Abstract A sequence is said to be k-automatic if the nth term of this sequence is generated by a finite state machine with n in base k as input. Regular sequences were first defined by Allouche and Shallit as a generalization of automatic sequences. Given a prime p and a polynomial f(x) ∈ Qp[x], we consider the sequence {vp(f(n))}n=0, where vp is the p-adic valuation. We show that this sequence...

In this paper we considered a moving from classical logic and Kolmogorov’s probability theory to non-classical p-adic valued logic and p-adic valued probability theory. Namely, we defined p-adic valued logic and further we constructed probability space for some ideals on truth values of p-adic valued logic. We proposed also p-adic valued inductive logic. Such a logic was considered for the firs...

In this article, we study p-adic torus periods for certain p-adic valued functions on Shimura curves coming from classical origin. We prove a p-adic Waldspurger formula for these periods, as a generalization of the recent work of Bertolini, Darmon, and Prasanna. In pursuing such a formula, we construct a new anti-cyclotomic p-adic L-function of Rankin– Selberg type. At a character of positive w...

The current paper establishes the sharp weak bounds of p-adic fractional Hardy operator. Furthermore, optimal type estimates for operator on central Morrey space are also acquired.

In this paper we investigate generalized Gibbs measure (GGM) for $$p$$ -adic Hard-Core (HC) model with a countable set of spin values on Cayley tree order $$k\geq 2$$ . This is defined by parameters $$\lambda_i$$ , $$i\in \mathbb N$$ We analyze functional equation which provides the consistency condition finite-dimensional distributions. Each solutions defines GGM version Kolmogorov’s theorem. ...

Let k be a locally compact non-discrete field with non-Archimedean valuation (Say, just the p-adic numbers Qp), O its ring of integers (say Zp) and P a generator for the maximal ideal of O (i.e. p). In future lectures we will see how Hecke algebras relate to the representation theory of reductive algebraic groups over these fields. The main example to keep in mind is G = GL(n, k). When I talk a...

For a polynomial f(x) in (Zp∩Q)[x] of degree d ≥ 3 let L(f⊗Fp;T ) be the L function of the exponential sum of f mod p. Let NP(f ⊗ Fp) denote the Newton polygon of L(f⊗Fp; T ). Let HP(f) denote the Hodge polygon of f , which is the lower convex hull in R2 of the points (n, n(n+1) 2d ) for 0 ≤ n ≤ d−1. We prove that there is a Zariski dense subset U defined over Q in the space A of degree-d monic...

Consider the Weil sumWF,d(u) = ∑ x∈F ψ(x +ux), where F is a finite field of characteristic p, ψ is the canonical additive character of F , d is coprime to |F |, and u ∈ F . We say that WF,d(u) is threevalued when it assumes precisely three distinct values as u runs through F : this is the minimum number of distinct values in the nondegenerate case, and three-valuedWF,d are rare and desirable. W...