Related papers: Frobenius map and the $p$-adic Gamma function

Let $\mathbb{F}_{q}$ denote the finite field of order $q$ (a power of a prime $p$). We study the $p$-adic valuations for zeros of $L$-functions associated with exponential sums of the following family of Laurent polynomials…

(Dieudonn\'e and) Dwork's lemma gives a necessary and sufficient condition for an exponential of a formal power series $S(z)$ with coefficients in $Q_p$ to have coefficients in $Z_p$. We establish theorems on the $p$-adic valuation of the…

We study power series and analyticity in the quaternionic setting. We first consider a function f defined as the sum of a quaternionic power series centered at 0 in its domain of convergence (which is a ball B(0,R) centered at 0). At each…

Let $f$ be a newform of even weight $2\kappa$ for $D^\times$, where $D$ is a possibly split indefinite quaternion algebra over $\mathbb{Q}$. Let $K$ be a quadratic imaginary field splitting $D$ and $p$ an odd prime split in $K$. We extend…

We prove a version of Tao's algebraic regularity lemma for asymptotic classes in the context of graphons. We apply it to study expander difference polynomials over fields with powers of Frobenius.

We generalize our work on Carlitz prime power torsion extension to torsion extensions of Drinfeld modules of arbitrary rank. As in the Carlitz case, we give a description of these extensions in terms of evaluations of Anderson generating…

In a recent work, Maciej Do\l{}e\k{}ga and the author have given a formula of the expansion of the Jack polynomial $J^{(\alpha)}_\lambda$ in the power-sum basis as a non-orientability generating series of bipartite maps whose edges are…

A detailed analysis of the remainder obtained by truncating the Euler series up to the $n$th-order term is presented. In particular, by using an approach recently proposed by Weniger, asymptotic expansions of the remainder, both in inverse…

We exploit an ansatz in order to construct power series expansions for pairs of conjugate functions defined on domains of Euclidean $3$--space. Convergence properties of the resulting series are investigated. Entire solutions which are not…

The goal of this article is to construct explicitly a p-adic family of representations (which are dihedral representations), to construct their associated (phi,Gamma)-modules by writing down explicit matrices for phi and for the action of…

In the research, with aid of the Fa\`a di Bruno formula, be virtue of several identities for the Bell polynomials of the second kind, with help of two combinatorial identities, by means of the (logarithmically) complete monotonicity of…

Let $\Omega$ be a finite symmetric subset of GL$_n(\mathbb{Z}[1/q_0])$, and $\Gamma:=\langle \Omega \rangle$. Then the family of Cayley graphs $\{{\rm Cay}(\pi_m(\Gamma),\pi_m(\Omega))\}_m$ is a family of expanders as $m$ ranges over fixed…

The generating function which counts partitions with the Plancherel measure (and its q-deformed version), can be rewritten as a matrix integral, which allows to compute its asymptotic expansion to all orders. There are applications in…

Given the complement of a hyperplane arrangement, let $\Gamma$ be the closure of the graph of the map inverting each of its defining linear forms. The characteristic polynomial manifests itself in the Hilbert series of $\Gamma$ in two…

Two linear recurrences exhibit mirror symmetry connecting the constants $e$ and $\pi$. When parametrized, their asymptotic connection constants extend to meromorphic functions satisfying additive functional equations with rational…

A graphical expansion formula for non-commutative matrix integrals with values in a finite-dimensional real or complex von Neumann algebra is obtained in terms of ribbon graphs and their non-orientable counterpart called Moebius graphs. The…

Let $f$ be a modular form which is non-ordinary at $p$. Kim and Loeffler have recently constructed two-variable $p$-adic $L$-functions associated to $f$. In the case where $a_p=0$, they showed that, as in the one-variable case, Pollack's…

We find an inverse factorial series expansion for the ratio of products of gamma functions whose arguments are linear functions of the variable. We a give recurrence relation for the coefficients in terms of the N{\o}rlund-Bernoulli…

We establish asymptotic formulas for sums of reciprocals of primes in arithmetic progressions, generalizing recent results on multiple Mertens evaluations by Tenenbaum, Qi, and Hu. Specifically, for any fixed constant $K>0$, we derive…

Starting from the Weierstrass elliptic function, we study the associated Frobenius structure, incorporating the perspective of derived categories, particularly that of homological mirror symmetry. Given a deformation of the Weierstrass…