Counting cusps of subgroups of

Let be a number field with real places and complex places, and let be the ring of integers of . The quotient has cusps, where is the class number of . We show that under the assumption of the generalized Riemann hypothesis that if is not or an imaginary quadratic field and if , then has infinitely many maximal subgroups with cusps. A key element in the proof is a connection to Artin's Primitive Root Conjecture.

     

A remark on irregularity of the -Neumann problem on non-smooth domains

It is an observation due to J. J. Kohn that for a smooth bounded pseudoconvex domain in there exists such that the -Neumann operator on maps (the space of -forms with coefficient functions in -Sobolev space of order ) into itself continuously. We show that this conclusion does not hold without the smoothness assumption by constructing a bounded pseudoconvex domain in , smooth except at one point, whose -Neumann operator is not bounded on for any .

     

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.

     

Tauberian type theorem for operators with interpolation spectrum for Hölder classes

We consider an invertible operator on a Banach space whose spectrum is an interpolating set for Hölder classes. We show that if , , with and , then for all , assuming that satisfies suitable regularity conditions. When is a Hilbert space and (i.e. is a contraction), we show that under the same assumptions, is unitary and this is sharp.

     

On the triple jump of the set of atoms of a Boolean algebra

We prove the following result concerning the degree spectrum of the atom relation on a computable Boolean algebra. Let be a computable Boolean algebra with infinitely many atoms and be the Turing degree of the atom relation of . If is a c.e. degree such that , then there is a computable copy of where the atom relation has degree . In particular, for every c.e. degree , any computable Boolean algebra with infinitely many atoms has a computable copy where the atom relation has degree .

     

Enriched Reedy categories

We define the notion of an enriched Reedy category and show that if is a -Reedy category for some symmetric monoidal model category and is a -model category, the category of -functors and -natural transformations from to is again a model category.

     

On the plane section of an integral curve in positive characteristic

If is an integral curve and an algebraically closed field of characteristic 0, it is known that the points of the general plane section of are in uniform position. From this it follows easily that the general minimal curve containing is irreducible. If char, the points of may not be in uniform position. However, we prove that the general minimal curve containing is still irreducible.

     

Dynamics of the function and the Green-Tao theorem on arithmetic progressions in the primes

Let be the set of all positive integers , where are primes and possibly two, but not all three of them are equal. For any , define a function by where is the largest prime factor of . It is clear that if , then . For any , define , for . An element is semi-periodic if there exists a nonnegative integer and a positive integer such that . We use ind to denote the least such nonnegative integer . Wushi Goldring [Dynamics of the function and primes, J. Number Theory 119(2006), 86-98] proved that any element is semi-periodic. He showed that there exists such that , ind, and conjectured that ind can be arbitrarily large.

In this paper, it is proved that for any we have ind , and the Green-Tao Theorem on arithmetic progressions in the primes is employed to confirm Goldring's above conjecture.

     

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.

     

Dahlberg's bilinear estimate for solutions of divergence form complex elliptic equations

We consider divergence form elliptic operators , defined in , where the coefficient matrix is , uniformly elliptic, complex and -independent. Using recently obtained results concerning the boundedness and invertibility of layer potentials associated to such operators, we show that if in , then for any vector-valued we have the bilinear estimate

where and where is the usual non-tangential maximal operator. The result is new even in the case of real symmetric coefficients and generalizes an analogous result of Dahlberg for harmonic functions on Lipschitz graph domains. We also identify the domain of the generator of the Poisson semigroup for the equation in

     

Cofinality changes required for a large set of unapproachable ordinals below

In , assume that is a strong limit cardinal and . Let be the set of approachable ordinals less than . An open question of M. Foreman is whether can be non-stationary in some and preserving extension of . It is shown here that if is such an outer model, then is infinite, for each positive integer .

     

On the cofiniteness of local cohomology modules

In this note we show that if is an ideal of a Noetherian ring and is a finitely generated -module, then for any minimax submodule of the -module is finitely generated, whenever the modules are minimax. As a consequence, it follows that the associated primes of are finite. This generalizes the main result of Brodmann and Lashgari (2000).

     

Homogeneous polynomials on strictly convex domains

We consider a circular, bounded, strictly convex domain with boundary of class . For any compact subset of we construct a sequence of homogeneous polynomials on which are big at each point of . As an application for any circular subset of type we construct a holomorphic function which is square integrable on and such that where denotes unit disc in .

     

Stable algebras of entire functions

Suppose that and belong to the algebra generated by the rational functions and an entire function of finite order on and that has algebraic polar variety. We show that either or , where is a polynomial and are rational functions. In the latter case, belongs to the algebra generated by the rational functions, and .

The stability property is related to the problem of algebraic dependence of entire functions over the ring of polynomials. The case of algebraic dependence over of two entire or meromorphic functions on is completely resolved in this paper.

     

Radial limits of inner functions and Bloch spaces

Let be an inner function in the unit ball , . Assume that

where and is the radial derivative. Then, for all , the set has a non-zero real Hausdorff -content, and it has a non-zero complex Hausdorff -content.

     

On zeros of Eisenstein series for genus zero Fuchsian groups

Let SL be a genus zero Fuchsian group of the first kind with as a cusp, and let be the holomorphic Eisenstein series of weight on that is nonvanishing at and vanishes at all the other cusps (provided that such an Eisenstein series exists). Under certain assumptions on and on a choice of a fundamental domain , we prove that all but possibly of the nontrivial zeros of lie on a certain subset of . Here is a constant that does not depend on the weight, is the upper half-plane, and is the canonical hauptmodul for

     

Characterizing indecomposable plane continua from their complements

We show that a plane continuum is indecomposable iff has a sequence of not necessarily distinct complementary domains satisfying the double-pass condition: for any sequence of open arcs, with and , there is a sequence of shadows , where each is a shadow of , such that . Such an open arc divides into disjoint subdomains and , and a shadow (of ) is one of the sets .

     

On the - boundedness of operators

We prove that if is in , is a Banach space, and is a linear operator defined on the space of finite linear combinations of -atoms in with the property that

then admits a (unique) continuous extension to a bounded linear operator from to . We show that the same is true if we replace -atoms by continuous -atoms. This is known to be false for -atoms.

     

An elementary proof of the law of quadratic reciprocity over function fields

Let and be relatively prime monic irreducible polynomials in (). In this paper, we give an elementary proof for the following law of quadratic reciprocity in :

where is the Legendre symbol.