Volume 2009, Issue 629

Let F be a non-Archimedean locally compact field and let D be a central division algebra over F . Let π 1 and π 2 be respectively two smooth irreducible representations of GL( n 1 , D ) and GL( n 2 , D ), n 1 , n 2 ≧ 0. In this article, we give some sufficient conditions on π 1 and π 2 so that the parabolically induced representation of π 1 ⊗ π 2 to GL( n 1 + n 2 , D ) has a unique irreducible quotient. In the case where π 1 is a cuspidal representation, we compute the Zelevinsky's parameters of such a quotient in terms of the parameters of π 2 . This is the key point for making explicit Howe correspondence for dual pairs of type II (cf. [Mínguez, thesis, Orsay 2006]).

We give a factorization of averages of Borcherds forms over CM points associated to a quadratic form of signature ( n , 2). As a consequence of this result, we are able to state a theorem like that of Gross and Zagier about which primes can occur in this factorization. One remarkable phenomenon we observe is that the regularized theta lift of a weakly holomorphic modular form is always finite.

Let X be a projective non-singular quartic hypersurface of dimension 39 or more, which is defined over ℚ. We show that X (ℚ) is non-empty provided that X (ℝ) is non-empty and X has p -adic points for every prime p .

For each integer n ≧ 2, we construct an irreducible, smooth, complex projective variety M of dimension n , whose fundamental group has infinitely generated homology in degree n + 1 and whose universal cover is a Stein manifold, homotopy equivalent to an infinite bouquet of n -dimensional spheres. This non-finiteness phenomenon is also reflected in the fact that the homotopy group π n ( M ), viewed as a module over ℤπ 1 ( M ), is free of infinite rank. As a result, we give a negative answer to a question of Kollár on the existence of quasi-projective classifying spaces (up to commensurability) for the fundamental groups of smooth projective varieties. To obtain our examples, we develop a complex analog of a method in geometric group theory due to Bestvina and Brady.

We investigate formal ribbons on curves. Roughly speaking, formal ribbon is a family of locally linearly compact vector spaces on a curve. We establish a one-to-one correspondence between formal ribbons on curves plus some geometric data and some subspaces of two-dimensional local fields.

In this paper we study the question of surjectivity of the power maps g ↦ g n on p -adic algebraic groups. We give a complete solution of this question in terms of the orders of certain compact analytic subgroups. We next study the p -adic one parameter subgroups of p -adic algebraic groups and find conditions, when the union of all such one parameter groups fills up the entire group. We show that there is a close relation between the above two problems. We also obtain similar results on groups defined over rational numbers.

Higher order group cohomology is defined and first properties are given. Using modular symbols, an Eichler-Shimura homomorphism is constructed mapping spaces of higher order cusp forms to higher order cohomology groups.