Let $k$ be a global field, $\bar{k}$ a separable closure of $k$, and ${{G}_{k}}$ the absolute Galois group Gal$(\bar{k}/k)$ of $\bar{k}$ over $k$. For every $\sigma \,\in \,{{G}_{K}}$, let ${{\bar{k}}^{\sigma }}$ be the fixed subfield of $\bar{k}$ under $\sigma$. Let $E/k$ be an elliptic curve over $k$. It is known that the Mordell–Weil group $E({{\bar{k}}^{\sigma }})$ has infinite rank. We present a new proof of this fact in the following two cases. First, when $k$ is a global function field of odd characteristic and $E$ is parametrized by a Drinfeld modular curve, and secondly when $k$ is a totally real number field and $E/k$ is parametrized by a Shimura curve. In both cases our approach uses the non-triviality of a sequence of Heegner points on $E$ defined over ring class fields.

We solve several multi-parameter families of binomial Thue equations of arbitrary degree; for example, we solve the equation

$${{5}^{u}}{{x}^{n}}-{{2}^{r}}{{3}^{5}}{{y}^{n}}=\pm 1,$$

in non-zero integers $x,y$ and positive integers $u,r,s$ and $n\ge 3$. Our approach uses several Frey curves simultaneously, Galois representations and level-lowering, new lower bounds for linear forms in 3 logarithms due to Mignotte and a famous theorem of Bennett on binomial Thue equations.

Let $A={{({{a}_{j,k}})}_{j,k\ge 1}}$ be a non-negative matrix. In this paper, we characterize those $A$ for which ${{\left\| A \right\|}_{E,F}}$ are determined by their actions on decreasing sequences, where $E$ and $F$ are suitable normed Riesz spaces of sequences. In particular, our results can apply to the following spaces: ${{\ell }_{p}}$, $d(w,p)$, and ${{\ell }_{p}}(w)$. The results established here generalize ones given by Bennett; Chen, Luor, and Ou; Jameson; and Jameson and Lashkaripour.

We say that an abelian variety over a $p$-adic field $K$ has anisotropic reduction $(\text{AR})$ if the special fiber of its Néron minimal model does not contain a nontrivial split torus. This includes all abelian varieties with potentially good reduction and, in particular, those with complex or quaternionic multiplication. We give a bound for the size of the $K$-rational torsion subgroup of a $g$-dimensional $\text{AR}$ variety depending only on $g$ and the numerical invariants of $K$ (the absolute ramification index and the cardinality of the residue field). Applying these bounds to abelian varieties over a number field with everywhere locally anisotropic reduction, we get bounds which, as a function of $g$, are close to optimal. In particular, we determine the possible cardinalities of the torsion subgroup of an $\text{AR}$ abelian surface over the rational numbers, up to a set of 11 values which are not known to occur. The largest such value is 72.

We prove a characteristic free version of Weyl’s theorem on polarization. Our result is an exact analogue of Weyl’s theorem, the difference being that our statement is about separating invariants rather than generating invariants. For the special case of finite group actions we introduce the concept of cheap polarization, and show that it is enough to take cheap polarizations of invariants of just one copy of a representation to obtain separating vector invariants for any number of copies. This leads to upper bounds on the number and degrees of separating vector invariants of finite groups.

This is the third in a series of works devoted to spectral asymptotics for non-selfadjoint perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2, having a periodic classical flow. Assuming that the strength $\epsilon$ of the perturbation is in the range ${{h}^{2}}\ll \epsilon \ll {{h}^{1/2}}$ (and may sometimes reach even smaller values), we get an asymptotic description of the eigenvalues in rectangles $[-1/C,1/C]+i\epsilon [{{F}_{0}}-1/C,{{F}_{0}}+1/C],C\gg 1$, when $\epsilon {{F}_{0}}$ is a saddle point value of the flow average of the leading perturbation.

We study the case of an Axiom $\text{A}$ holomorphic non-degenerate (hence non-invertible) map $f:{{\mathbb{P}}^{2}}\mathbb{C}\to {{\mathbb{P}}^{2}}\mathbb{C}$, where ${{\mathbb{P}}^{2}}\mathbb{C}$ stands for the complex projective space of dimension 2. Let $\Lambda $ denote a basic set for $f$ of unstable index 1, and $x$ an arbitrary point of $\Lambda $; we denote by ${{\delta }^{s}}(x)$ the Hausdorff dimension of $\text{W}_{r}^{s}(x) \cap \,\Lambda$, where $r$ is some fixed positive number and $\text{W}_{r}^{s}(x)$ is the local stable manifold at $x$ of size $r;{{\delta }^{s}}(x)$ is called the stable dimension at$x$. Mihailescu and Urbański introduced a notion of inverse topological pressure, denoted by ${{P}^{-}}$, which takes into consideration preimages of points. Manning and McCluskey studied the case of hyperbolic diffeomorphisms on real surfaces and give formulas for Hausdorff dimension. Our non-invertible situation is different here since the local unstable manifolds are not uniquely determined by their base point, instead they depend in general on whole prehistories of the base points. Hence our methods are different and are based on using a sequence of inverse pressures for the iterates of $f$, in order to give upper and lower estimates of the stable dimension. We obtain an estimate of the oscillation of the stable dimension on $\Lambda $. When each point $x$ from $\Lambda $ has the same number ${d}'$ of preimages in $\Lambda $, then we show that ${{\delta }^{s}}(x)$ is independent of $x$; in fact ${{\delta }^{s}}(x)$ is shown to be equal in this case with the unique zero of the map $t\to P(t{{\phi }^{s}}-\log {d}')$. We also prove the Lipschitz continuity of the stable vector spaces over $\Lambda $; this proof is again different than the one for diffeomorphisms (however, the unstable distribution is not always Lipschitz for conformal non-invertible maps). In the end we include the corresponding results for a real conformal setting.

For a general vector field we exhibit two Hilbert spaces, namely the space of so called closed functions and the space of exact functions and we calculate the codimension of the space of exact functions inside the larger space of closed functions. In particular we provide a new approach for the known cases: the Glauber field and the second-order Ginzburg–Landau field and for the case of the fourth-order Ginzburg–Landau field.

The Jiang–Su algebra $Z$ has come to prominence in the classification program for nuclear ${{C}^{*}}$-algebras of late, due primarily to the fact that Elliott’s classification conjecture in its strongest form predicts that all simple, separable, and nuclear ${{C}^{*}}$-algebras with unperforated $\text{K}$-theory will absorb $Z$ tensorially, i.e., will be $Z$-stable. There exist counterexamples which suggest that the conjecture will only hold for simple, nuclear, separable and $Z$-stable ${{C}^{*}}$-algebras. We prove that virtually all classes of nuclear ${{C}^{*}}$-algebras for which the Elliott conjecture has been confirmed so far consist of $Z$-stable ${{C}^{*}}$-algebras. This follows in large part from the following result, also proved herein: separable and approximately divisible ${{C}^{*}}$-algebras are $Z$-stable.