Table of contents

Let be an integral, symmetric, positive definite matrix of order with an even diagonal. For the theta series of of degree

where runs through all integral matrices and is a point of the Siegel upper halfplane of degree , the congruence subgroup of the group is found, with respect to which is a Siegel modular form with a multiplicator system (the analog of the group ). The analogous problem is solved for theta series of degree with spherical functions. The appropriate multiplicator systems are computed for even . Bibliography: 5 items.

Normal extensions of a given number field , which are unramified outside a given set  of divisors and are for a fixed prime  closed under -extensions, are considered in the paper. It is assumed that contains all Archimedean places and all prime divisors of . The cohomology group is described, and it is proved that the cohomological -dimension of the Galois group does not exceed 2. Bibliography: 10 items.

The semisimplicity of -adic representations corresponding to one-dimensional étale cohomology, and a proof of the Tate's conjecture about homomorphisms of abelian varieties, are derived from the Tate's finiteness conjecture on isogenies of polarized abelian varieties. Bibliography: 5 items.

In this paper we show that a Kähler K3 surface containing 16 nonsingular rational curves which do not intersect one another is a Kummer surface. We also give a direct proof of the global Torelli theorem for Kummer surfaces and develop a criterion for a surface to be Kummer which refines the criterion in the paper A Torelli theorem for algebraic K3 surfaces by I. I. Pjateckii-Šapiro and I. R. Šafarevič (Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 530-572 = Math. USSR-Izv. 5 (1971), 547-588; MR 44 #1666). Bibliography: 8 items.

It is proved that a topological versal deformation exists for almost any germ of a smooth mapping; the exceptional germs constitute a set of codimension infinity in the space of all germs. Bibliography: 7 items.

A complete classification of quaternionic Riemannian spaces (that is, spaces with the holonomy group , ) which admit a transitive solvable group of motions is given. It turns out that the rank of these spaces does not exceed four and that all spaces whose rank is less than four are symmetric. The spaces of rank four are in natural one-to-one correspondence with the Clifford modules of Atiyah, Bott and Shapiro. In this correspondence, the simplest Clifford modules, which are connected with division algebras, are mapped to symmetric spaces of exceptional Lie groups. Other Clifford modules, which are obtained from the simplest with help of tensor products, direct sums and restrictions, correspond to nonsymmetric spaces. Bibliography: 17 items.

By means of the method described in the author's paper "Quantization" (Math. USSR Izv. 8 (1974), 1109-1165), we construct the quantization of a classical mechanics whose phase space is a classical complex symmetric space. We establish the important qualitative differences between the quantization of such mechanics and the quantization of ordinary mechanics with plane phase spaces: for all the spaces considered, except for the sphere, Planck's constant is bounded above. Moreover, in the compact case Planck's constant takes on only discrete values. Bibliography: 17 items.

We strengthen, and prove by a new method, the principle of localization in the theory of orthogonal polynomials, which was previously discussed by G. Freud. We sharpen a theorem of Geronimus on Steklov orthogonal polynomials. Bibliography: 8 items.

For the derivative of a polynomial with real coefficients we obtain an inequality which involves the distribution of the zeros of the polynomial. It is shown that for polynomials with arbitrary complex coefficients the inequality holds only under additional hypotheses. Bibliography: 3 items.

In this paper necessary and sufficient conditions are found for imbeddings of the form . It is proved that in the one-dimensional case the corresponding condition on the modulus of continuity of a monotone function is not only sufficient but also necessary for . In connection with this the existence is established of a monotone function in with preassigned order of modulus of continuity. Bibliography: 10 items.