相关论文: $p$-adic Periods and Selmer Scheme Images

We discuss how non-commutative fundamental groups could eventually contribute to algorithms for finding rational points on hyperbolic curves.

Time-periodic weak solutions for a coupled hyperbolic-parabolic system are obtained. A linear heat and wave equation are considered on two respective $d$-dimensional spatial domains that share a common $(d-1)$-dimensional interface…

We establish necessary and sufficient conditions for the realization of mapping schemata as post-critically finite polynomials, or more generally, as post-critically finite polynomial maps from a finite union of copies of the complex…

In the late 1980's Sullivan initiated a programme to prove quasisymmetric rigidity in one-dimensional dynamics: interval or circle maps that are topologically conjugate are quasisymmetrically conjugate (provided some obvious necessary…

The Asymptotic Iteration Method (AIM) is a technique for solving analytically and approximately the linear second-order differential equation, especially the eigenvalue problems that frequently appear in theoretical and mathematical…

We illustrate a completely analytic approach to Mel'nikov theory, which is based on a suitable extension of a classical method, and which is parallel and -- at least in part -- complementary to the standard procedure. This approach can be…

Let $X/\mathbb{Q}$ be a curve of genus $g \ge 2$ with Jacobian $J$ and let $\ell$ be a prime of good reduction. Using Selmer varieties, Kim defines a decreasing sequence \[ X(\mathbb{Q}_\ell) \supseteq X(\mathbb{Q}_\ell)_1 \supseteq…

We study a certain polytope arising from embedding the Hamiltonian cycle problem in a discounted Markov decision process. The Hamiltonian cycle problem can be reduced to finding particular extreme points of a certain polytope associated…

We reformulate the theory of p-adic iterated integrals on semistable curves using the unipotent log rigid fundamental group. This fundamental group carries Frobenius and monodromy operators whose basic properties are established. By…

We describe how one can use the "Selmer group Chabauty" method developed by the author to show that certain hyperelliptic curves of the form $$ C \colon y^2 = x^N + h(x)^2 \,, $$ where $N = 2g + 1$ is odd, $h \in \mathbb{Z}[x]$ with…

We explore the combination theorem for a group G splitting as a graph of relatively hyperbolic groups. Using the fine graph approach to relative hyperbolicity, we find short proofs of the relative hyperbolicity of G under certain…

This paper establishes limit theorems and quantitative statistical stability for a class of piecewise partially hyperbolic maps that are not necessarily continuous nor locally invertible. By employing a flexible functional-analytic…

Explicit formulas are obtained for the number of periodic points and maximum tail length of split polynomial maps over finite fields for affine and projective space. This work includes a detailed analysis of the structure of the directed…

Mumford showed that Schottky subgroups of $PGL(2,K)$ give rise to certain curves, now called Mumford curves, over a non-Archimedean field K. Such curves are foundational to subjects dealing with non-Archimedean varieties, including…

Let $X$ be a smooth projective curve of genus $\geq2$ over a number field. A natural variant of Grothendieck's Section Conjecture postulates that every section of the fundamental exact sequence for $X$ which everywhere locally comes from a…

We generalize a well known periodicity lemma from the case of free groups to the case of acylindrically hyperbolic groups. This generalization will be used later to describe solutions of certain equations in acylindrically hyperbolic groups…

This note addresses some questions that arise in the series of works by Kyoji Saito on the growth functions of graphs. We study "hyperbolike" graphs, which include Cayley graphs of hyperbolic groups. We generalize some well-known results on…

Model sets (or cut and project sets) provide a familiar and commonly used method of constructing and studying nonperiodic point sets. Here we extend this method to situations where the internal spaces are no longer Euclidean, but instead…

We prove the conjectured compatibility of $p$-adic fundamental lines with specializations at motivic points for a wide class of $p$-adic families of $p$-adic Galois representations (for instance, the families which arise from $p$-adic…

We construct inducing schemes for general multi-dimensional piecewise expanding maps where the base transformation is Gibbs-Markov and the return times have exponential tails. Such structures are a crucial tool in proving statistical…