Let $E$ be an elliptic curve defined over an imaginary quadratic field $F$ of class number $1$. No systematic construction of global points on such an $E$ is known. In this article, we present a $p$-adic analytic construction of points on $E$, which we conjecture to be global, defined over ring class fields of a suitable relative quadratic extension $K/F$. The construction follows ideas of Darmon to produce an analog of Heegner points, which is especially interesting since none of the geometry of modular parametrizations extends to this setting. We present some computational evidence for our construction

We show that if $f$ is a $1$-quasiconformal map defined on an open subset of a Carnot group $G$, then composition with $f$ preserves $Q$-harmonic functions. We combine this with a regularity theorem for $Q$-harmonic functions and an algebraic regularity theorem for maps between Carnot groups to show that $f$ is smooth. We give some applications to the study of rigidity

We study a refined version of Linnik's problem on the asymptotic behavior of the number of representations of integers $m$ by an integral polynomial as $m$ tends to infinity. Assuming that the polynomials arise from invariant theory, we reduce the question to the study of limiting behavior of measures invariant under unipotent flows. Our main tool is then Ratner's theorem on the uniform distribution of unipotent flows, in a form refined by Dani and Margulis [DM2]

We prove an abstract Birkhoff normal form theorem for Hamiltonian partial differential equations (PDEs). The theorem applies to semilinear equations with nonlinearity satisfying a property that we call tame modulus. Such a property is related to the classical tame inequality by Moser. In the nonresonant case we deduce that any small amplitude solution remains very close to a torus for very long times. We also develop a general scheme to apply the abstract theory to PDEs in one space dimensions, and we use it to study some concrete equations (nonlinear wave (NLW) equation, nonlinear Schrödinger (NLS) equation) with different boundary conditions. An application to an NLS equation on the $d$-dimensional torus is also given. In all cases we deduce bounds on the growth of high Sobolev norms. In particular, we get lower bounds on the existence time of solutions

Let $R$ be a complete discrete valuation ring (DVR) of mixed characteristic $(0,p)$ with field of fractions $K$ containing the $p$th roots of unity. This article is concerned with semistable models of $p$-cyclic covers of the projective line $C\to {\mathbb{P}}_K^1$. We start by providing a new construction of a semistable model of $C$ in the case of an equidistant branch locus. If the cover is given by the Kummer equation $Z^p=f(X_0)$, we define what we call the monodromy polynomial $L(Y)$ of $f(X_0)$, a polynomial with coefficients in $K$. Its zeros are key to obtaining a semistable model of $C$. As a corollary, we obtain an upper bound for the minimal extension $K\prime /K$, over which a stable model of the curve $C$ exists. Consider the polynomial $L(Y)\Pi (Y^p-f(y_i))$, where the $y_i$ range over the zeros of $L(Y)$. We show that the splitting field of this polynomial always contains $K\prime$ and that, in some instances, the two fields are equal

The purpose of this work is to define a derived Hall algebra $\mathrm{DH}(T)$, associated to any differential graded (DG) category $T$ (under some finiteness conditions), generalizing the Hall algebra of an abelian category. Our main theorem states that $\mathrm{DH}(T)$ is associative and unital. When the associated triangulated category $[T]$ is endowed with a t-structure with heart $A$, it is shown that $\mathrm{DH}(T)$ contains the usual Hall algebra $H(A)$. We also prove an explicit formula for the derived Hall numbers purely in terms of invariants of the triangulated category associated to $T$. As an example, we describe the derived Hall algebra of a hereditary abelian category