Volume 2007, Issue 603

We consider certain Massey products in the cohomology of a Galois extension of fields with coefficients in p -power roots of unity. We prove formulas for these products both in general and in the special case that the Galois extension in question is the maximal extension of a number field unramified outside a set of primes S including those above p and any archimedean places. We then consider those ℤ p -Kummer extensions L ∞ of the maximal p -cyclotomic extension K ∞ of a number field K that are unramified outside S . We show that Massey products describe the structure of a certain “decomposition-free” quotient of a graded piece of the maximal unramified abelian pro- p extension of L ∞ in which all primes above those in S split completely, with the grading arising from the augmentation filtration on the group ring of the Galois group of L ∞ / K ∞ . We explicitly describe examples of the maximal unramified abelian pro- p extensions of unramified outside p Kummer extensions of the cyclotomic field of all p -power roots of unity, for irregular primes p .

We give a purity result for exponential sums of the type , where k is a finite field of characteristic p , ψ : k → is a non-trivial additive character and ƒ ∈ k [ x 1 ,…, x n ] is a polynomial whose highest degree homogeneous form splits as a product of factors defining a divisor with normal crossings in ℙ n −1 .

We develop a theory of quasi-coherent and constructible sheaves on algebraic stacks correcting a mistake in the recent book of Laumon and Moret-Bailly. We study basic cohomological properties of such sheaves, and prove stack-theoretic versions of Grothendieck's Fundamental Theorem for proper morphisms, Grothendieck's Existence Theorem, Zariski's Connectedness Theorem, as well as finiteness theorems for proper push forwards of coherent and constructible sheaves. We also explain how to define a derived pullback functor which enables one to carry through the construction of a cotangent complex for a morphism of algebraic stacks due to Laumon and Moret-Bailly.

We define a complex connection on a real hypersurface of ℂ n +1 which is naturally inherited from the ambient space. Using a system of Codazzi-type equations, we classify connected real hypersurfaces in ℂ n +1 , n ≧ 2, which are Levi umbilical and have non zero constant Levi curvature. It turns out that such surfaces are contained either in a sphere or in the boundary of a complex tube domain with spherical section.

The standard realization of the Hecke algebra on classical holomorphic cusp forms and the corresponding period polynomials is well known. In this article we consider a nonstandard realization of the Hecke algebra on Maass cusp forms for the Hecke congruence subgroups Γ 0 ( n ). We show that the vector valued period functions derived recently by Hilgert, Mayer and Movasati as special eigenfunctions of the transfer operator for Γ 0 ( n ) are indeed related to the Maass cusp forms for these groups. This leads also to a simple interpretation of the “Hecke like” operators of these authors in terms of the aforementioned nonstandard realization of the Hecke algebra on the space of vector valued period functions.

Let X be a projective minimal Gorenstein 3-fold of general type with canonical singularities. We prove that the 5-canonical map is birational onto its image.

In this paper we construct a Rankin-Selberg integral which represents the Spin 10 × St L -function attached to the group GSO 10 × PGL 2 . We use this integral representation to give some equivalent conditions for a generic cuspidal representation on GSO 10 to be a functorial lift from the group G 2 × PGL 2 .

We show the following theorem: Let X be an irreducible compact Hermitian symmetric manifold of complex dimension n whose bisectional curvature is ( s − 1)-nondegenerate. Then in X there exists no smooth Levi-flat CR manifold M with real co-dimension n – s and CR dimension s ≧ 2, such that the determinant ℂ-line bundle of is smoothly trivial.