These notes (somewhat revised from the version presented at the 2007 AWS) present a few facets of the relationship between p-adic analysis, algebraic de Rham cohomology, and zeta functions of algebraic varieties. A key theme is the explicit, computable nature of these constructions, which makes them suitable for numerical calculations. For instance, if you ask the computer algebra system Magma ...

Let p be a fixed odd prime. Throughout this paper Zp,Qp,C, andCp, 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 N be the set of natural numbers and Z N ∪ {0}. Let νp be the normalized exponential valuation of Cp with |p|p p−νp p p−1 see 1–14 . When one speaks of ...

— We study the geometry of germs of definable (semialgebraic or subanalytic) sets over a p-adic field from the metric, differential and measure geometric point of view. We prove that the local density of such sets at each of their points does exist. We then introduce the notion of distinguished tangent cone with respect to some open subgroup with finite index in the multiplicative group of our ...

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...

Quantum arithmetics provides a possible resolution of a long-lasting challenge of finding a mathematical justification for the canonical identification mapping p-adics to reals playing a key role in TGD in particular in p-adic mass calculations. p-Adic numbers have p-adic pinary expansions ∑ anp n satisfying an < p. of powers p n to be products of primes p1 < p satisfying an < p for ordinary p-...

We present an algorithm to determine if the L-series associated to an automorphic representation and the one associated to an elliptic curve over an imaginary quadratic field agree. By the work of Harris-Soudry-Taylor, Taylor, and Berger-Harcos, we can associate to an automorphic representation a family of compatible -adic representations. Our algorithm is based on Faltings-Serre’s method to pr...

In this paper, we investigate some properties for the q-tangent numbers and polynomials. By using these properties, we give some interesting identities on the 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 of algebraic ...

Let p be a fixed odd prime. Throughout this paper, Zp,Qp,Cp will, respectively, denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of algebraic closure of Qp. The p-adic absolute value | |p on Cp is normalized so that |p|p 1/p. Let Z>0 be the set of natural numbers and Z≥0 Z>0 ∪ {0}. As is well known, the Bernoulli polynomials Bn x are defined by the ge...

The purpose of this paper is to derive formulae for the sums of products of the q-Euler polynomials and numbers, since many identities can be obtained from our sums of products of the q-Euler polynomials and numbers. In 1 , Simsek evaluated the complete sums for the Euler numbers and polynomials and obtained some identities related to Euler numbers and polynomials from his complete sums, and Ja...

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 ...