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226. 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.


227. 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.

228. 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

229. 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

230. 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

231. 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

232. 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

233. 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

234. 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

235. CMB 1997 (vol 40 pp. 81)

Movahhedi, A.; Salinier, A.
Une caractérisation des corps satisfaisant le théorème 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
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