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201. 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

202. CMB 1998 (vol 41 pp. 335)

 Codecà, P.; Nair, M.

203. 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

204. 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

205. 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

206. 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

207. 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, $q$-exponentialsCategories:33D05, 11A55, 11K70

208. 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

209. 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

210. 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

211. 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

212. 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

213. CMB 1997 (vol 40 pp. 364)

Narayanan, Sridhar
 On the non-vanishing of a certain class of Dirichlet series In this paper, we consider Dirichlet series with Euler products of the form $F(s) = \prod_{p}{\bigl(1 + {a_p\over{p^s}}\bigr)}$ in $\Re(s) > 1$, and which are regular in $\Re(s) \geq 1$ except for a pole of order $m$ at $s = 1$. We establish criteria for such a Dirichlet series to be non-vanishing on the line of convergence. We also show that our results can be applied to yield non-vanishing results for a subclass of the Selberg class and the Sato-Tate conjecture. Categories:11Mxx, 11M41

214. CMB 1997 (vol 40 pp. 376)

Gross, Benedict H.; Savin, Gordan
 The dual pair $PGL_3 \times G_2$ Let $H$ be the split, adjoint group of type $E_6$ over a $p$-adic field. In this paper we study the restriction of the minimal representation of $H$ to the closed subgroup $PGL_3 \times G_2$. Categories:22E35, and, 50, 11F70

215. CMB 1997 (vol 40 pp. 214)

Mollin, R. A.; Goddard, B.; Coupland, S.
 Polynomials of quadratic type producing strings of primes The primary purpose of this paper is to provide necessary and sufficient conditions for certain quadratic polynomials of negative discriminant (which we call Euler-Rabinowitsch type), to produce consecutive prime values for an initial range of input values less than a Minkowski bound. This not only generalizes the classical work of Frobenius, the later developments by Hendy, and the generalizations by others, but also concludes the line of reasoning by providing a complete list of all such prime-producing polynomials, under the assumption of the generalized Riemann hypothesis ($\GRH$). We demonstrate how this prime-production phenomenon is related to the exponent of the class group of the underlying complex quadratic field. Numerous examples, and a remaining conjecture, are also given. Categories:11R11, 11R09, 11R29

216. CMB 1997 (vol 40 pp. 81)

Movahhedi, A.; Salinier, A.
 Une caractÃ©risation des corps satisfaisant le thÃ©orÃ¨me de l'axe principal Resum\'e. On caract\'erise les corps $K$ satisfaisant le th\'eor\eme de l'axe principal \a l'aide de propri\'et\'es des formes carac\-t\'erisation de ces m\^emes corps due \a Waterhouse, on retrouve \a partir de l\`a, de fa\c{c}on \'el\'ementaire, un r\'esultat de Becker selon lequel un pro-$2$-groupe qui se r\'ealise comme groupe de Galois absolu d'un tel corps $K$ est engendr\'e par des involutions. ABSTRACT. We characterize general fields $K$, satisfying the Principal Axis Theorem, by means of properties of trace forms of the finite extensions of $K$. From this and Waterhouse's characterization of the same fields, we rediscover, in quite an elementary way, a result of Becker according to which a pro-$2$-group which occurs as the absolute Galois group of such a field $K$, is generated by Categories:11E10, 12D15

217. CMB 1997 (vol 40 pp. 72)

Lee, Min Ho
 Generalized Siegel modular forms and cohomology of locally symmetric varieties We generalize Siegel modular forms and construct an exact sequence for the cohomology of locally symmetric varieties which plays the role of the Eichler-Shimura isomorphism for such generalized Siegel modular forms. Categories:11F46, 11F75, 22E40
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