Table of contents

Let u be a smooth function on a manifold M. In this paper necessary and sufficient conditions are obtained for u to be the real part of a CR-function on M. These conditions comprise a system of linear differential equations in this function, whose order depends on the location of M with respect to the complex structure. Bibliography: 8 titles.

In this paper a derivation is given of a generalized Selberg trace formula corresponding to the odd eigenfunctions of the Laplace-Beltrami operator in the space , where the discrete group is and is the upper halfplane (the Dirichlet problem on half of the fundamental domain). As an application a generalization is obtained of Minakshisundaram's formula:

(1)

This paper is devoted to the problem of stability of the homology or Riemannian manifolds. An effective criterion is obtained for the stability of homology classes of arbitrary compact Riemannian manifolds. Bibliography: 4 titles.

In this paper Fano 3-folds of the principal series in are studied. A classification is given of trivial (i.e. containing a trigonal canonical curve) 3-folds of this kind. Among all Fano 3-folds of the principal series these are distinguished by the property that they are not the intersection of the quadrics containing them. It turns out that the genus of such 3-folds does not exceed 10. Fano 3-folds of genus one (i.e. with ) containing a line are described. It is proved that they exist for and . Their rationality for and is established by direct construction. Bibliography: 21 titles.

Theorems are proved on the consistency with , for , of each of the following three propositions: (1) there exists an L-minimal (in particular, nonconstructive) such that and , but every of class with constructive code is itself constructive; (2) there exist such that their -degrees differ by a formula from , but not by formulas from with constants from ( and are said to differ by a formula ; (3) there exists an infinite, but Dedekind finite, set of class , whereas there are no such sets of class . The proof uses Cohen's forcing method. Bibliography: 17 titles.

A complete birational classification of algebraic tori with a biquadratic splitting field is obtained in this paper. It is shown that any torus of the indicated type is birationally equivalent to a direct product of n copies (n ≥ 0) of a special three-dimensional torus T and an affine space A^m. An affirmative answer to one of Zariski's conjectures is also obtained for tori of this type. Up till now a birational classification of tori has been known only in the case of a metacyclic splitting field (i.e. in the case where all Sylow subgroups of the Galois group of the splitting field are cyclic). Bibliography: 12 titles.

Using an isomorphism of Eichler-Shimura type, the algebraicity is proved, up to a common constant factor, of the values in the center of the critical strip of the L-series connected with a "new" modular cusp form on G_A for an imaginary quadratic field twisted with arbitrary Dirichlet characters of this field. Bibliography: 8 titles.

This paper studies the problem of left divisibility in a semigroup given by a single defining relation of the form a = bA, where A is an arbitrary word in the alphabet a, b. Solvability of the problems of equality and left divisibility of words is proved for a semigroup given by a defining relation of the form a = (bA)^na, where n > 1. Bibliography: 4 titles.