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226. 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

227. 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

228. 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

229. 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

230. 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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