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176. CJM 2001 (vol 53 pp. 310)

Ito, Hiroshi
On a Product Related to the Cubic Gauss Sum, III
We have seen, in the previous works [5], [6], that the argument of a certain product is closely connected to that of the cubic Gauss sum. Here the absolute value of the product will be investigated.

Keywords:Gauss sum, Lagrange resolvent
Categories:11L05, 11R33

177. CJM 2001 (vol 53 pp. 414)

Rivat, Joël; Sargos, Patrick
Nombres premiers de la forme $\floor{n^c}$
For $c>1$ we denote by $\pi_c(x)$ the number of integers $n \leq x$ such that $\floor{n^c}$ is prime. In 1953, Piatetski-Shapiro has proved that $\pi_c(x) \sim \frac{x}{c\log x}$, $x \rightarrow +\infty$ holds for $c<12/11$. Many authors have extended this range, which measures our progress in exponential sums techniques. In this article we obtain $c < 1.16117\dots\;$.

Categories:11L07, 11L20, 11N05

178. CJM 2001 (vol 53 pp. 122)

Levy, Jason
A Truncated Integral of the Poisson Summation Formula
Let $G$ be a reductive algebraic group defined over $\bQ$, with anisotropic centre. Given a rational action of $G$ on a finite-dimensional vector space $V$, we analyze the truncated integral of the theta series corresponding to a Schwartz-Bruhat function on $V(\bA)$. The Poisson summation formula then yields an identity of distributions on $V(\bA)$. The truncation used is due to Arthur.

Categories:11F99, 11F72

179. CJM 2001 (vol 53 pp. 98)

Khuri-Makdisi, Kamal
On the Curves Associated to Certain Rings of Automorphic Forms
In a 1987 paper, Gross introduced certain curves associated to a definite quaternion algebra $B$ over $\Q$; he then proved an analog of his result with Zagier for these curves. In Gross' paper, the curves were defined in a somewhat {\it ad hoc\/} manner. In this article, we present an interpretation of these curves as projective varieties arising from graded rings of automorphic forms on $B^\times$, analogously to the construction in the Satake compactification. To define such graded rings, one needs to introduce a ``multiplication'' of automorphic forms that arises from the representation ring of $B^\times$. The resulting curves are unions of projective lines equipped with a collection of Hecke correspondences. They parametrize two-dimensional complex tori with quaternionic multiplication. In general, these complex tori are not abelian varieties; they are algebraic precisely when they correspond to $\CM$ points on these curves, and are thus isogenous to a product $E \times E$, where $E$ is an elliptic curve with complex multiplication. For these $\CM$ points one can make a relation between the action of the $p$-th Hecke operator and Frobenius at $p$, similar to the well-known congruence relation of Eichler and Shimura.

Category:11F

180. CJM 2001 (vol 53 pp. 33)

Borwein, Peter; Choi, Kwok-Kwong Stephen
Merit Factors of Polynomials Formed by Jacobi Symbols
We give explicit formulas for the $L_4$ norm (or equivalently for the merit factors) of various sequences of polynomials related to the polynomials $$ f(z) := \sum_{n=0}^{N-1} \leg{n}{N} z^n. $$ and $$ f_t(z) = \sum_{n=0}^{N-1} \leg{n+t}{N} z^n. $$ where $(\frac{\cdot}{N})$ is the Jacobi symbol. Two cases of particular interest are when $N = pq$ is a product of two primes and $p = q+2$ or $p = q+4$. This extends work of H{\o}holdt, Jensen and Jensen and of the authors. This study arises from a number of conjectures of Erd\H{o}s, Littlewood and others that concern the norms of polynomials with $-1,1$ coefficients on the disc. The current best examples are of the above form when $N$ is prime and it is natural to see what happens for composite~$N$.

Keywords:Character polynomial, Class Number, $-1,1$ coefficients, Merit factor, Fekete polynomials, Turyn Polynomials, Littlewood polynomials, Twin Primes, Jacobi Symbols
Categories:11J54, 11B83, 12-04

181. CJM 2000 (vol 52 pp. 1121)

Ballantine, Cristina M.
Ramanujan Type Buildings
We will construct a finite union of finite quotients of the affine building of the group $\GL_3$ over the field of $p$-adic numbers $\mathbb{Q}_p$. We will view this object as a hypergraph and estimate the spectrum of its underlying graph.

Keywords:automorphic representations, buildings
Category:11F70

182. CJM 2000 (vol 52 pp. 1269)

Spriano, Luca
Well Ramified Extensions of Complete Discrete Valuation Fields with Applications to the Kato Conductor
We study extensions $L/K$ of complete discrete valuation fields $K$ with residue field $\oK$ of characteristic $p > 0$, which we do not assume to be perfect. Our work concerns ramification theory for such extensions, in particular we show that all classical properties which are true under the hypothesis {\it ``the residue field extension $\oL/\oK$ is separable''} are still valid under the more general hypothesis that the valuation ring extension is monogenic. We also show that conversely, if classical ramification properties hold true for an extension $L/K$, then the extension of valuation rings is monogenic. These are the ``{\it well ramified}'' extensions. We show that there are only three possible types of well ramified extensions and we give examples. In the last part of the paper we consider, for the three types, Kato's generalization of the conductor, which we show how to bound in certain cases.

Categories:11S, 11S15, 11S20

183. CJM 2000 (vol 52 pp. 737)

Gan, Wee Teck
An Automorphic Theta Module for Quaternionic Exceptional Groups
We construct an automorphic realization of the global minimal representation of quaternionic exceptional groups, using the theory of Eisenstein series, and use this for the study of theta correspondences.

Categories:11F27, 11F70

184. CJM 2000 (vol 52 pp. 804)

Kottwitz, Robert E.; Rogawski, Jonathan D.
The Distributions in the Invariant Trace Formula Are Supported on Characters
J.~Arthur put the trace formula in invariant form for all connected reductive groups and certain disconnected ones. However his work was written so as to apply to the general disconnected case, modulo two missing ingredients. This paper supplies one of those missing ingredients, namely an argument in Galois cohomology of a kind first used by D.~Kazhdan in the connected case.

Categories:22E50, 11S37, 10D40

185. CJM 2000 (vol 52 pp. 673)

Balog, Antal; Wooley, Trevor D.
Sums of Two Squares in Short Intervals
Let $\calS$ denote the set of integers representable as a sum of two squares. Since $\calS$ can be described as the unsifted elements of a sieving process of positive dimension, it is to be expected that $\calS$ has many properties in common with the set of prime numbers. In this paper we exhibit ``unexpected irregularities'' in the distribution of sums of two squares in short intervals, a phenomenon analogous to that discovered by Maier, over a decade ago, in the distribution of prime numbers. To be precise, we show that there are infinitely many short intervals containing considerably more elements of $\calS$ than expected, and infinitely many intervals containing considerably fewer than expected.

Keywords:sums of two squares, sieves, short intervals, smooth numbers
Categories:11N36, 11N37, 11N25

186. CJM 2000 (vol 52 pp. 833)

Mináč, Ján; Smith, Tara L.
W-Groups under Quadratic Extensions of Fields
To each field $F$ of characteristic not $2$, one can associate a certain Galois group $\G_F$, the so-called W-group of $F$, which carries essentially the same information as the Witt ring $W(F)$ of $F$. In this paper we investigate the connection between $\wg$ and $\G_{F(\sqrt{a})}$, where $F(\sqrt{a})$ is a proper quadratic extension of $F$. We obtain a precise description in the case when $F$ is a pythagorean formally real field and $a = -1$, and show that the W-group of a proper field extension $K/F$ is a subgroup of the W-group of $F$ if and only if $F$ is a formally real pythagorean field and $K = F(\sqrt{-1})$. This theorem can be viewed as an analogue of the classical Artin-Schreier's theorem describing fields fixed by finite subgroups of absolute Galois groups. We also obtain precise results in the case when $a$ is a double-rigid element in $F$. Some of these results carry over to the general setting.

Categories:11E81, 12D15

187. CJM 2000 (vol 52 pp. 613)

Ou, Zhiming M.; Williams, Kenneth S.
Small Solutions of $\phi_1 x_1^2 + \cdots + \phi_n x_n^2 = 0$
Let $\phi_1,\dots,\phi_n$ $(n\geq 2)$ be nonzero integers such that the equation $$ \sum_{i=1}^n \phi_i x_i^2 = 0 $$ is solvable in integers $x_1,\dots,x_n$ not all zero. It is shown that there exists a solution satisfying $$ 0 < \sum_{i=1}^n |\phi_i| x_i^2 \leq 2 |\phi_1 \cdots \phi_n|, $$ and that the constant 2 is best possible.

Keywords:small solutions, diagonal quadratic forms
Category:11E25

188. CJM 2000 (vol 52 pp. 369)

Granville, Andrew; Mollin, R. A.; Williams, H. C.
An Upper Bound on the Least Inert Prime in a Real Quadratic Field
It is shown by a combination of analytic and computational techniques that for any positive fundamental discriminant $D > 3705$, there is always at least one prime $p < \sqrt{D}/2$ such that the Kronecker symbol $\left(D/p\right) = -1$.

Categories:11R11, 11Y40

189. CJM 2000 (vol 52 pp. 31)

Chan, Heng Huat; Liaw, Wen-Chin
On Russell-Type Modular Equations
In this paper, we revisit Russell-type modular equations, a collection of modular equations first studied systematically by R.~Russell in 1887. We give a proof of Russell's main theorem and indicate the relations between such equations and the constructions of Hilbert class fields of imaginary quadratic fields. Motivated by Russell's theorem, we state and prove its cubic analogue which allows us to construct Russell-type modular equations in the theory of signature~$3$.

Categories:33D10, 33C05, 11F11

190. CJM 2000 (vol 52 pp. 47)

Chinburg, T.; Kolster, M.; Snaith, V. P.
Comparison of $K$-Theory Galois Module Structure Invariants
We prove that two, apparently different, class-group valued Galois module structure invariants associated to the algebraic $K$-groups of rings of algebraic integers coincide. This comparison result is particularly important in making explicit calculations.

Categories:11S99, 19F15, 19F27

191. CJM 2000 (vol 52 pp. 172)

Mao, Zhengyu; Rallis, Stephen
Cubic Base Change for $\GL(2)$
We prove a relative trace formula that establishes the cubic base change for $\GL(2)$. One also gets a classification of the image of base change. The case when the field extension is nonnormal gives an example where a trace formula is used to prove lifting which is not endoscopic.

Categories:11F70, 11F72

192. CJM 1999 (vol 51 pp. 1258)

Baake, Michael; Moody, Robert V.
Similarity Submodules and Root Systems in Four Dimensions
Lattices and $\ZZ$-modules in Euclidean space possess an infinitude of subsets that are images of the original set under similarity transformation. We classify such self-similar images according to their indices for certain 4D examples that are related to 4D root systems, both crystallographic and non-crystallographic. We encapsulate their statistics in terms of Dirichlet series generating functions and derive some of their asymptotic properties.

Categories:11S45, 11H05, 52C07

193. CJM 1999 (vol 51 pp. 1307)

Johnson, Norman W.; Weiss, Asia Ivić
Quadratic Integers and Coxeter Groups
Matrices whose entries belong to certain rings of algebraic integers can be associated with discrete groups of transformations of inversive $n$-space or hyperbolic $(n+1)$-space $\mbox{H}^{n+1}$. For small $n$, these may be Coxeter groups, generated by reflections, or certain subgroups whose generators include direct isometries of $\mbox{H}^{n+1}$. We show how linear fractional transformations over rings of rational and (real or imaginary) quadratic integers are related to the symmetry groups of regular tilings of the hyperbolic plane or 3-space. New light is shed on the properties of the rational modular group $\PSL_2 (\bbZ)$, the Gaussian modular (Picard) group $\PSL_2 (\bbZ[{\it i}])$, and the Eisenstein modular group $\PSL_2 (\bbZ[\omega ])$.

Categories:11F06, 20F55, 20G20, 20H10, 22E40

194. CJM 1999 (vol 51 pp. 1020)

Kozlov, Dmitry N.
On Functions Satisfying Modular Equations for Infinitely Many Primes
In this paper we study properties of the functions which satisfy modular equations for infinitely many primes. The two main results are: \begin{enumerate} \item[1)] every such function is analytic in the upper half plane; \item[2)] if such function takes the same value in two different points $z_1$ and $z_2$ then there exists an $f$-preserving analytic bijection between neighbourhoods of $z_1$ and $z_2$. \end{enumerate}

Category:11Mxx

195. CJM 1999 (vol 51 pp. 952)

Deitmar, Anton; Hoffmann, Werner
On Limit Multiplicities for Spaces of Automorphic Forms
Let $\Gamma$ be a rank-one arithmetic subgroup of a semisimple Lie group~$G$. For fixed $K$-Type, the spectral side of the Selberg trace formula defines a distribution on the space of infinitesimal characters of~$G$, whose discrete part encodes the dimensions of the spaces of square-integrable $\Gamma$-automorphic forms. It is shown that this distribution converges to the Plancherel measure of $G$ when $\Ga$ shrinks to the trivial group in a certain restricted way. The analogous assertion for cocompact lattices $\Gamma$ follows from results of DeGeorge-Wallach and Delorme.

Keywords:limit multiplicities, automorphic forms, noncompact quotients, Selberg trace formula, functional calculus
Categories:11F72, 22E30, 22E40, 43A85, 58G25

196. CJM 1999 (vol 51 pp. 835)

Kim, Henry H.
Langlands-Shahidi Method and Poles of Automorphic $L$-Functions: Application to Exterior Square $L$-Functions
In this paper we use Langlands-Shahidi method and the result of Langlands which says that non self-conjugate maximal parabolic subgroups do not contribute to the residual spectrum, to prove the holomorphy of several \emph{completed} automorphic $L$-functions on the whole complex plane which appear in constant terms of the Eisenstein series. They include the exterior square $L$-functions of $\GL_n$, $n$ odd, the Rankin-Selberg $L$-functions of $\GL_n\times \GL_m$, $n\ne m$, and $L$-functions $L(s,\sigma,r)$, where $\sigma$ is a generic cuspidal representation of $\SO_{10}$ and $r$ is the half-spin representation of $\GSpin(10, \mathbb{C})$. The main part is proving the holomorphy and non-vanishing of the local normalized intertwining operators by reducing them to natural conjectures in harmonic analysis, such as standard module conjecture.

Categories:11F, 22E

197. CJM 1999 (vol 51 pp. 771)

Flicker, Yuval Z.
Stable Bi-Period Summation Formula and Transfer Factors
This paper starts by introducing a bi-periodic summation formula for automorphic forms on a group $G(E)$, with periods by a subgroup $G(F)$, where $E/F$ is a quadratic extension of number fields. The split case, where $E = F \oplus F$, is that of the standard trace formula. Then it introduces a notion of stable bi-conjugacy, and stabilizes the geometric side of the bi-period summation formula. Thus weighted sums in the stable bi-conjugacy class are expressed in terms of stable bi-orbital integrals. These stable integrals are on the same endoscopic groups $H$ which occur in the case of standard conjugacy. The spectral side of the bi-period summation formula involves periods, namely integrals over the group of $F$-adele points of $G$, of cusp forms on the group of $E$-adele points on the group $G$. Our stabilization suggests that such cusp forms---with non vanishing periods---and the resulting bi-period distributions associated to ``periodic'' automorphic forms, are related to analogous bi-period distributions associated to ``periodic'' automorphic forms on the endoscopic symmetric spaces $H(E)/H(F)$. This offers a sharpening of the theory of liftings, where periods play a key role. The stabilization depends on the ``fundamental lemma'', which conjectures that the unit elements of the Hecke algebras on $G$ and $H$ have matching orbital integrals. Even in stating this conjecture, one needs to introduce a ``transfer factor''. A generalization of the standard transfer factor to the bi-periodic case is introduced. The generalization depends on a new definition of the factors even in the standard case. Finally, the fundamental lemma is verified for $\SL(2)$.

Categories:11F72, 11F70, 14G27, 14L35

198. CJM 1999 (vol 51 pp. 266)

Deitmar, Anton; Hoffman, Werner
Spectral Estimates for Towers of Noncompact Quotients
We prove a uniform upper estimate on the number of cuspidal eigenvalues of the $\Ga$-automorphic Laplacian below a given bound when $\Ga$ varies in a family of congruence subgroups of a given reductive linear algebraic group. Each $\Ga$ in the family is assumed to contain a principal congruence subgroup whose index in $\Ga$ does not exceed a fixed number. The bound we prove depends linearly on the covolume of $\Ga$ and is deduced from the analogous result about the cut-off Laplacian. The proof generalizes the heat-kernel method which has been applied by Donnelly in the case of a fixed lattice~$\Ga$.

Categories:11F72, 58G25, 22E40

199. CJM 1999 (vol 51 pp. 225)

Betke, U.; Böröczky, K.
Asymptotic Formulae for the Lattice Point Enumerator
Let $M$ be a convex body such that the boundary has positive curvature. Then by a well developed theory dating back to Landau and Hlawka for large $\lambda$ the number of lattice points in $\lambda M$ is given by $G(\lambda M) =V(\lambda M) + O(\lambda^{d-1-\varepsilon (d)})$ for some positive $\varepsilon(d)$. Here we give for general convex bodies the weaker estimate \[ \left| G(\lambda M) -V(\lambda M) \right | \le \frac{1}{2} S_{\Z^d}(M) \lambda^{d-1}+o(\lambda^{d-1}) \] where $S_{\Z^d}(M)$ denotes the lattice surface area of $M$. The term $S_{\Z^d}(M)$ is optimal for all convex bodies and $o(\lambda^{d-1})$ cannot be improved in general. We prove that the same estimate even holds if we allow small deformations of $M$. Further we deal with families $\{P_\lambda\}$ of convex bodies where the only condition is that the inradius tends to infinity. Here we have \[ \left| G(P_\lambda)-V(P_\lambda) \right| \le dV(P_\lambda,K;1)+o \bigl( S(P_\lambda) \bigr) \] where the convex body $K$ satisfies some simple condition, $V(P_\lambda,K;1)$ is some mixed volume and $S(P_\lambda)$ is the surface area of $P_\lambda$.

Categories:11P21, 52C07

200. CJM 1999 (vol 51 pp. 10)

Chacron, M.; Tignol, J.-P.; Wadsworth, A. R.
Tractable Fields
A field $F$ is said to be tractable when a condition described below on the simultaneous representation of quaternion algebras holds over $F$. It is shown that a global field $F$ is tractable i{f}f $F$ has at most one dyadic place. Several other examples of tractable and nontractable fields are given.

Categories:12E15, 11R52
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