All dihedral division algebras of degree five are cyclic

In 1982 Rowen and Saltman proved that every division algebra which is split by a dihedral extension of degree of the center, odd, is in fact cyclic. The proof requires roots of unity of order in the center. We show that for , this assumption can be removed. It then follows that , the -torsion part of the Brauer group, is generated by cyclic algebras, generalizing a result of Merkurjev (1983) on the and torsion parts.

     

A note on the injective dimension of local cohomology modules

For a Noetherian ring we call an -module cofinite if there exists an ideal of such that is -cofinite; we show that every cofinite module satisfies . As an application we study the question which local cohomology modules satisfy . There are two situations where the answer is positive. On the other hand, we present two counterexamples, the failure in these two examples coming from different reasons.

     

A quasiconformal mapping class group acting trivially on the asymptotic Teichmüller space

For an analytically infinite Riemann surface , the quasiconformal mapping class group always acts faithfully on the ordinary Teichmüller space . However in this paper, an example of is constructed for which acts trivially on its asymptotic Teichmüller space .

     

On the normal bundle of submanifolds of

Let be a submanifold of dimension of the complex projective space . We prove results of the following type.i) If is irregular and , then the normal bundle is indecomposable. ii) If is irregular, and , then is not the direct sum of two vector bundles of rank . iii) If , and is decomposable, then the natural restriction map is an isomorphism (and, in particular, if is embedded Segre in , then is indecomposable). iv) Let and , and assume that is a direct sum of line bundles; if assume furthermore that is simply connected and is not divisible in . Then is a complete intersection. These results follow from Theorem 2.1 below together with Le Potier's vanishing theorem. The last statement also uses a criterion of Faltings for complete intersection. In the case when this fact was proved by M. Schneider in 1990 in a completely different way.

     

Binomial coefficients and quadratic fields

Let be a real quadratic field with discriminant where is an odd prime. For we determine modulo in terms of a Lucas sequence, the fundamental unit and the class number of .

     

On the convergence in capacity on compact Kahler manifolds and its applications

The main aim of the present note is to study the convergence in on a compact Kahler mainfold . The obtained results are used to study global extremal functions and describe the -pluripolar hull of an -pluripolar subset in .

     

On a congruence of Blichfeldt concerning the order of finite groups

We show that if is a finite group, a conjugacy class of and , are the distinct elements in the multiset (here is the value of on any element of ), then

This is a dual to a generalization of a theorem of Blichfeldt stating that if is a finite group, a generalized character and are the distinct values of , then

We also observe that in Blichfeldt's congruence can be replaced, with a minor adjustment, by any rational value of . A similar change can be done to the first congruence above.

     

The depth of an ideal with a given Hilbert function

Let denote the polynomial ring in variables over a field with each . Let be a homogeneous ideal of with and the Hilbert function of the quotient algebra . Given a numerical function satisfying for some homogeneous ideal of , we write for the set of those integers such that there exists a homogeneous ideal of with and with . It will be proved that one has either for some or .

     

Extensions by spaces of continuous functions

We present two complementary results on the splitting of exact sequences having the form . The first one characterizes the Banach spaces such that for every compact space . The second is a nonlinear generalization of Zippin's criterion for the extension of -valued operators.

     

Polynomials with roots in for all

Let be a monic polynomial in with no rational roots but with roots in for all , or equivalently, with roots mod for all . It is known that cannot be irreducible but can be a product of two or more irreducible polynomials, and that if is a product of irreducible polynomials, then its Galois group must be a union of conjugates of proper subgroups. We prove that for any , every finite solvable group that is a union of conjugates of proper subgroups (where all these conjugates have trivial intersection) occurs as the Galois group of such a polynomial, and that the same result (with ) holds for all Frobenius groups. It is also observed that every nonsolvable Frobenius group is realizable as the Galois group of a geometric, i.e. regular, extension of .

     

Simple real rank zero algebras with locally Hausdorff spectrum

Let be a unital, simple, separable -algebra with real rank zero, stable rank one, and weakly unperforated ordered group. Suppose, also, that can be locally approximated by type I algebras with Hausdorff spectrum and bounded irreducible representations (the bound being dependent on the local approximating algebra). Then is tracially approximately finite dimensional (i.e., has tracial rank zero).

Hence, is an -algebra with bounded dimension growth and is determined by -theoretic invariants.

The above result also gives the first proof for the locally case.

     

Iterating the Cesàro operators

The discrete Cesàro operator associates to a given complex sequence the sequence , where . When is a convergent sequence we show that converges under the sup-norm if, and only if, . For its adjoint operator , we establish that converges for any .

The continuous Cesàro operator, , has two versions: the finite range case is defined for and the infinite range case for . In the first situation, when is continuous we prove that converges under the sup-norm to the constant function . In the second situation, when is a continuous function having a limit at infinity, we prove that converges under the sup-norm if, and only if, .

     

On exceptional eigenvalues of the Laplacian for

An explicit Dirichlet series is obtained, which represents an analytic function of in the half-plane except for having simple poles at points that correspond to exceptional eigenvalues of the non-Euclidean Laplacian for Hecke congruence subgroups by the relation for . Coefficients of the Dirichlet series involve all class numbers of real quadratic number fields. But, only the terms with for sufficiently large discriminants contribute to the residues of the Dirichlet series at the poles , where is the multiplicity of the eigenvalue for . This may indicate (I'm not able to prove yet) that the multiplicity of exceptional eigenvalues can be arbitrarily large. On the other hand, by density theorem the multiplicity of exceptional eigenvalues is bounded above by a constant depending only on .

     

Annihilating a subspace of with the sign of a continuous function

Let be a -finite, nonatomic, Baire measure space. Let be a finite dimensional subspace of . There is a bounded, continuous function, , defined on , such that

(1) for all , and (2) almost everywhere.

     

Boundary -algebras for acylindrical groups

Let be an infinite, locally finite tree with more than two ends. Let be an acylindrical uniform lattice. Then the boundary algebra is a simple Cuntz-Krieger algebra whose K-theory is determined explicitly.

     

A rigidity theorem for holomorphic generators on the Hilbert ball

We present a rigidity property of holomorphic generators on the open unit ball of a Hilbert space . Namely, if is the generator of a one-parameter continuous semigroup on such that for some boundary point , the admissible limit - , then vanishes identically on .

     

Distribution of Farey fractions in residue classes and Lang-Trotter conjectures on average

We prove that the set of Farey fractions of order , that is, the set , is uniformly distributed in residue classes modulo a prime provided for any fixed . We apply this to obtain upper bounds for the Lang-Trotter conjectures on Frobenius traces and Frobenius fields ``on average' over a one-parametric family of elliptic curves.

     

Detecting completeness from Ext-vanishing

Motivated by work of C. U. Jensen, R.-O. Buchweitz, and H. Flenner, we prove the following result. Let be a commutative noetherian ring and an ideal in the Jacobson radical of . Let be the -adic completion of . If is a finitely generated -module such that for all , then is -adically complete.

     

Pointwise Hardy inequalities and uniformly fat sets

We prove that it is equivalent for domain in to admit the pointwise -Hardy inequality, have uniformly -fat complement, or satisfy a uniform inner boundary density condition.

     

On Generic differential -extensions

Let be an algebraically closed field with trivial derivation and let denote the differential rational field , with , , , , differentially independent indeterminates over . We show that there is a Picard-Vessiot extension for a matrix equation , with differential Galois group , with the property that if is any differential field with field of constants , then there is a Picard-Vessiot extension with differential Galois group if and only if there are with well defined and the equation giving rise to the extension .