Modular forms and differential operators

In 1956, Rankin described which polynomials in the derivatives of modular forms are again modular forms, and in 1977, H Cohen defined for eachn ≥ 0 a bilinear operation which assigns to two modular formsf andg of weightk andl a modular form [f, g]_n of weightk +l + 2n. In the present paper we study these “Rankin-Cohen brackets” from two points of view. On the one hand we give various explanations of their modularity and various algebraic relations among them by relating the modular form theory to the theories of theta series, of Jacobi forms, and of pseudodifferential operators. In a different direction, we study the abstract algebraic structure (“RC algebra”) consisting of a graded vector space together with a collection of bilinear operations [,]_n of degree + 2n satisfying all of the axioms of the Rankin-Cohen brackets. Under certain hypotheses, these turn out to be equivalent to commutative graded algebras together with a derivationS of degree 2 and an element Φ of degree 4, up to the equivalence relation (∂,Φ) ~ (∂ - ϕE, Φ - ϕ² + ∂(ϕ)) where ϕ is an element of degree 2 andE is the Fuler operator (= multiplication by the degree). Dedicated to the memory of Professor K G Ramanathan     

The Radon-Nikodym theorem for von neumann algebras

Let ϕ be a faithful normal semi-finite weight on a von Neumann algebraM. For each normal semi-finite weight ϕ onM, invariant under the modular automorphism group Σ of ϕ, there is a unique self-adjoint positive operatorh, affiliated with the sub-algebra of fixed-points for Σ, such that ϕ=ϕ(h·). Conversely, each suchh determines a Σ-invariant normal semi-finite weight. An easy application of this non-commutative Radon-Nikodym theorem yields the result thatM is semi-finite if and only if Σ consists of inner automorphisms. Partially supported by NSF Grant # 28976 X. Partially supported by NSF Grant # GP-28737 This revised version was published online in November 2006 with corrections to the Cover Date.     

On the topological type of anticonformal square roots of automorphisms of Riemann surfaces

Let X be a compact Riemann surface and f be a conformal automorphism of X of order n. An anticonformal square root of f is an anticonformal automorphism g of X such that g²=f. If g₁ and g₂ are two anticonformal square roots of f we study the problem of whether g₁ and g₂ have the same topological type, i. e., if there exists a homeomorphism h:X→X such that g₁=hg₂h⁻¹. We use techniques of noneuclidean crystallographic (NEC) groups and the topological classification of orientation reversing maps of finite period on surfaces given in [C1] and [Y]. Partially supported by DGICYT PB92-0716 and EC project CHRX-CT93-408     

Quantitative Generalized Bertini-Sard Theorem for Smooth Affine Varieties

Let X ⊂ ℂⁿ be a smooth affine variety of dimension n – r and let f = (f₁,..., f_m): X → ℂ_m be a polynomial dominant mapping. We prove that the set K(f) of generalized critical values of f (which always contains the bifurcation set B(f) of f) is a proper algebraic subset of ℂ_m. We give an explicit upper bound for the degree of a hypersurface containing K(f). If I(X)—the ideal of X—is generated by polynomials of degree at most D and deg f_i ≤ d, then K(f) is contained in an algebraic hypersurface of degree at most (d + (m – 1)(d – 1)+(D – 1)r)_n-rD_r. In particular if X is a hypersurface of degree D and f: X → ℂ is a polynomial of degree d, then f has at most (d + D – 1)_n-1D generalized critical values. This bound is asymptotically optimal for f linear. We give an algorithm to compute the set K(f) effectively. Moreover, we obtain similar results in the real case.     

On a maximal torus in subgroups of a general linear group

Let k be a field, K/k a finite extension of it of degree n. We denote G=Aut(kK), G_o=Aut(k K) and fix in K a basis ω₁,...,ω_n over k. In this basis, to any automorphism group of kK there corresponds a matrix group, which is denoted by the same symbol. Let G′≤G., In this paper, the conditions under which G′⊎G_o is a maximal torus in G′ are studied. The calculation of N_G′(G′⊎G_o) is carried out, provided that thee conditions are fulfilled. The case G′=SL (kK) is of particular interset. It is known that for Galois extensions and for extensions of algebraic number fields, G′⊎G_o is a maximal torus in G′. Bibligraphy: 2 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995. pp. 15–22.     

Iteration of mapping classes and limits of hyperbolic 3-manifolds

Let ϕ∈Mod(S) be an element of the mapping class group of a surface S. We classify algebraic and geometric limits of sequences {Q(ϕ ⁱ X,Y)} _i=1 ^∞ of quasi-Fuchsian hyperbolic 3-manifolds ranging in a Bers slice. When ϕ has infinite order with finite-order restrictions, there is an essential subsurface D _ϕ⊂S so that the geometric limits have homeomorphism type S×ℝ-D _ϕ×{0}. Typically, ϕ has pseudo-Anosov restrictions, and D _ϕ has components with negative Euler characteristic; these components correspond to new asymptotically periodic simply degenerate ends of the geometric limit. We show there is an s≥1 depending on ϕ and bounded in terms of S so that {Q(ϕ ^si X,Y)} _i=1 ^∞ converges algebraically and geometrically, and we give explicit quasi-isometric models for the limits. Oblatum 4-I-1999 & 19-VII-2000?Published online: 30 October 2000     

Abelian groups admitting fixed-point-free automorphisms of order qn

For a wide class of Abelian groups, necessary and sufficient conditions under which a group admits an automorphism of order qⁿ are found; we also present necessary and sufficient conditions under which a group admits an automorphism ϕ of order qⁿ such that ϕ^qm is a fixed-point-free automorphism for some m < n. Translated from Matematicheskie Zametki, Vol. 21, No. 2, pp. 239–249, February, 1977.     

Comparison between the best approximations by lacunary and ordinary polynomials

We denote E_n(f) and E _k ⁿ (f) the best uniform approximations to a continuous function f defined on [a,b] by the sets of algebraic polynomials of degree ≤n and algebraic polynomials of degree ≤n with the coefficients of x^k (k≤n) being zero. In this paper, in cases of r<k and r≥k while [a, b]=[−1,1] (or r<k,k≤r<2k and r>2k while [a,b]=[0, 1]), we separately discuss the condtions for r-times continuously differentiable function f which enables .     

The continuity of superposition operators on some sequence spaces defined by moduli

Let λ and μ be solid sequence spaces. For a sequence of modulus functions Φ = (ϕ k) let λ(Φ) = {x = (x _k ): (ϕk(|x _k |)) ∈ λ}. Given another sequence of modulus functions Ψ = (ψ_k), we characterize the continuity of the superposition operators P _f from λ(Φ) into μ (Ψ) for some Banach sequence spaces λ and μ under the assumptions that the moduli ϕk (k ∈ ℕ) are unbounded and the topologies on the sequence spaces λ(Φ) and μ(Ψ) are given by certain F-norms. As applications we consider superposition operators on some multiplier sequence spaces of Maddox type. This research was supported by Estonian Science Foundation Grant 5376.     

A Matrix Poincaré formula for holomorphic automorphisms of quadrics of higher codimension. Real Associative Quadrics

In this paper we describe the holomorphic automorphisms for two infinite series of Hermitian quadrics: quadrics of real co-dimension 2 in ℂⁿ⁺² and a special class of quadrics of co-dimension n in ℂ²ⁿ with large automorphism groups (Real Associative Quadrics). We give explicit formulas of the automorphisms. They are rational maps of degree not exceeding the co-dimension.     

Una nota sui gruppi policiclici che ammettono un automorfismo con numero di Reidemeister finito

LetG be a group and ϕ an automorphism ofG. Two elementsx, y ∈ G are called ϕ-conjugate if there existsg ∈ G such thatx=g ⁻¹ yg ^θ. It is easily verified that the ϕ-conjugation is an equivalence relation; the numberR(ϕ) of ϕ-classes ofG is called the Reidemeister number of the automorphism ϕ. In this paper we prove that if a polycyclic groupsG admits an automorphism ϕ of ordern such thatR(ϕ)<∞, thenG contains a subgroup of finite index with derived length at most 2 ⁿ⁻¹ .

     

Double extensions of dynamical systems and constructing mixing filtrations

Let T: X→X be an automorphism (a measurable invertible measure-preserving transformation) of a probability space (X, F, μ) and let two μ-symmetric Markov generators A_u and A_s acting on the space L₂=L₂ (X, F, μ) be “eigenfunctions” of the automorphism T with eigenvaluesθ _u > 1 andθ _s < 1, respectively. We construct an extension of the automorphism T having increasing and decreasing filtrations by means of a transformation on the path space of these processes. Under additional conditions, we give an estimate of the maximal correlation coefficient between the δ-fields chosen from these filtrations. Hyperbolic toral automorphisms are considered as an example. Applications to limit theorems are given. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 244, 1997, pp. 61–72. Translated by M. I. Gordin.     

A problem on simultaneous approximation and a conjecture of Hasson

In 1980, M. Hasson raised a conjecture as follows: Let N≥1, then there exists a function f₀(x)∈C _[−1,1] ^2N , for N+1≤k≤2N, such that p _n ^(k) (f₀,1)→f ₀ ^(k) (1), n→∞, where p_n(f,x) is the algebraic polynomial of best approximation of degree ≤n to f(x). In this paper, a, positive answer to this conjecture is given.     

Finite groups with an almost regular automorphism of order four

P. Shumyatsky’s question 11.126 in the “Kourovka Notebook” is answered in the affirmative: it is proved that there exist a constant c and a function of a positive integer argument f(m) such that if a finite group G admits an automorphism ϕ of order 4 having exactly m fixed points, then G has a normal series G ⩾ H ⩽ N such that |G/H| ⩽ f(m), the quotient group H/N is nilpotent of class ⩽ 2, and the subgroup N is nilpotent of class ⩽ c (Thm. 1). As a corollary we show that if a locally finite group G contains an element of order 4 with finite centralizer of order m, then G has the same kind of a series as in Theorem 1. Theorem 1 generalizes Kovács’ theorem on locally finite groups with a regular automorphism of order 4, whereby such groups are center-by-metabelian. Earlier, the first author proved that a finite 2-group with an almost regular automorphism of order 4 is almost center-by-metabelian. The proof of Theorem 1 is based on the authors’ previous works dealing in Lie rings with an almost regular automorphism of order 4. Reduction to nilpotent groups is carried out by using Hall-Higman type theorems. The proof also uses Theorem 2, which is of independent interest, stating that if a finite group S contains a nilpotent subgroup T of class c and index |S: T | = n, then S contains also a characteristic nilpotent subgroup of class ⩽ c whose index is bounded in terms of n and c. Previously, such an assertion has been known for Abelian subgroups, that is, for c = 1. __________ Translated from Algebra i Logika, Vol. 45, No. 5, pp. 575–602, September–October, 2006.     

When Polarizations Generate

Let G be a reductive complex algebraic group and V a finite-dimensional G-module. From elements of the invariant algebra C[V]^G we obtain, by polarization, elements of C[kV]^G, where k ≥ 1 and kV denotes the direct sum of k copies of V. For G simple, our main result is the classification of the G-modules V and integers k ≥ 2 such that polarizations generate C[kV]^G.     

The Computation of Invariant Fields and a Constructive Version of a Theorem by Rosenlicht

Let G be an algebraic group acting on an irreducible variety X. We present an algorithm for computing the invariant field k(X)^G. Moreover, we give a constructive version of a theorem of Rosenlicht, which says that almost all orbits can be separated by rational invariants. More precisely, we give an algorithm for computing a nonempty open subset of X with a geometric quotient.     

Degrees of irreducible characters of the suzuki 2-groups

The degrees of the irreducible characters of the Suzuki 2-groupsA(m,ϕ) are described. Assume that the order of an automorphism ϕ of the fieldGF (2 ^m ) isk>1,G=A(m,ϕ), and cd (G) is the set of degrees of the irreducible characters ofG. Ifk is odd, then cd(G)={1, 2( ^m−m/k)/2 }, and ifk=2, then cd(G)={1, 2 ^m/2 }. Ifk is even andk≠2 then cd(G)={1,2 ^m/2 , 2 ^m/2−m/k }, and the groupG has (2 ^m −1)2 ^m/k /(2 ^m/k +1) characters of degree2 ^m/2 and (2^m−1)2^2m/k/(2^m/k+1) characters of degree 2^m/2−m/k. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 258–263, August, 1999.     

On squares in polynomial products

Let f(X) ? \mathbb Z[X]{f(X) \in \mathbb {Z}[X]} be an irreducible polynomial of degree D ≥ 2 and let N be a sufficiently large positive integer. We estimate the number of positive integers n ≤ N such that the product

Conjugacy classes of non-connected semisimple algebraic groups

Consider a non-connected algebraic group G = G ⋉ Γ with semisimple identity component G and a subgroup of its diagram automorphisms Γ. The identity component G acts on a fixed exterior component Gτ, id ≠ τ ∈ Γ by conjugation. In this paper we will describe the conjugacy classes and the invariant theory of this action. Let T be a τ -stable maximal torus of G and its Weyl group W. Then the quotient space Gτ//G is isomorphic to (T/(1 − τ )(T))/W^τ. Furthermore, exploiting the Jordan decomposition, the reduced fibres of this quotient map are naturally associated bundles over semisimple G-orbits. Similar to Steinberg's connected and simply connected case [22] and under additional assumptions on the fundamental group of G, a global section to this quotient map exists. The material presented here is a synopsis of the Ph.D thesis of the author, cf. [15].