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176. CMB 2000 (vol 43 pp. 304)

Darmon, Henri; Mestre, Jean-François
Courbes hyperelliptiques à multiplications réelles et une construction de Shih
Soient $r$ et $p$ deux nombres premiers distincts, soit $K = \Q(\cos \frac{2\pi}{r})$, et soit $\F$ le corps r\'esiduel de $K$ en une place au-dessus de $p$. Lorsque l'image de $(2 - 2\cos \frac{2\pi}{r})$ dans $\F$ n'est pas un carr\'e, nous donnons une construction g\'eom\'etrique d'une extension r\'eguliere de $K(t)$ de groupe de Galois $\PSL_2 (\F)$. Cette extension correspond \`a un rev\^etement de $\PP^1_{/K}$ de \og{} signature $(r,p,p)$ \fg{} au sens de [3, sec.~6.3], et son existence est pr\'edite par le crit\`ere de rigidit\'e de Belyi, Fried, Thompson et Matzat. Sa construction s'obtient en tordant la representation galoisienne associ\'ee aux points d'ordre $p$ d'une famille de vari\'et\'es ab\'eliennes \`a multiplications r\'eelles par $K$ d\'ecouverte par Tautz, Top et Verberkmoes [6]. Ces vari\'et\'es ab\'eliennes sont d\'efinies sur un corps quadratique, et sont isog\`enes \`a leur conjugu\'e galoisien. Notre construction g\'en\'eralise une m\'ethode de Shih [4], [5], que l'on retrouve quand $r = 2$ et $r = 3$. Let $r$ and $p$ be distinct prime numbers, let $K = \Q(\cos \frac{2\pi}{r})$, and let $\F$ be the residue field of $K$ at a place above $p$. When the image of $(2 - 2\cos \frac{2\pi}{r})$ in $\F$ is not a square, we describe a geometric construction of a regular extension of $K(t)$ with Galois group $\PSL_2 (\F)$. This extension corresponds to a covering of $\PP^1_{/K}$ of ``signature $(r,p,p)$'' in the sense of [3, sec.~6.3], and its existence is predicted by the rigidity criterion of Belyi, Fried, Thompson and Matzat. Its construction is obtained by twisting the mod $p$ galois representation attached to a family of abelian varieties with real multiplications by $K$ discovered by Tautz, Top and Verberkmoes [6]. These abelian varieties are defined in general over a quadratic field, and are isogenous to their galois conjugate. Our construction generalises a method of Shih [4], [5], which one recovers when $r = 2$ and $r = 3$.

Categories:11G30, 14H25

177. CMB 2000 (vol 43 pp. 236)

Voloch, José Felipe
On a Question of Buium
We prove that $\{(n^p-n)/p\}_p \in \prod_p \mathbf{F}_p$, with $p$ ranging over all primes, is independent of $1$ over the integers, assuming a conjecture in elementary number theory generalizing the infinitude of Mersenne primes. This answers a question of Buium. We also prove a generalization.

Category:11A07

178. CMB 2000 (vol 43 pp. 239)

Yu, Gang
On the Number of Divisors of the Quadratic Form $m^2+n^2$
For an integer $n$, let $d(n)$ denote the ordinary divisor function. This paper studies the asymptotic behavior of the sum $$ S(x) := \sum_{m\leq x, n\leq x} d(m^2 + n^2). $$ It is proved in the paper that, as $x \to \infty$, $$ S(x) := A_1 x^2 \log x + A_2 x^2 + O_\epsilon (x^{\frac32 + \epsilon}), $$ where $A_1$ and $A_2$ are certain constants and $\epsilon$ is any fixed positive real number. The result corrects a false formula given in a paper of Gafurov concerning the same problem, and improves the error $O \bigl( x^{\frac53} (\log x)^9 \bigr)$ claimed by Gafurov.

Keywords:divisor, large sieve, exponential sums
Categories:11G05, 14H52

179. CMB 2000 (vol 43 pp. 218)

Mollin, R. A.; van der Poorten, A. J.
Continued Fractions, Jacobi Symbols, and Quadratic Diophantine Equations
The results herein continue observations on norm form equations and continued fractions begun and continued in the works \cite{chows}--\cite{mol}, and \cite{mvdpw}--\cite{schinz}.

Categories:11R11, 11D09, 11R29, 11R65

180. CMB 2000 (vol 43 pp. 115)

Schmutz Schaller, Paul
Perfect Non-Extremal Riemann Surfaces
An infinite family of perfect, non-extremal Riemann surfaces is constructed, the first examples of this type of surfaces. The examples are based on normal subgroups of the modular group $\PSL(2,{\sf Z})$ of level $6$. They provide non-Euclidean analogues to the existence of perfect, non-extremal positive definite quadratic forms. The analogy uses the function {\it syst\/} which associates to every Riemann surface $M$ the length of a systole, which is a shortest closed geodesic of $M$.

Categories:11H99, 11F06, 30F45

181. CMB 1999 (vol 42 pp. 427)

Berndt, Bruce C.; Chan, Heng Huat
Ramanujan and the Modular $j$-Invariant
A new infinite product $t_n$ was introduced by S.~Ramanujan on the last page of his third notebook. In this paper, we prove Ramanujan's assertions about $t_n$ by establishing new connections between the modular $j$-invariant and Ramanujan's cubic theory of elliptic functions to alternative bases. We also show that for certain integers $n$, $t_n$ generates the Hilbert class field of $\mathbb{Q} (\sqrt{-n})$. This shows that $t_n$ is a new class invariant according to H.~Weber's definition of class invariants.

Keywords:modular functions, the Borweins' cubic theta-functions, Hilbert class fields
Categories:33C05, 33E05, 11R20, 11R29

182. CMB 1999 (vol 42 pp. 441)

Berrizbeitia, P.; Elliott, P. D. T. A.
Product Bases for the Rationals
A sequence of positive rationals generates a subgroup of finite index in the multiplicative positive rationals, and group product representations by the sequence need only a bounded number of terms, if and only if certain related sequences have densities uniformly bounded from below.

Categories:11N99, 11N05

183. CMB 1999 (vol 42 pp. 263)

Choie, Youngju; Lee, Min Ho
Mellin Transforms of Mixed Cusp Forms
We define generalized Mellin transforms of mixed cusp forms, show their convergence, and prove that the function obtained by such a Mellin transform of a mixed cusp form satisfies a certain functional equation. We also prove that a mixed cusp form can be identified with a holomorphic form of the highest degree on an elliptic variety.

Categories:11F12, 11F66, 11M06, 14K05

184. CMB 1999 (vol 42 pp. 393)

Savin, Gordan
A Class of Supercuspidal Representations of $G_2(k)$
Let $H$ be an exceptional, adjoint group of type $E_6$ and split rank 2, over a $p$-adic field $k$. In this article we discuss the restriction of the minimal representation of $H$ to a dual pair $\PD^{\times}\times G_2(k)$, where $D$ is a division algebra of dimension 9 over $k$. In particular, we discover an interesting class of supercuspidal representations of $G_2(k)$.

Categories:22E35, 22E50, 11F70

185. CMB 1999 (vol 42 pp. 129)

Baker, Andrew
Hecke Operations and the Adams $E_2$-Term Based on Elliptic Cohomology
Hecke operators are used to investigate part of the $\E_2$-term of the Adams spectral sequence based on elliptic homology. The main result is a derivation of $\Ext^1$ which combines use of classical Hecke operators and $p$-adic Hecke operators due to Serre.

Keywords:Adams spectral sequence, elliptic cohomology, Hecke operators
Categories:55N20, 55N22, 55T15, 11F11, 11F25

186. CMB 1999 (vol 42 pp. 25)

Brown, Tom C.; Graham, Ronald L.; Landman, Bruce M.
On the Set of Common Differences in van der Waerden's Theorem on Arithmetic Progressions
Analogues of van der Waerden's theorem on arithmetic progressions are considered where the family of all arithmetic progressions, $\AP$, is replaced by some subfamily of $\AP$. Specifically, we want to know for which sets $A$, of positive integers, the following statement holds: for all positive integers $r$ and $k$, there exists a positive integer $n= w'(k,r)$ such that for every $r$-coloring of $[1,n]$ there exists a monochromatic $k$-term arithmetic progression whose common difference belongs to $A$. We will call any subset of the positive integers that has the above property {\em large}. A set having this property for a specific fixed $r$ will be called {\em $r$-large}. We give some necessary conditions for a set to be large, including the fact that every large set must contain an infinite number of multiples of each positive integer. Also, no large set $\{a_{n}: n=1,2,\dots\}$ can have $\liminf\limits_{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}} > 1$. Sufficient conditions for a set to be large are also given. We show that any set containing $n$-cubes for arbitrarily large $n$, is a large set. Results involving the connection between the notions of ``large'' and ``2-large'' are given. Several open questions and a conjecture are presented.

Categories:11B25, 05D10

187. CMB 1999 (vol 42 pp. 78)

González, Josep
Fermat Jacobians of Prime Degree over Finite Fields
We study the splitting of Fermat Jacobians of prime degree $\ell$ over an algebraic closure of a finite field of characteristic $p$ not equal to $\ell$. We prove that their decomposition is determined by the residue degree of $p$ in the cyclotomic field of the $\ell$-th roots of unity. We provide a numerical criterion that allows to compute the absolutely simple subvarieties and their multiplicity in the Fermat Jacobian.

Categories:11G20, 14H40

188. CMB 1999 (vol 42 pp. 68)

Gittenberger, Bernhard; Thuswaldner, Jörg M.
The Moments of the Sum-Of-Digits Function in Number Fields
We consider the asymptotic behavior of the moments of the sum-of-digits function of canonical number systems in number fields. Using Delange's method we obtain the main term and smaller order terms which contain periodic fluctuations.

Categories:11A63, 11N60

189. CMB 1998 (vol 41 pp. 488)

Sun, Heng
Remarks on certain metaplectic groups
We study metaplectic coverings of the adelized group of a split connected reductive group $G$ over a number field $F$. Assume its derived group $G'$ is a simply connected simple Chevalley group. The purpose is to provide some naturally defined sections for the coverings with good properties which might be helpful when we carry some explicit calculations in the theory of automorphic forms on metaplectic groups. Specifically, we \begin{enumerate} \item construct metaplectic coverings of $G({\Bbb A})$ from those of $G'({\Bbb A})$; \item for any non-archimedean place $v$, show the section for a covering of $G(F_{v})$ constructed from a Steinberg section is an isomorphism, both algebraically and topologically in an open subgroup of $G(F_{v})$; \item define a global section which is a product of local sections on a maximal torus, a unipotent subgroup and a set of representatives for the Weyl group.

Categories:20G10, 11F75

190. CMB 1998 (vol 41 pp. 335)

191. CMB 1998 (vol 41 pp. 328)

Mollin, R. A.
Class number one and prime-producing quadratic polynomials revisited
Over a decade ago, this author produced class number one criteria for real quadratic fields in terms of prime-producing quadratic polynomials. The purpose of this article is to revisit the problem from a new perspective with new criteria. We look at the more general situation involving arbitrary real quadratic orders rather than the more restrictive field case, and use the interplay between the various orders to provide not only more general results, but also simpler proofs.

Categories:11R11, 11R09, 11R29

192. CMB 1998 (vol 41 pp. 158)

Gaál, István
Power integral bases in composits of number fields
In the present paper we consider the problem of finding power integral bases in number fields which are composits of two subfields with coprime discriminants. Especially, we consider imaginary quadratic extensions of totally real cyclic number fields of prime degree. As an example we solve the index form equation completely in a two parametric family of fields of degree $10$ of this type.

Categories:11D57, 11R21

193. CMB 1998 (vol 41 pp. 187)

Loh, W. K. A.
Exponential sums on reduced residue systems
The aim of this article is to obtain an upper bound for the exponential sums $\sum e(f(x)/q)$, where the summation runs from $x=1$ to $x=q$ with $(x,q)=1$ and $e(\alpha)$ denotes $\exp(2\pi i\alpha)$. We shall show that the upper bound depends only on the values of $q$ and $s$, %% where $s$ is the number of terms in the polynomial $f(x)$.

Category:11L07

194. CMB 1998 (vol 41 pp. 15)

Brown, Tom; Shiue, Peter Jau-Shyong; Yu, X. Y.
Sequences with translates containing many primes
Garrison [3], Forman [2], and Abel and Siebert [1] showed that for all positive integers $k$ and $N$, there exists a positive integer $\lambda$ such that $n^k+\lambda$ is prime for at least $N$ positive integers $n$. In other words, there exists $\lambda$ such that $n^k+\lambda$ represents at least $N$ primes. We give a quantitative version of this result. We show that there exists $\lambda \leq x^k$ such that $n^k+\lambda$, $1\leq n\leq x$, represents at least $(\frac 1k+o(1)) \pi(x)$ primes, as $x\rightarrow \infty$. We also give some related results.

Category:11A48

195. CMB 1998 (vol 41 pp. 71)

Hurrelbrink, Jurgen; Rehmann, Ulf
Splitting patterns and trace forms
The splitting pattern of a quadratic form $q$ over a field $k$ consists of all distinct Witt indices that occur for $q$ over extension fields of $k$. In small dimensions, the complete list of splitting patterns of quadratic forms is known. We show that {\it all\/} splitting patterns of quadratic forms of dimension at most nine can be realized by trace forms.

Keywords:Quadratic forms, Witt indices, generic splitting.
Category:11E04

196. CMB 1998 (vol 41 pp. 86)

Lubinsky, D. S.
On \lowercase{$q$}-exponential functions for \lowercase{$|q| =1$}
We discuss the $q$-exponential functions $e_q$, $E_q$ for $q$ on the unit circle, especially their continuity in $q$, and analogues of the limit relation $ \lim_{q\rightarrow 1}e_q((1-q)z)=e^z$.

Keywords:$q$-series, $q$-exponentials
Categories:33D05, 11A55, 11K70

197. CMB 1998 (vol 41 pp. 125)

Boyd, David W.
Uniform approximation to Mahler's measure in several variables
If $f(x_1,\dots,x_k)$ is a polynomial with complex coefficients, the Mahler measure of $f$, $M(f)$ is defined to be the geometric mean of $|f|$ over the $k$-torus $\Bbb T^k$. We construct a sequence of approximations $M_n(f)$ which satisfy $-d2^{-n}\log 2 + \log M_n(f) \le \log M(f) \le \log M_n(f)$. We use these to prove that $M(f)$ is a continuous function of the coefficients of $f$ for polynomials of fixed total degree $d$. Since $M_n(f)$ can be computed in a finite number of arithmetic operations from the coefficients of $f$ this also demonstrates an effective (but impractical) method for computing $M(f)$ to arbitrary accuracy.

Categories:11R06, 11K16, 11Y99

198. CMB 1997 (vol 40 pp. 402)

Carpenter, Jenna P.
On the Preservation of Root Numbers and the Behavior of Weil Characters Under Reciprocity Equivalence
This paper studies how the local root numbers and the Weil additive characters of the Witt ring of a number field behave under reciprocity equivalence. Given a reciprocity equivalence between two fields, at each place we define a local square class which vanishes if and only if the local root numbers are preserved. Thus this local square class serves as a local obstruction to the preservation of local root numbers. We establish a set of necessary and sufficient conditions for a selection of local square classes (one at each place) to represent a global square class. Then, given a reciprocity equivalence that has a finite wild set, we use these conditions to show that the local square classes combine to give a global square class which serves as a global obstruction to the preservation of all root numbers. Lastly, we use these results to study the behavior of Weil characters under reciprocity equivalence.

Categories:11E12, 11E08

199. CMB 1997 (vol 40 pp. 385)

Bae, Sunghan; Kang, Pyung-Lyun
Elliptic units and class fields of global function fields
Elliptic units of global function fields were first studied by D.~Hayes in the case that $\deg\infty$ is assumed to be $1$, and he obtained some class number formulas using elliptic units. We generalize Hayes' results to the case that $\deg\infty$ is arbitrary.

Categories:11R58, 11G09

200. CMB 1997 (vol 40 pp. 498)

Selvaraj, Chikkanna; Selvaraj, Suguna
Matrix transformations based on Dirichlet convolution
This paper is a study of summability methods that are based on Dirichlet convolution. If $f(n)$ is a function on positive integers and $x$ is a sequence such that $\lim_{n\to \infty} \sum_{k\le n} {1\over k}(f\ast x)(k) =L$, then $x$ is said to be {\it $A_f$-summable\/} to $L$. The necessary and sufficient condition for the matrix $A_f$ to preserve bounded variation of sequences is established. Also, the matrix $A_f$ is investigated as $\ell - \ell$ and $G-G$ mappings. The strength of the $A_f$-matrix is also discussed.

Categories:11A25, 40A05, 40C05, 40D05
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