We prove that certain Gibbs measures on subshifts of finite type are nonsingular and ergodic for certain countable equivalence relations, including the orbit relation of the adic transformation (the same as equality after a permutation of finitely many coordinates). The relations we consider are defined by cocycles taking values in groups, including some nonabelian ones. This generalizes (half ...

Throughout this paper, we always make use of the following notations: C denotes the set of complex numbers, Zp denotes the ring of p-adic rational integers, Qp denotes the field of p-adic rational numbers, and Cp denotes the completion of algebraic closure of Qp. Let νp be the normalized exponential valuation of Cp with |p|p = p−νp(p) = p−1. When one talks of q-extension, q is considered in man...

We consider (4+D)-dimensional Kaluza-Klein cosmological model with two scaling factors, as real, p-adic and adelic quantum mechanical one. One of the scaling factors corresponds to the D-dimensional internal space, and second one to the 4-dimensional universe. In standard quantum cosmology, i.e. over the field of real numbers R, it leads to dynamical compactification of additional dimensions an...

We formulate and prove an analogue of the noncommutative Iwasawa main conjecture for `-adic Lie extensions of a separated scheme X of finite type over a finite field of characteristic prime to `.

We associate a canonical virtual motive to definable sets over a field of characteristic zero. We use this construction to to show that very general p-adic integrals are canonically interpolated by motivic ones.

For a variety over a local field, Bloch proposed a conjectural formula for the alternating sum of Artin conductor of `adic cohomology. We prove that the formula is valid modulo 2 if the variety has even dimension.

Let p be a prime number and let K be a finite extension of Qp. Let R be the valuation ring of K, P the maximal ideal of R, and K̄ = R/P the residue field of K. Let q denote the cardinality of K̄, so K̄ ≃ Fq. For z in K, let ord z denote the valuation of z, and set |z| = q . Let f be a non constant element of K[x1, . . . , xm]. The p-adic Igusa local zeta function Z(s) associated to f (relative to ...

We present a rewiew and also new possible applications of p-adic numbers to pre-spacetime physics. It is shown that instead of the extension IR → Qp , which is usually implied in padic quantum field theory, it is possible to build a model based on the IR → Qp, where p=n+2 extension and get rid of loop divergences. It is also shown that the concept of mass naturally arises in p-adic models as in...

Given a p-adic representation of the Galois group of a local field, we show that its Galois cohomology can be computed using the associated étale (φ,Γ)-module over the Robba ring; this is a variant of a result of Herr. We then establish analogues, for not necessarily étale (φ,Γ)-modules over the Robba ring, of the Euler-Poincaré characteristic formula and Tate local duality for p-adic represent...

The l-adic parabolic cohomology groups attached to noncongruence subgroups of SL2(Z) are finite-dimensional l-adic representations of Gal(Q/K) for some number field K. We exhibit examples (with K = Q) for which the primitive parts give Galois representations whose images are open subgroups of the full group of symplectic similitudes (of arbitrary dimension). The determination of the image of th...