Related papers: Dwork crystals II

A broader definition of generalized truncations of graphs is introduced followed by an exploration of some standard concepts and parameters with regard to generalized truncations.

We develop a $p$-adic theory of periods for 1-motives, extending the classical theory of complex periods into the non-archimedean setting. For 1-motives with good reduction over $p$-adic local fields, we construct a $p$-adic integration…

We prove two transformations for the $p$-adic hypergeometric functions which can be described as $p$-adic analogues of a Euler's transformation and a transformation of Clausen. We first evaluate certain character sums, and then relate them…

For an odd prime $p$ and a positive integer $n$, let ${_n}G_n[\cdots]_p$ denote McCarthy's $p$-adic hypergeometric function. In this article, we prove $p$-adic analogue of certain classical hypergeometric identities and using these…

(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…

This is an expository paper which gives a quick introduction to Dwork's conjecture about p-adic meromorphic continuation of his unit root zeta function arising from algebraic geometry. Special emphasis is given to the case of elliptic…

A geometrization of a Kronecker $h$-regular multi-time Lagrangian function with partial derivatives of order one is described, in the sense of d-connections, d-torsions and d-curvatures.

We introduce a $p$-adic $L$-function $\mathscr L_{A/L}$ associated to an ordinary elliptic curve $A$ over a global function field $K$ of characteristic $p$ together with a $\mathbb{Z}_{p}^{d}$-extension $L/K$, $d=0$ allowed, unramified…

These are notes of my lecture courses given in the summer of 2024 in the School on Number Theory and Physics at ICTP in Trieste and in the 27th Brazilian Algebra Meeting at IME-USP in S\~ao Paulo. We give an elementary account of $p$-adic…

In this article, a new class of operators, termed Ad-contractions, is introduced to extend the framework of A-contractions to the setting of dislocated metric spaces. Fixed point results are established for single mappings, sequences of…

We state and analyze a generalization of the "truncation trick" suggested by Gourdon and Sebah to improve the performance of power series evaluation by binary splitting. It follows from our analysis that the values of D-finite functions…

We define L-functions for the class of real-analytic modular forms recently introduced by F. Brown. We establish their main properties and construct the analogue of period polynomial in cases of special interest, including those of modular…

We discuss recent developments in $p$-adic geometry, ranging from foundational results such as the degeneration of the Hodge-to-de Rham spectral sequence for "compact $p$-adic manifolds" over new period maps on moduli spaces of abelian…

Geometric zeta functions of Ihara and Hashimoto are generalized to higher rank. The $p$-adic version of the Patterson conjecture is proven.

With the help of Wick rotation over $p$-adic numbers $\mathbb{Q}_p$, the $p$-adic version of Euclidean $\textrm{dS}_2$ space(noted as $p\textrm{dS}_2$) is obtained based on $p\textrm{AdS}_2$($p$-adic version of Euclidean $\textrm{AdS}_2$…

Convergence properties of random ergodic averages have been extensively studied in the literature. In these notes, we exploit a uniform estimate by Cohen \& Cuny who showed convergence of a series along randomly perturbed times for…

We provide several new $q$-congruences for truncated basic hypergeometric series, mostly of arbitrary order. Our results include congruences modulo the square or the cube of a cyclotomic polynomial, and in some instances, parametric…

We consider a system of polynomials $T_{s}(z,q)\in\mathbb{Z}[z,q]$ which appear as truncations of the K-theoretic vertex function for the cotangent bundles over Grassmannians $T^{*}Gr(k,n)$. We prove that these polynomials satisfy a natural…

There is a notion of $p$-adic period coming from the crystalline Frobenius automorphism of the de Rham cohomology of an algebraic variety. In this paper, we consider sequences of $p$-adic periods, one for each prime. We study the sequences…

We study the power mean inequality of the generalized trigonometric and hyperbolic functions with two parameters. The generalized $p$-trigonometric and $(p, q)$-trigonometric functions were introduced by P. Lindqvist and S. Takeuchi,…