Volume 2005, Issue 584

In this paper we give an explicit formula for the mass of a quadratic form in n ≧ 3 variables with respect to a maximal lattice over an arbitrary number field k , and use this to find the mass of many a-maximal lattices. We make the minor technical assumption that locally the determinant of the form is a unit up to a square if n is odd. The corresponding formula for k totally real was recently computed by Shimura [ G. Shimura , An exact mass formula for orthogonal groups, Duke Math. J. 97 (1999), no. 1, 1–66].

We estimate the number of solutions of certain diagonal congruences involving factorials. We use these results to bound exponential sums with products of two factorials n ! m ! and also derive asymptotic formulas for the number of solutions of various congruences with factorials. For example, we prove that the products of two factorials n  ! m  ! with max{ n , m  } <  p 1/2+ ε are uniformly distributed modulo p , and that any residue class modulo p  is representable in the form m  ! n  ! +  n 1 ! + …+  n 47 ! with max{ m ,  n ,  n 1 , … ,  n 47 } <  p 1350/1351+ ε  .

We completely describe all semi-stable torsion free sheaves of degree zero on nodal cubic curves using the technique of Fourier-Mukai transforms. The Fourier-Mukai images of such sheaves are torsion sheaves of finite length, which we compute explicitly. We show that the twist functors, which are associated to the structure sheaf and the structure sheaf k ( p 0 ) of a smooth point p 0 , generate an SL(2, ℤ)-action (up to shifts) on the bounded derived category of coherent sheaves on any Weierstraß cubic.

For any n  ≧ 2, let F ∈ ℤ [ x 1 , … , x n ] be a form of degree d ≧ 2, which produces a geometrically irreducible hypersurface in ℙ n–1 . This paper is concerned with the number N ( F ; B ) of rational points on F  = 0 which have height at most B . For any ε > 0 we establish the estimate N ( F ; B ) = O ( B n− 2+ ε ), whenever either n ≦ 5 or the hypersurface is not a union of lines. Here the implied constant depends at most upon d , n and ε .

We prove Calderón and Zygmund type estimates for a class of elliptic problems whose model is the non-homogeneous p  ( x  )-Laplacean system Under optimal continuity assumptions on the function p  ( x  ) > 1 we prove that Our estimates are motivated by recent developments in non-Newtonian fluidmechanics and elliptic problems with non-standard growth conditions, and are the natural, ‘‘non-linear’’ counterpart of those obtained by Diening and Růžička [ L. Diening and M. Růžička , Calderón-Zygmund operators on generalized Lebesgue spaces L p (‧) and problems related to fluid dynamics, J. reine angew. Math. 563 (2003), 197–220] in the linear case.

We study Gorenstein liaison of codimension two subschemes of an arithmetically Gorenstein scheme X . Our main result is a criterion for two such subschemes to be in the same Gorenstein liaison class, in terms of the category of ACM sheaves on X . As a consequence we obtain a criterion for X  to have the property that every codimension 2 arithmetically Cohen-Macaulay subscheme is in the Gorenstein liaison class of a complete intersection. Using these tools we prove that every arithmetically Gorenstein subscheme of ℙ n is in the Gorenstein liaison class of a complete intersection and we are able to characterize the Gorenstein liaison classes of curves on a nonsingular quadric threefold in ℙ 4 .

Chains of minimal degree rational curves have been used as an important tool in the study of Fano manifolds. Their own geometric properties, however, have not been studied much. The goal of the paper is to introduce an infinitesimal method to study chains of minimal rational curves via varieties of minimal rational tangents and their higher secants. For many examples of Fano manifolds this method can be used to compute the minimal length of chains needed to join two general points. One consequence of our computation is a bound on the multiplicities of divisors at a general point of the moduli of stable bundles of rank two on a curve.

We show that many algebraic actions of higher-rank abelian groups on zero-dimensional compact abelian groups are mutually disjoint. The proofs exploit differences in the entropy geometry arising from subdynamics and a form of Abramov-Rokhlin formula for half-space entropies.

The existence of non-linearizable isotropic CR-automorphisms of Levi non-degenerate hypersurfaces in complex space is characteristic for quadrics. In this paper we discover elliptic CR-manifolds of CR-dimension and codimension two that are not related to the quadric and that have non-linearizable isotropic automorphisms. This is a new and unexpected phenomenon. A complete description of such manifolds is given.