We develop a p-adic version of the so-called Grothendieck-Teichmüller theory (which studies Gal(Q̄/Q) by means of its action on profinite braid groups or mapping class groups). For every place v of Q̄, we give some geometrico-combinatorial descriptions of the local Galois group Gal(Q̄v/Qv) inside Gal(Q̄/Q). We also show that Gal(Q̄p/Qp) is the automorphism group of an appropriate π1-functor in p-adi...

We consider adelic approach to strings and spatial noncommutativity. Path integral method to string amplitudes is emphasized. Uncertainties in spatial measurements in quantum gravity are related to noncommutativity between coordinates. p-Adic and adelic Moyal products are introduced. In particular, p-adic and adelic counterparts of some real noncommutative scalar solitons are constructed.

The function ez shows that a complex entire function can indeed omit one value. Lately, it has become fashionable to prove p -adic versions of value distribution theorems, of which Picard’s Theorem is an example, though not a recent one. More recent examples can be found in the works listed in the references section. Recall that the p -adic absolute value | |p on the rational number field Q is ...

Conjecture 1 (Langlands). To each reductive group G over a number field K, each automorphic (complex) representation π of G, and each finite dimensional representation r of the (complex) group G, there is defined an automorphic L-function L(s, π, r), which enjoys an analytic continuation and functional equation generalizing the Riemann zeta function πΓ(s/2)ζ(s) (or Artin’s L-function L(s, σ), w...

Suppose K/Q is a totally real extension of degree d = [K : Q]. Let Qp denote the completion of with respect to the p-adic valuation when p is a rational prime number and let Q,, denote IR when the valuation is the usual absolute value. This latter case is thought of as corresponding to the case where the prime p is 'infinite'. Suppose : K->QP is a (Q-)linear form. We say that is p-ad...

In this paper, we investigate some properties for the (h, q)-tangent numbers and polynomials. By using these properties, we obtain some interesting identities on the (h, q)-tangent polynomials and Bernstein polynomials. Throughout this paper, let p be a fixed odd prime number. The symbol, Zp, Qp and Cp denote the ring of p-adic integers, the field of p-adic rational numbers and the completion o...

In quantum-mechanical experiments, as well as in all measurements, numerical results belong to the field of rational numbers Q. In principle, the corresponding theoretical models could be made using only Q, but it would missed usual effectiveness and beauty of mathematical analysis. So, instead of Q one traditionally applies the field of real numbers R in classical mechanics and the field of co...

One purpose of this paper is to define the twisted q-Bernoulli numbers by using p-adic invariant integrals on Zp. Finally, we construct the twisted q-zeta function and q-L-series which interpolate the twisted q-Bernoulli numbers.

We prove local results on the $p$-adic density of elliptic curves over $\mathbb{Q}_p$ with different reduction types, together global densities $\mathbb{Q}$ specified types at one or more (including infinitely many) primes. These include: integral Weierstrass equations which are minimal models semistable (that is, square-free conductor) is $1/\zeta(2)\approx60.79\%$, same as integers; $\zeta(10...