Volume 2004, Issue 577

We prove the equivalence of two fundamental properties of algebraic stacks: being a quotient stack in a strong sense, and the resolution property, which says that every coherent sheaf is a quotient of some vector bundle. Moreover, we prove these properties in the important special case of orbifolds whose associated algebraic space is a scheme.

We carry out (with technical modifications) some cases of a procedure proposed by R. Langlands in Beyond Endoscopy . This gives a new proof of the classification of “dihedral forms” on GL(2), avoiding endoscopic methods: instead, it uses an infinite limiting process in the trace formula. We also explain the relationship of Langlands' idea to the results of [10] that count automorphic forms associated to Galois representations. We prove results of a similar nature over an arbitrary number field. The main tool is a version of the relative trace formula, namely a slight variant (for PGL(2)) of the Kuznetsov-Bruggeman-Miatello formula.

We classify pointed Hopf algebras with finite Gelfand-Kirillov dimension, which are domains, whose groups of group-like elements are finitely generated and abelian, and whose infinitesimal braidings are positive.

We prove a Moser-type theorem for self-dual harmonic 2-forms on closed 4-manifolds, and use it to classify local forms on neighborhoods of singular circles on which the 2-form vanishes. Removing neighborhoods of the circles, we obtain a symplectic manifold with contact boundary—we show that the contact form on each S 1 × S 2 , after a slight modification, must be one of two possibilities.

Let N/K be a finite abelian extension of a quadratic imaginary number field K . Let p denote a prime ideal in K of residue characteristic p such that p splits in K /ℚ and p splits completely in N/K . In analogy to a result of Solomon in the cyclotomic case we construct an elliptic p-unit in N and express its valuation in terms of a p -adic logarithm. As an application we prove the Equivariant Tamagawa Number Conjecture formulated by Burns and Flach for the pair ( h 0 (Spec( N )), ℤ[Gal( N/K )]) for a natural family of genus extensions N/K .

In a previous article, we have proved a result asserting the existence of a compatible family of Galois representations containing a given crystalline irreducible odd two-dimensional representation. We apply this result to establish new cases of the Fontaine-Mazur conjecture, namely, an irreducible Barsotti-Tate λ-adic 2-dimensional Galois representation unramified at 3 and such that the traces a p of the images of Frobenii verify ℚ({a 2 p }) = ℚ always comes from an abelian variety. We also show the non-existence of irreducible Barsotti-Tate 2-dimensional Galois representations of conductor 1 and apply this to the irreducibility of Galois representations on level 1 genus 2 Siegel cusp forms.

Let ℰ → C be an elliptic surface defined over a number field K . For a finite covering C′ → C defined over K , let ℰ′ = ℰ × C C′ be the corresponding elliptic surface over C′ . In this paper we give a strong upper bound for the rank of ℰ′ ( C′ / K ) in the case of geometrically abelian unramified coverings C′ → C and under the assumption that the Tate conjecture is true for ℰ′ / K . In the case that C is an elliptic curve and the map C′ = C → C is the multiplication-by- n map, the bound for rank( ℰ′ ( C′ / K )) takes the form O ( n k /log log n ), which may be compared with the elementary bound of O ( n 2 ).

In this paper we discuss a pore scale model for crystal dissolution and precipitation in porous media. We consider first general domains, for which existence of weak solutions is proven. For the particular case of strips we show that free boundaries occur in the form of dissolution/precipitation fronts. As the ratio between the thickness and the length of the strip vanishes we obtain the upscaled reactive solute transport model proposed in [12].

Given a real-analytic Riemannian manifold X there is a canonical complex structure, which is compatible with the canonical complex structure on T*X and makes the leaves of the Riemannian foliation on TX into holomorphic curves, on its tangent bundle. A Grauert tube over X of radius r , denoted as T r X , is the collection of tangent vectors of X of length less than r equipped with this canonical complex structure. In this article, we prove the following two rigidity properties of Grauert tubes. First, for any real-analytic Riemannian manifold such that r max > 0, we show that the identity component of the automorphism group of T r X is isomorphic to the identity component of the isometry group of X provided that r < r max . Secondly, let X be a homogeneous Riemannian manifold and let the radius r < r max , then the automorphism group of T r X is isomorphic to the isometry group of X and there is a unique Grauert tube representation for such a complex manifold T r X .