Related papers: The wild McKay correspondence and $p$-adic measure…
In the paper we completely describe the set of all solutions of a recursive equation, arising from the Bethe lattice models over $p$-adic numbers.
Results in $p$-adic transcendence theory are applied to two problems in the Chabauty-Coleman method. The first is a question of McCallum and Poonen regarding repeated roots of Coleman integrals. The second is to give lower bounds on the…
Previous works have shown that certain weight $2$ newforms are $p$-adic limits of weakly holomorphic modular forms under repeated application of the $U$-operator. The proofs of these theorems originally relied on the theory of harmonic…
About 20 years ago, J-P.~Serre announced a bound on the trace of elements of compact Lie groups under the adjoint representation together with related results, provided indications of his proofs, and invited a better proof. This note…
We prove a $p$-adic version of the Integral Geometry Formula for averaging the intersection of two $p$-adic projective algebraic sets. We apply this result to give bounds on the number of points in the modulo $p^m$ reduction of a projective…
Using the theta correspondence, we extend the classification of irreducible representations of quasi-split unitary groups (the so-called local Langlands correspondence, which was established in an early paper of Mok) to non quasi-split…
We use a method of Buzzard to study p-adic families of different types of modular forms - classical, over imaginary quadratic fields and totally real fields. In the case of totally real fields of even degree, we get local constancy of…
An extension of Szemer\'edi's Theorem is proved for sets of positive density in approximate lattices in general locally compact and second countable abelian groups. As a consequence, we establish a recent conjecture of Klick, Strungaru and…
In the first paper of this sequence, we provided an explicit hypergeometric modularity method by combining different techniques from the classical, $p$-adic, and finite field settings. In this article, we explore an application of this…
We present a new structure called the "conservative matrix field", initially developed to elucidate and provide insight into the methodologies employed by Ap\'ery's in his proof of the irrationality of the Riemann zeta function at 3. This…
Ergodicity for local and nonlocal stochastic singular $p$-Laplace equations is proven, without restriction on the spatial dimension and for all $p\in[1,2)$. This generalizes previous results from [Gess, T\"{o}lle; J. Math. Pures Appl.,…
We describe our online database of finite extensions of the p-adic numbers, and how it can be used to facilitate local analysis of number fields.
Physical self-adjoint extensions and their spectra of the one-dimensional Dirac type Hamiltonian operator in which both the mass and velocity are constant except for a finite jump at one point of the real axis are correctly found. Different…
In \cite{Miller-Akin1999}, Miller and Akin investigated the invariant measures for correspondences, which are also known as upper semi-continuous set-valued maps. Recently, the variational principle and thermodynamic formalism for forward…
The restriction and Kakeya problems in Euclidean space have received much attention in the last few decades, and are related to many problems in harmonic analysis, PDE, and number theory. In this paper we initiate the study of these…
We examine the properties of distributions with the density of the form: $% \frac{2A_{n}c^{n-2}\sqrt{c^{2}-x^{2}}}{\pi \prod_{j=1}^{n}(c(1+a_{j}^{2})-2a_{j}x)},$ where $c,a_{1},\ldots ,a_{n}$ are some parameters and $A_{n}$ a suitable…
We show how our recent results on compositions of d.c. functions (and mappings) imply positive results on extensions of d.c. functions (and mappings). Examples answering two natural relevant questions are presented. Two further theorems,…
Using periodic points we study a notion of entropy with values in the p-adic numbers. This is done for actions of countable discrete residually finite groups $\Gamma$. For suitable $\Gamma = \mathbb{Z}^d$-actions we obtain p-adic analogues…
We study $p$-adic model of hard spheres with three states on the Cayley tree. We show that there exist three translation-invariant $p$-adic Gibbs measures and two periodic measures on a Cayley tree of oreder two.
We consider the action of Mandelbrot multiplicative cascades on probability measures supported on a symbolic space. For general probability measures, we obtain almost a sharp criterion of non-degeneracy of the limiting measure; it relies on…