Serre in the 1970s was the first to formalize such a question on the way to constructing p-adic L-functions, by way of developing the notion of a p-adic modular form to be the p-adic limit of some compatible family of q-expansions of classical modular forms. Katz came along fairly soon afterwards and generalized the theory to a much more geometric context, and showed that Serre’s p-adic forms e...

coming from the cokernels of the rst column. These are cyclic (pro-cyclic in the limit) and are generated by the image of q t. So, lim A(K t)=NA(K n;t) = ?=< (q t ; L t =K t) > where (q t ; L t =K t) is the norm residue symbol of q t for the extension L t =K t. We write Q t for the image of (q t ; L t =K t) under the isomorphism ? ! Z p induced by sending to 1. Note that although Q t is not can...

R. Hofer and A. Winterhof proved that the 2-adic complexity of two-prime (binary) generator period pq with two odd primes $$p\ne q$$ is close to its it can attain maximum in many cases. When applied producing quaternary sequences, we need determine 4-adic complexity. It there are only possible values for generator, which at least $$pq-1-\max \{\log _4(pq^2),\log _4(p^2q)\}$$ . Examples p q $$5\...

§0. Introduction. In this paper, we define the module D̃(V ) of distributions with rational poles on a finite dimensional rational vector space a V . This is an infinite dimensional vector space over Q endowed with a natural action of the reductive group GV := Aut(V ). Indeed, this action extends to a natural action of the adelic group GV (AQ). For each prime p, we define we define the notion of...

1. Hensel’s lemma 2. Metric definition of p-adic integers Zp and p-adic rationals Qp 3. Elementary/clumsy definitions of adeles A and ideles J 4. Uniqueness of objects characterized by mapping properties 5. Existence of limits 6. Zp and Ẑ as limits 7. Qp and A as colimits 8. Abelian solenoids (R×Qp)/Z[1/p] and A/Q 9. Non-abelian solenoids and SL2(Q)\SL2(A) Although we will also give the more ty...

We consider a nearest-neighbor inhomogeneous p-adic Potts (with q ≥ 2 spin values) model on the Cayley tree of order k ≥ 1. The inhomogeneity means that the interaction Jxy couplings depend on nearest-neighbors points x, y of the Cayley tree. We study (p− adic) Gibbs measures of the model. We show that (i) if q / ∈ pN then there is unique Gibbs measure for any k ≥ 1 and ∀Jxy with |Jxy | < p . (...

Let p be a fixed prime. Throughout this paper Z, Zp, Qp, and Cp will, respectively, denote the ring of rational integers, the ring of p-adic rational integers, the field of p-adic rational numbers and the completion of algebraic closure of Qp, cf.[1, 2, 3]. Let vp be the normalized exponential valuation of Cp with |p| = p −vp(p) = p and let a+ pZp = {x ∈ Zp|x ≡ a( mod p N )}, where a ∈ Z lies i...