In this paper we are concerned with the problem of counting rational points of bounded height on rational cubic surfaces. For most such surfaces this question appears much too hard for current methods. We shall therefore examine a particular example, namely the surface

formula here

for which the problem is tractable, though by no means trivial. This surface contains the lines X0=Xi=0, for i=1, 2, 3, which we shall exclude from consideration. We therefore define

formula here

where an integer vector is said to be primitive if its components have no common factor. The function [Nscr ](H) counts points with the most naive height available. This suffices, however, to bring out the key features of the problem. We may observe at once that the condition x0[les ]H is redundant, and that it suffices to assume that (x1, x2, x3) is primitive.

Our principal result is the following theorem.

In an earlier paper [2], I gave explicit rules for calculating the Brauer group of a Del Pezzo surface of degree 3 or 4. Recent developments have made it desirable to extend these to the construction of the corresponding Azumaya algebras, so as to be able to calculate the associated Brauer–Manin obstruction to weak approximation or even to the Hasse principle. That is the main purpose of this note, which needs to be read in conjunction with [2]. The special case of diagonal cubic surfaces has already been treated in [1], though the process described there is not an algorithm in the strict sense; however, in numerical cases it is probably more efficient than the algorithm described in this note would be.

It has been conjectured that equation

formula here

has only finitely many solutions. We observe that (x, y, n, q)=(3, 11, 5, 2), (7, 20, 4, 2) and (18, 7, 3, 3) are solutions of (1). These are the only known solutions and perhaps (1) has no other solution. Ljunggren [8] proved in 1943 that (1) with q=2 has no solution other than x=3, y=11, n=5 and x=7, y=20, n=4. Shorey and Tijdeman [16] confirmed the conjecture if x is fixed. Let z>1 be an integer. The main purpose of this paper is to show that (1) has no solution if x is restricted to the infinite set of squares z2 with z[ges ]32 and z∈{2, 3, 4, 8, 9, 16, 27}.

If Q is a finitely generated abelian group, k a field, V a finitely generated kQ-module Bieri and Groves have proved that [otimes ]mV is a finitely generated as a kQ-module with diagonal Q-action if and only if ∧iV is finitely generated as a kQ-module for all 1[les ]i[les ]m. We generalize this result by showing that if the mth exterior power of V or the mth symmetric power of V is finitely generated as a kQ-module so is the mth tensor power of V. Further we show the equivalence between the finite generation of symmetric and tensor powers in the case when the ground ring is a PID of characteristic 0.

Suppose that f and g are transcendental entire functions of finite order, without wandering domains, and suppose that f∘g=g∘f. Then the Julia sets of f and g coincide.

As usual, let π(x) denote the number of prime numbers [les ]x and ψ(x) the well known Chebyshev's function. Let E(x) denote either (ψ(x)−x)/√x or (π(x)−li x)/(√x/log x), x[ges ]2. The study of E occupies a central place in the theory of primes. A classical result of Littlewood [7] states that E(x)=Ω±(log log log x) as x tends to infinity, showing in particular that E is unbounded. We expect rather erratic behaviour of E, but still one can wonder if it belongs to one of the classic function spaces [Xscr ], necessarily containing some unbounded functions. Let us extend definition of E(x) for x<2 by putting E(x)=0. A natural question is if it belongs to BMO, the space of functions with bounded mean oscillation, see, e.g. [2]. A locally integrable function f on the real line belongs to BMO if there exists a constant C such that for every bounded interval I⊂R we have

formula here

with a suitable constant αI∈R. [mid ]I[mid ] denotes here the length of I. Without any loss in generality one can take

formula here

the average of f over I (cf. [2], chapter VI). BMO is important and intensely studied in the complex analysis. It is obvious that BMO is larger than the space of bounded (measurable) functions and thus it seems a natural candidate for [Xscr ]. E∈BMO would mean that E behaves in a certain predictable way. Otherwise, we obtain another confirmation of the big irregularity in the distribution of primes.

We describe the construction of explicit representations of some of the exceptional covers of generic groups, that is, covers whose existence is not explained by the general theory of such groups.

We use certain derivations of a polynomial ring, that do not leave any proper non-zero ideal invariant, to construct simple non-holonomic modules over the nth Weyl algebra. This approach extends to the rings of differential operators of other smooth affine varieties, like smooth quadric surfaces.

We define Castelnuovo–Mumford regularity for graded modules over non-commutative graded algebras. Two fundamental commutative results are generalized to the non-commmutative case: a vanishing-theorem by Mumford, and a theorem on linear resolutions and syzygies by Eisenbud and Goto. The generalizations deal with sufficiently well-behaved algebras (i.e. so-called quantum polynomial algebras).

We go on to define Castelnuovo–Mumford regularity for sheaves on a non-commutative projective scheme, as defined by Artin. Again, a version of Mumford's vanishing-theorem is proved, and we use it to generalize a result by Martin, Migliore and Nollet, on degrees of generators of graded saturated ideals in polynomial algebras, to quantum polynomial algebras.

Finally, we generalize a practical result of Schenzel which determines the regularity of a module in terms of certain Tor-modules.

This paper asks: given a vector bundle ξ and a line bundle λ over the same base space, are λ[otimes ]ξ and ξ equivalent? We concentrate on real bundles ξ. Although the question is sensible in its own right, we explain in Section 2 our immediate motivation for studying it. In Section 3 we make some general comments about the question, the most significant being that under certain restrictions the answer depends on the stable class of ξ rather than on ξ itself (Proposition 3·4).

The rest of the paper tackles an interesting special case. To state the main result, let P(ℝn+1) denote n-dimensional real projective space, H the Hopf line bundle over it, and an+1 the order of the reduced Grothendieck group [wavy overbar]KO0(P(ℝn+1)).

We generalize a theorem of R. Thomas, which sometimes allows one to tell by inspection that a finitely presented group G is infinite. Groups to which his theorem applies have presentations with not too many more relators than generators, with at least some of the relators being proper powers. Our generalization provides lower bounds for the ranks of the abelianizations of certain normal subgroups of G in terms of their indices. We derive Thomas's theorem as a special case.

Definition [4]. Let A be a noetherian ring, [afr ] an ideal of A and M an A-module. M is said to be [afr ]-cofinite if M has support in V([afr ]) and ExtiA(A/[afr ], M) is a finite A-module for each i.

Remark. (a) If 0→M′→M→M″ →0 is exact and two of the modules in the sequence are [afr ]-cofinite, then so is the third one.

This has the following consequence, which will be used several times.

(b) If f[ratio ]M→N is a homomorphism between two [afr ]-cofinite modules and one of the three modules Ker f, Im f and Coker f is [afr ]-cofinite, then all three of them are [afr ]-cofinite.

Example [5, remark 1·3]. If A is local with maximal ideal [mfr ], then an A-module is [mfr ]-cofinite if and only if it is an artinian A-module.

The contact between a surface in Euclidean space and the family of spheres carries a good deal of useful geometrical information about the surface. The centres of those spheres having degenerate tangency with the surface (in Arnold's notation of type A2) form the focal set, and the set of points on the surface where there is a contact of type A3, the so-called ridge curve, is of some interest in the field of computer vision, as a robust feature of the surface. These ridges come together at umbilics, where the contact between the surface and the unique sphere of curvature is of type D. A dual notion, that of a subparabolic curve is also of some interest. In this paper we describe the way in which the ridge and subparabolic curves can evolve in a generic 1-parameter family of surfaces.

A link map is a map from a union of spheres into another sphere with pairwise disjoint images. Two link maps are link homotopic if one can be deformed into the other by a homotopy through link maps. This rather crude equivalence relation was introduced in 1954 by Milnor[13] for classical links and an algorithm for the classification problem in this setting was given in 1990 by Habegger and Lin[4]. In the past years, link homotopy in higher dimensions has also attracted much renewed interest. For link maps in a large metastable range, the corresponding classification problem has been reduced to standard questions in homotopy theory by Koschorke[9, 10].

In this paper we consider link maps of 2-spheres into the 4-sphere. This is another interesting case which differs considerably from the two cases above because of the failure of Whitney's trick in dimension four. Here the first contribution was made in 1984 by Fenn and Rolfsen[2] who established the first non-trivial link map of two 2- spheres into the 4-sphere. Their idea has been generalized by Kirk[6] to define his σ-invariant that distinguishes infinitely many different link homotopy classes. For several years, Kirk's invariant remained the strongest invariant for link maps into the 4-sphere. Only recently, it became known that Kirk's invariant suffices to classify two-component link maps in dimension four [12].

In this paper, we will prove that if A is a C*-algebra with an effective coaction ε by a compact quantum group, then the fixed point algebra and the reduced crossed product are Morita equivalent. As an application, we prove an imprimitivity type theorem for crossed products of coactions by discrete Kac C*-algebras.

Let (Vi, ti) be a pair consisting of a solid torus Vi and an unknotted arc ti properly imbedded in Vi for i=1 and 2. Taking the union of these pairs we obtain a pair (M, K) of the 3-dimensional sphere or a lens space M and a knot K. We call such a knot K a genus one 1-bridge knot. In this paper we determine when a genus one 1-bridge knot is the trivial knot or a cabled knot.

In this paper we investigate separation properties in the dual Ĝ of a connected, simply connected, nilpotent Lie group G. Following [4, 19], we are particularly interested in the question of when the group G is quasi-standard, in which case the group C*-algebra C*(G) may be represented as a continuous bundle of C*-algebras over a locally compact, Hausdorff, space such that the fibres are primitive throughout a dense subset. The same question for other classes of locally compact groups has been considered previously in [1, 5, 18]. Fundamental to the study of quasi-standardness is the relation of inseparability in Ĝ[ratio ]π∼σ in Ĝ if π and σ cannot be separated by disjoint open subsets of Ĝ. Thus we have been led naturally to consider also the set sep (Ĝ) of separated points in Ĝ (a point in a topological space is separated if it can be separated by disjoint open subsets from each point that is not in its closure).

Let M be a compact, orientable 3-manifold with ∂M a torus. If r is a slope on ∂M (the isotopy class of an unoriented essential simple loop), then we can form the closed 3-manifold M(r) by gluing a solid torus Vr to M along their boundaries in such a way that r bounds a disc in Vr. We say that M(r) is obtained from M by r-Dehn filling.

Assume now that M contains no essential sphere, disc, torus or annulus. Then, by Thurston's Geometrization Theorem for Haken manifolds [T1, T2], M is hyperbolic, in the sense that int M has a complete hyperbolic structure of finite volume. Furthermore, M(r) is hyperbolic for all but finitely many r [T1, T2] and the precise nature of the set of exceptional slopes E(M)={r: M(r) is not hyperbolic} has been the subject of a considerable amount of investigation. The maximal observed value of e(M)=[mid ]E(M)[mid ] (the cardinality of E(M)) is 10, realized, apparently uniquely, by the exterior of the figure eight knot [T1].

Let Δ(r1, r2) denote as usual the minimal geometric intersection number of two slopes r1 and r2. If [Sscr ] is any set of slopes, then clearly any upper bound for Δ([Sscr ])=max{Δ(r1, r2): r1, r2∈[Sscr ]} gives one for [mid ][Sscr ][mid ]. For example, one can check (using [GLi, lemma 2·1]) that for 1[les ]Δ([Sscr ])[les ]10, the maximum values of [mid ][Sscr ][mid ] are as given in Table 1.

In particular, any upper bound for Δ(M)=Δ(E(M)) gives a corresponding bound for e(M). (The maximal observed value of Δ(M) is 8, realized by the figure eight knot exterior and the figure eight sister manifold [T1, HW].)

If M(r) is not hyperbolic, then it is either reducible (contains an essential sphere), toroidal (contains an essential torus), a small Seifert fibre space (one with base S2 and at most three singular fibres), or a counterexample to the Geometrization Conjecture [T1, T2]. A survey of the presently known upper bounds on the distances Δ(r1, r2) between various classes of exceptional slopes r1 and r2, and the maximal values realized by known examples, is given in [Go2]. (See also [Wu2] for a discussion of the additional cases that arise when M has more than one boundary component.) In the present note we prove the following theorem, which deals with one further pair of possibilities.

We present a new class of curves which are self-approaching in the following sense. For any three consecutive points a, b, c on the curve the point b is closer to c than a to c. This is a generalisation of curves with increasing chords which are self- approaching in both directions. We show a tight upper bound of 5·3331… for the length of a self-approaching curve over the distance between its endpoints.

The Funk transform is the integral transform from the space of smooth even functions on the unit sphere S2⊂ℝ3 to itself defined by integration over great circles. One can regard this transform as a limit in a certain sense of the Penrose transform from [Copf ]ℙ2 to [Copf ]ℙ*ast;2. We exploit this viewpoint by developing a new proof of the bijectivity of the Funk transform which proceeds by considering the cohomology of a certain involutive (or formally integrable) structure on an intermediate space. This is the simplest example of what we hope will prove to be a general method of obtaining results in real integral geometry by means of complex holomorphic methods derived from the Penrose transform.