the Principal Series for Sp~2(F) over a p-adic 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 Tadić 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

60. 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

61. CJM 2010 (vol 62 pp. 543)

Hare, Kevin G.
More Variations on the Sierpiński Sieve
This paper answers a question of Broomhead, Montaldi and Sidorov about the existence of gaskets of a particular type related to the Sierpiński 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

62. 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 non-singularity 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 square-free 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, non-singularity, integral basis, genus, signature of a place, class number
Categories:14H05, 11R58, 14H45, 11G20, 11G30, 11R16, 11R29

63. 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 Dieudonné 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 Bruhat--Tits 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

64. CJM 2010 (vol 62 pp. 563)

Ishii, Taku
Whittaker Functions on Real Semisimple Lie Groups of Rank Two
We give explicit formulas for Whittaker functions on real semisimple Lie groups of real rank two belonging to the class one principal series representations. By using these formulas we compute certain archimedean zeta integrals.

Categories:11F70, 22E30

65. CJM 2009 (vol 62 pp. 582)

Konyagin, Sergei V.; Pomerance, Carl; Shparlinski, Igor E.
On the Distribution of Pseudopowers
An x-pseudopower 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

66. 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 Galois-equivariant $L$-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, Rubin-Stark conjecture
Categories:11R42, 11F30

67. CJM 2009 (vol 62 pp. 400)

Prasanna, Kartik
On p-Adic Properties of Central L-Values 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 semi-stable elliptic curve $E$ of conductor $N$. A consideration of the conjecture of Birch and Swinnerton-Dyer 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 half-integral weight. In this context, we formulate a conjecture on congruences of half-integral weight forms and explain its relevance to the problem of $p$-indivisibility of $L$-values of quadratic twists.

Categories:11F40, 11F67, 11G05

68. CJM 2009 (vol 62 pp. 456)

Yang, Tonghai
The Chowla—Selberg 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

69. CJM 2009 (vol 61 pp. 1383)

Wambach, Eric
Integral Representation for $U_{3} \times \GL_{2}$
Gelbart and Piatetskii-Shapiro constructed various integral representations of Rankin--Sel\-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

70. 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

71. 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 Hermite--Pad\'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 Kubota--Leopold $p$-adic zeta function.

Categories:41A10, 41A21, 11J72

72. 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, bi-quadratic, and tri-quadratic 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

73. CJM 2009 (vol 61 pp. 1118)

Pontreau, Corentin
Petits points d'une surface
Pour toute sous-vari\'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 quasi-optimale 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 sous-tores 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 normalisée, groupe multiplicatif, problème de Lehmer, petits points
Categories:11G50, 11J81, 14G40

74. CJM 2009 (vol 61 pp. 828)

Howard, Benjamin
Twisted Gross--Zagier Theorems
The theorems of Gross--Zagier and Zhang relate the N\'eron--Tate 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), 803--818.

Categories:11G18, 14G35

75. 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 quasi-split. 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
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