We show that the cohomology of complexes of solutions of exponantial type associated to holonomic algebraic D-modules is constructible. We also compute the Euler–Poincaré index of such complexes.

A new family of sheaves has been recently studied by M. Kashiwara and P. Schapira generalizing to constructible sheaves the notion of moderate and formal cohomology. We prove comparison theorems when we regard these sheaves as solutions of a $\cal D$-module. These results are natural generalizations of those of Y. Laurent and the author.

Let R be a local Artin ring with maximal ideal $\frak m$ and residue class field of characteristic p > 0. We show that every finite flat group scheme over R is annihilated by its rank, whenever $\frak m$p = p$\frak m$ = 0. This implies that any finite flat group scheme over an Artin ring the square of whose maximal ideal is zero, is annihilated by its rank.

An interesting theory arises when the classical theory of modular forms is expanded to include differential analogs of modular forms. One of the main motivations for expanding the theory of modular forms is the existence of differential modular forms with a remarkable property, called isogeny covariance, that classical modular forms cannot possess. Among isogeny covariant differential modular forms there exists a particular modular form that plays a central role in the theory. The main result presented in the article will be the explicit computation modulo p of this fundamental isogeny covariant differential modular form.

We define a 4-parameter family of generically irreducible and inequivalent representations of the Witt Lie algebra on which the infinitesimal rotation operator acts semisimply with infinite-dimensional eigenspaces. They are deformations of the (generically indecomposable) representations on spaces of polynomial differential operators between two spaces of tensor densities on S1, which are constructed by composing each such differential operator with the action of a rotation by a fixed angle.

We estimate some sums of the shape S(Xβ1,…, Xβm): = [sum ]1 [les ] d1 [les ] Xβ1… [sum ]1 [les ] dm [les ] Xβm ƒ(d1,…, dm), when m ∈ $\Bbb N$ and f is a nonnegative arithmetical function. We relate them to the behaviour of the associated Dirichlet series

F(s1,…, sm) = ∞[sum ]d1 = 1 … ∞[sum ]dm = 1 f(d1,…, dm)/d1s1 … dmsm. The main aim of this work is to develop analytic tools to count the rational points of bounded height on toric varieties.

The root discriminant of a number field of degree n is the nth root of the absolute value of its discriminant. Let R2m be the minimal root discriminant for totally complex number fields of degree 2m, and put α0 = lim infm R2m. One knows that α0 [ges ] 4πeγ ≈ 22.3, and, assuming the Generalized Riemann Hypothesis, α0 [ges ] 8πeγ ≈ 44.7. It is of great interest to know if the latter bound is sharp. In 1978, Martinet constructed an infinite unramified tower of totally complex number fields with small constant root discriminant, demonstrating that α0 < 92.4. For over twenty years, this estimate has not been improved. We introduce two new ideas for bounding asymptotically minimal root discriminants, namely, (1) we allow tame ramification in the tower, and (2) we allow the fields at the bottom of the tower to have large Galois closure. These new ideas allow us to obtain the better estimate α0 < 83.9.

In this paper we define a kind of generalized spherical functions on Sp(2, $\Bbb R$). We call it ‘Fourier–Jacobi type’, since it can be considered as a generalized Whittaker model associated with the Jacobi maximal parabolic subgroup. Also we give the multiplicity theorem and an explicit formula of these functions for discrete series representations of Sp(2, $\Bbb R$).

Let X/C be a projective algebraic manifold, and further let CHk(X)Q be the Chow group of codimension k algebraic cycles on X, modulo rational equivalence. By considering Q-spreads of cycles on X and the corresponding cycle map into absolute Hodge cohomology, we construct a filtration {F[ell ]}[ell ] [ges ] 0 on CHk(X)Q of ‘Bloch-Beilinson’ type. In the event that a certain conjecture of Jannsen holds (related to the Bloch-Beilinson conjecture on the injectivity, modulo torsion, of the Abel–Jacobi map for smooth proper varieties over Q), this filtration truncates. In particular, his conjecture implies that Fk+1 = 0.

Given a finite set A of integral vectors and a parameter vector, Gel'fand, Kapranov, and Zelevinskii defined a system of differential equations, called an A-hypergeometric (or a GKZ hypergeometric) system. Classifying the parameters according to the D-isomorphism classes of their corresponding A-hypergeometric systems is one of the most fundamental problems in the theory. In this paper we give a combinatorial answer for the problem under the assumption that the finite set A lies in a hyperplane off the origin, and illustrate it in two particularly simple cases: the normal case and the monomial curve case.

We study a dual pair of general linear Lie superalgebras in the sense of R. Howe. We give an explicit multiplicity-free decomposition of a symmetric and skew-symmetric algebra (in the super sense) under the action of the dual pair and present explicit formulas for the highest-weight vectors in each isotypic subspace of the symmetric algebra. We give an explicit multiplicity-free decomposition into irreducible gl(m|n)-modules of the symmetric and skew-symmetric algebras of the symmetric square of the natural representation of gl(m|n). In the former case, we also find explicit formulas for the highest-weight vectors. Our work unifies and generalizes the classical results in symmetric and skew-symmetric models and admits several applications.

Multivariate hypergeometric functions associated with toric varieties were introduced by Gel'fand, Kapranov and Zelevinsky. Singularities of such functions are discriminants, that is, divisors projectively dual to torus orbit closures. We show that most of these potential denominators never appear in rational hypergeometric functions. We conjecture that the denominator of any rational hypergeometric function is a product of resultants, that is, a product of special discriminants arising from Cayley configurations. This conjecture is proved for toric hypersurfaces and for toric varieties of dimension at most three. Toric residues are applied to show that every toric resultant appears in the denominator of some rational hypergeometric function.

Let $\cal X$ be a projective, geometrically irreducible, non-singular, algebraic curve defined over a finite field Fq2 of order q2. If the number of Fq2-rational points of $\cal X$ satisfies the Hasse–Weil upper bound, then $\cal X$ is said to be Fq2-maximal. For a point P0 ∈ $\cal X$(Fq2), let π be the morphism arising from the linear series $\cal D$: = |(q + 1)P0|, and let N: = dim($\cal D$). It is known that N [ges ] 2 and that π is independent of P0 whenever $\cal X$ is Fq2-maximal.

We show that over polynomial extensions of normal affine domains of dimension two over perfect fields (char. ≠ 2) of cohomological dimension [les ] 1, all finitely generated projective modules are cancellative, thus answering a question of Weibel affirmatively in the case of polynomial extensions.