Let K=ℚ(√−3). Assume E is an elliptic curve defined over K, with complex multiplication by OK, the ring of integers of K. Ω∈[Copf]X is a minimal period of E. For each prime v of K, let cv denote the Tamagawa number [E(Kv)[ratio ]E0(Kv)]. Let L(E/K, s) denote the Hasse–Weil L-function of E/K. [lfloor][mid ][rfloor](E/K)=ker{H1(K, E)→[oplus ]vH1(Kv, E)} denotes the Tate-Shafarevic group of E/K.

In recent years there has been an interest in using elliptic logarithms to find integral points on elliptic curves defined over the rationals, see [23], [17], [6] and [12]. This has been partly due to work of David [5], who gave an explicit lower bound for linear forms in elliptic logarithms. Previously, integral points on elliptic curves had been found by Siegel's method; that is, a reduction to a set of Thue equations which could be solved, in principle, by the methods in [19]. For examples of this method see [3], [7], [16], [18], [21], [22] and [8]. Other techniques can be used to find all integral points in some special cases, see, for instance, [14].

Triquotient maps of topological spaces were introduced by E. Michael as a natural generalization of both open and proper surjections. We introduce the notion of localic triquotient map. Our main result is that localic triquotient maps are effective descent maps. This generalizes the corresponding results for proper surjections (J. Vermeulen) and open surjections (A. Joyal and M. Tierney). Further results concern stability of triquotiency under various operations, for instance, arbitrary products and filtered (inverse) limits. Among the applications are a new constructive proof of Tychonoff's theorem, and a new result on stability of open surjections under filtered limits.

Let PI (respectively CPI) be the convex compact set of all unital positive (respectively completely positive) linear maps from the matrix algebra Mm([Copf]) into Mn([Copf]). We show that maximal faces of CPI correspond to one dimensional subspaces of the vector space Mm, n([Copf]). Furthermore, a maximal face of CPI lies on the boundary of PI if and only if the corresponding subspace is generated by a rank one matrix.

Let R be a commutative ring, Γ a finite group acting on R, and let k∈ℕ be invertible in R. Generalizing a definition of Kervaire, we construct an Adams operation ψk on the Grothendieck group and on the higher K-theory of projective modules over the twisted group ring R#Γ. For this, we generalize Atiyah's cyclic power operations and use shuffle products in higher K-theory. For the Grothendieck group, we show that ψk is multiplicative and that it commutes with base change, with the Cartan homomorphism, and with ψl for any other l which is invertible in R.

We study the question of whether the Morava K-theory of the classifying space of an elementary abelian group V is a permutation module (in either of two distinct senses, defined below) for the automorphism group of V. We use Brauer characters and computer calculations. Our algorithm for finding permutation submodules of modules for p-groups may be of independent interest.

In [7], the second author proved that there is an integer k such that every element of a finite non-abelian simple group S is a product of k commutators in S. The motivation for proving this result came from a model-theoretic question about simple groups. The proof depended on the classification of the finite simple groups, a theorem of Malle, Saxl and Weigel [5] which shows that in many finite simple classical groups S there is a real conjugacy class R such that S=R3∪{1}, and an ultraproduct argument. Here we shall use a similar combination of ideas to prove the following result.

It is shown that the analogue of the Helson–Reiter theorem for ideals in group algebras of locally compact abelian groups holds for a class of groups including all divisible nilpotent discrete groups.

Given a locally compact group G and a probability measure μ on G it is of interest to know, in various situations, whether there exist divergent sequences {gn} such that {gn μg−1n is relatively compact (see for example [DM3] and [DS]); this phenomenon may be viewed as ‘collapsing’ of the measure. It is the purpose of this note to prove Theorem 1 below and give certain applications to the asymptotic behaviour of concentration functions.

We completely classify all the twistor harmonic (non-minimal) Lagrangian immersions of compact surfaces in the complex projective plane [Copf]ℙ2, i.e. those Lagrangian immersions such that their twistor lifts to the twistor space over [Copf]ℙ2 are harmonic maps.

Some usual and important operations: change of variables, application of Fubini's theorem and derivation with respect to the isolated singularity (in the present work with respect to the origin of the spherical coordinates (r, θ)) are studied for the following singular integral

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where α∈C, Re(α)[les ]−n, j∈ℕ, α∈L1([sum ]n, C) and the symbol fp∫Ω, U means an integration on the set Ω in the finite part sense of Hadamard with respect to the domain configuration U. Moreover, applications to integral operators are outlined.

Let f∈L1(ℝd), and let fˆ be its Fourier integral. We study summability of the l-1 partial integral S(1)R, d(f; x)=∫[mid ]v[mid ][les ]R eiv·x fˆ(v)dv, x∈ℝd; note that the integral ranges over the l1-ball in ℝd centred at the origin with radius R>0. As a central result we prove that for δ[ges ]2d−1 the l-1 Riesz (R, δ) means of the inverse Fourier integral are positive, the lower bound being best possible. Moreover, we will give an l-1 analogue of Schoenberg's modification of Bochner's theorem on positive definite functions on ℝd as well as an extention of Polya's sufficiency condition.

Let X be a compact totally ordered space made into a semigroup by the multiplication xy=max{x, y}. Suppose that there is a continuous regular Borel measure μ on X with supp μ=X. Then the space L1(μ) of μ-integrable functions becomes a Banach algebra when provided with convolution as multiplication. The second dual L1(μ)** therefore has two Arens multiplications, each making it a Banach algebra. We shall always consider L1(μ)** to have the first of these: if F, G∈L1(μ)** and F=w*−limi ϕi, G=w*−limj ψj, where (ϕi), (ψj) are bounded nets in L1(μ), then

formula here

A complete system (CS) [Jscr]={J1, ..., Jn} on a connected closed surface F is a collection of pairwise disjoint simple closed curves on F such that the surface obtained by cutting F open along [Jscr] is a 2-sphere with 2n-holes. Two CSs on F are equivalent if each can be obtained from the other via finite number of slides (defined in Section 1) and isotopies. Let M be a 3-manifold and F a boundary component of M of genus n. A CS of surfaces for M is a CS on F which bounds n pairwise disjoint incompressible orientable surfaces in M. When [Jscr] is a CS of discs on the boundary of a handlebody V, it is well known that any CS on F which is equivalent to [Jscr] is also a CS of discs for V. Our first result says that the same thing happens for a CS of surfaces for M, that is, if [Jscr] is a CS of surfaces for M, then any CS equivalent to [Jscr] is also a CS of surfaces for M. The following theorem is our main result on CS of surfaces in 3-manifolds: