Let p be a fixed prime number. Throughout this paper Zp,Qp,C, and Cp will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, the complex number field, and the completion of algebraic closure of Qp. Let vp be the normalized exponential valuation of Cp with |p|p p−vp p 1/p. When one talks of q-extension, q is variously considered as an indeterminate,...

a+ dpZp = {x ∈ X | x ≡ a (mod dp N )}, where a ∈ Z lies in 0 ≤ a < dp , see [1-21]. The p-adic absolute value in Cp is normalized so that |p|p = 1/p. When one talks of q-extension, q is variously considered as an indeterminate, a complex number q ∈ C or a p-adic number q ∈ Cp. If q ∈ Cp, then we assume |q − 1|p < p − 1 p−1 , so that q = exp(x log q) for |x|p ≤ 1. We use the notation [x]q = [x :...

Let p be a fixed odd prime number. Throughout this paper, Zp,Qp,C, and Cp denote the ring of p-adic rational integers, the field of p-adic rational numbers, the complex number field, and the completion of the algebraic closure of Qp, respectively. Let N be the set of natural numbers and Z N ∪ {0}. Let vp be the normalized exponential valuation of Cp with |p|p p−vp p 1/p. When one talks of q-ext...

The p-adic interpolation of modular forms on congruence subgroups of SL2(Z) has been succesfully used in the past to interpolate values of L-series. In [12], Serre interpolated the values at negative integers of the ζ-series of a totally real number field (in fact of L-series of powers of the Teichmuller character) by interpolating Eisenstein series, which are holomorphic modular forms, and loo...

In this article we prove a Grothendieck trace formula for L-functions of not necessarily commutative adic sheaves.

Let p be a fixed odd prime number. Throughout this paper, Zp, Qp, C, and Cp will, respectively, denote the ring of p-adic rational integers, the field of p-adic rationalnumbers, the complex number field, and the completion of the algebraic closure of Qp. Let vp be the normalized exponential valuation of Cp with |p|p p−vp p 1/p. When one talks about q-extension, q is variously considered as an i...

Let p be a fixed prime number. Throughout this paper, Zp, Qp, C, and Cp will, respectively, denote the ring of p-adic rational integer, the field of p-adic rational numbers, the complex number field, and the completion of algebraic closure of Qp. Let N be the set of natural numbers and Z {0} ∪ N. Let νp be the normalized exponential valuation of Cp with |p|p p−νp p p−1. When one talks of q-exte...

We consider a nearest-neighbor p-adic Potts (with q ≥ 2 spin values and coupling constant J ∈ Qp) model on the Cayley tree of order k ≥ 1. It is proved that a phase transition occurs at k = 2, q ∈ pN and p ≥ 3 (resp. q ∈ 2N, p = 2). It is established that for p-adic Potts model at k ≥ 3 a phase transition may occur only at q ∈ pN if p ≥ 3 and q ∈ 2N if p = 2.

We show that translations and dilations of a p–adic wavelet coincides (up to the multiplication by some root of one) with a vector from the known basis of discrete p–adic wavelets. In this sense the continuous p–adic wavelet transform coincides with the discrete p–adic wavelet transform. The p–adic multiresolution approximation is introduced and relation with the real multiresolution approximat...