In this paper, we study the Mordell-Weil group of an elliptic curve as a Galois module. We consider an elliptic curve $E$ defined over a number field $K$ whose Mordell-Weil rank over a Galois extension $F$ is 1, 2 or 3. We show that $E$ acquires a point (points) of infinite order over a field whose Galois group is one of ${{C}_{n}}\times {{C}_{m}}(n=1,\,2,\,3,\,4,\,6,\,m\,=\,1,\,2),\,{{D}_{n}}\times {{C}_{m}}(n=2,\,3,\,4,\,6,\,m=\,1,\,2),\,{{A}_{4}}\times {{C}_{m}}(m=1,\,2),\,{{S}_{4}}\,\times \,{{C}_{m}}(m=1,\,2)$ . Next, we consider the case where $E$ has complex multiplication by the ring of integers $\mathcal{O}$ of an imaginary quadratic field $\Re $ contained in $K$. Suppose that the $\mathcal{O}$-rank over a Galois extension $F$ is 1 or 2. If $\Re \ne \mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$ and ${{h}_{\Re }}$ (class number of $\Re $) is odd, we show that $E$ acquires positive $\mathcal{O}$-rank over a cyclic extension of $K$ or over a field whose Galois group is one of $\text{S}{{\text{L}}_{2}}(\mathbb{Z}/3\mathbb{Z})$ , an extension of $\text{S}{{\text{L}}_{2}}(\mathbb{Z}/3\mathbb{Z})$ by $\mathbb{Z}/2\mathbb{Z}$, or a central extension by the dihedral group. Finally, we discuss the relation of the above results to the vanishing of $L$-functions.

A homogeneous real polynomial $p$ is hyperbolic with respect to a given vector $d$ if the univariate polynomial $t\,\mapsto \,p(x\,-\,td)$ has all real roots for all vectors $x$. Motivated by partial differential equations, Gårding proved in 1951 that the largest such root is a convex function of $x$, and showed various ways of constructing new hyperbolic polynomials. We present a powerful new such construction, and use it to generalize Gårding’s result to arbitrary symmetric functions of the roots. Many classical and recent inequalities follow easily. We develop various convex-analytic tools for such symmetric functions, of interest in interior-point methods for optimization problems over related cones.

Bivariate polynomials with a fixed leading term ${{x}^{m}}{{y}^{n}}$ , which deviate least from zero in the uniform or ${{L}^{2}}$-norm on the unit disk $D$ (resp. a triangle) are given explicitly. A similar problem in ${{L}^{p}}$ , $1\,\le \,p\,\le \,\infty $ is studied on $D$ in the set of products of linear polynomials.

A contractive $n$-tuple $A\,=\,({{A}_{1}},...,{{A}_{n}})$ has a minimal joint isometric dilation $S\,=\,({{S}_{1}},...,{{S}_{n}})$ where the ${{S}_{i}}$’s are isometries with pairwise orthogonal ranges. This determines a representation of the Cuntz-Toeplitz algebra. When $A$ acts on a finite dimensional space, the wot-closed nonself-adjoint algebra $\mathfrak{S}$ generated by $S$ is completely described in terms of the properties of $A$. This provides complete unitary invariants for the corresponding representations. In addition, we show that the algebra $\mathfrak{S}$ is always hyper-reflexive. In the last section, we describe similarity invariants. In particular, an $n$-tuple $B$ of $d\,\times \,d$ matrices is similar to an irreducible $n$-tuple $A$ if and only if a certain finite set of polynomials vanish on $B$.

We generalise the two-sided construction of examples of pairs of subfactors of the hyperfinite $\text{I}{{\text{I}}_{1}}$ factor $R$ in $[\text{E1}]$—which arise by considering unitary braid representations with certain properties—to multi-sided pairs. We show that the index for the multi-sided pair can be expressed as a power of that for the two-sided pair. This construction can be applied to the natural examples—where the braid representations are obtained in connection with the representation theory of Lie algebras of types $A,\,B,\,C,\,D$. We also compute the (first) relative commutants.

We study when the spaces of Lorentz multipliers from ${{L}^{p,t\,}}\to \,{{L}^{p,s}}$ are distinct. Our main interest is the case when $s\,<\,t$, the Lorentz-improving multipliers. We prove, for example, that the space of multipliers which map ${{L}^{p,t\,}}\to \,{{L}^{p,s}}$ is different from those mapping ${{L}^{r,v}}\,\to \,{{L}^{r,u}}$ if either $r\,=\,p$ or ${p}'$ and $1/s\,-\,1/t\,\ne \,1/u\,-\,1/v$, or $r\,\ne \,p$ or ${p}'$. These results are obtained by making careful estimates of the Lorentz multiplier norms of certain linear combinations of Fejer or Dirichlet kernels. For the case when the first indices are different the linear combination we analyze is in the spirit of a Rudin-Shapiro polynomial.

We give a description of the monoid of Murray-von Neumann equivalence classes of projections for multiplier algebras of a wide class of $\sigma $-unital simple ${{C}^{*}}$-algebras $A$ with real rank zero and stable rank one. The lattice of ideals of this monoid, which is known to be crucial for understanding the ideal structure of the multiplier algebra $\mathcal{M}(A)$, is therefore analyzed. In important cases it is shown that, if $A$ has finite scale then the quotient of $\mathcal{M}(A)$ modulo any closed ideal $I$ that properly contains $A$ has stable rank one. The intricacy of the ideal structure of $\mathcal{M}(A)$ is reflected in the fact that $\mathcal{M}(A)$ can have uncountably many different quotients, each one having uncountably many closed ideals forming a chain with respect to inclusion.

For a dense ${{G}_{\delta }}$ set of real parameters $\theta $ in [0, 1] (containing the rationals) it is shown that the group ${{K}_{0}}({{A}_{\theta }}\,{{\rtimes }_{\sigma }}\,{{\mathbb{Z}}_{4}})$ is isomorphic to ${{\mathbb{Z}}^{9}}$, where ${{A}_{\theta }}$ is the rotation ${{\text{C}}^{*}}$-algebra generated by unitaries $U,\,V$ satisfying $VU\,=\,{{e}^{2\pi i\theta }}UV$ and $\sigma $ is the Fourier automorphism of ${{A}_{\theta }}$ defined by $\sigma (U)\,=\,V,\,\sigma (V)\,=\,{{U}^{-1}}$. More precisely, an explicit basis for ${{K}_{0}}$ consisting of nine canonical modules is given. (A slight generalization of this result is also obtained for certain separable continuous fields of unital ${{\text{C}}^{*}}$-algebras over [0, 1].) The Connes Chern character $\text{ch:}\,{{K}_{0}}({{A}_{\theta }}\,{{\rtimes }_{\sigma }}\,{{\mathbb{Z}}_{4}})\,\to \,{{H}^{\text{ev}}}{{({{A}_{\theta \,}}{{\rtimes }_{\sigma }}\,{{\mathbb{Z}}_{4}})}^{*}}$ is shown to be injective for a dense ${{G}_{\delta }}$ set of parameters $\theta $. The main computational tool in this paper is a group homomorphism $\mathbf{T}\,\text{:}\,{{K}_{0}}({{A}_{\theta }}\,{{\rtimes }_{\sigma }}\,{{\mathbb{Z}}_{4}})\,\to \,{{\mathbb{R}}^{8}}\,\times \,\mathbb{Z}$ obtained from the Connes Chern character by restricting the functionals in its codomain to a certain nine-dimensional subspace of ${{H}^{\text{ev}}}({{A}_{\theta }}\,{{\rtimes }_{\sigma }}\,{{\mathbb{Z}}_{4}})$. The range of $\mathbf{T}$ is fully determined for each $\theta $. (We conjecture that this subspace is all of ${{H}^{\text{ev}}}$.)