We consider the Dirichlet series Z(P,A;s) = [sum ]$_m∈ A ∩ Z^n$P$^-s$(m) (s ∈ C) where P ∈ R[X$_1$, …, X$_n$] and A is an open semi-algebraic subset of R$^n$. We will say that Z(P,A;s) exists if this multiple series is absolutely convergent. In this paper we study the existence and several properties of meromorphic continuations of such series, under certain assumption on P and A. As an application, we show the existence of a finite asymptotic expansion of the counting function with support in A: N$_p$(A,t):= [sharp ] m ∈ A ∩ Z$^n$ | P(m) [les ] t} when t → +∞.

In this paper we study reducibility of representations of split classical p-adic groups induced from self-contragredient supercuspidal represetation associated via Howe‘s construction to an admissible character, we show that in many cases Shahidi‘s criterion for reducibility of the induced representation reduces to a simple condition on the admissible character.

We prove, assuming the generalized Riemann hypothesis for imaginary quadratic fields, the following special case of a conjecture of Oort, concerning Zarsiski closures of sets of CM points in Shimura varieties. Let X be an irreducible algebraic curve in C$^2$, containing infinitely many points of which both coordinates are j-invariants of CM elliptic curves. Suppose that both projections from X to C are not constant. Then there is an integer m [ges ] 1such that X is the image, under the usual map, of the modular curve Y$_0$(m). The proof uses some number theory and some topological arguments.

There are infinitely many fundamentally distinct families of polarized Abelian fourfolds of Weil type with multiplication from the cyclotomic field of cube roots of unity. The Hodge conjecture is shown to hold at a sufficiently general fiber in any of these families.