We develop several algorithms for computations in Galois extensions of p-adic fields. Our algorithms are based on existing algorithms for number fields and are exact in the sense that we do not need to consider approximations to p-adic numbers. As an application we describe an algorithmic approach to prove or disprove various conjectures for local and global epsilon constants.

Let p be a fixed prime. 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, cf.[1, 4, 6, 10]. Let vp be the normalized exponential valuation of Cp with |p|p = p −vp(p) = p. When one talks of q-extension, q is variously considered as an...

We study the properties of Eisenstein-Kronecker numbers, which are related to special values of Hecke L-function of imaginary quadratic fields. We prove that the generating function of these numbers is a reduced (normalized or canonical in some literature) theta function associated to the Poincaré bundle of an elliptic curve. We introduce general methods to study the algebraic and p-adic proper...

We consider the following problem in the paper of Kim et al. 2010 : “Find Witt’s formula for Carlitz’s type q-Euler numbers.” We give Witt’s formula for Carlitz’s type q-Euler numbers, which is an answer to the above problem. Moreover, we obtain a new p-adic q-l-function lp,q s, χ for Dirichlet’s character χ, with the property that lp,q −n, χ En,χn,q − χn p p nqEn,χn,qp for n 0, 1, . . . using ...

This paper presents some basic theorems giving the structure of cyclic codes of length n over the ring of integers modulo pa and over the p-adic numbers, where p is a prime not dividing n. An especially interesting example is the 2-adic cyclic code of length 7 with generator polynomial X 3 + ,~X 2 + (L I)X -l, where )~ satisfies ~2 _ k + 2 = 0. This is the 2-adic generalization of both the bina...

We extend the map Exp for elliptic curves in short Weierstrass form over $ \mathbb{C} to Edwards local fields. Subsequently, we compute field \mathbb{Q}_{p} of p $-adic numbers.

Let p be a prime number with p ≡ 1(mod 2). Throughout this paper, Zp,Qp and Cp will 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 norm | · |p is normalized as |p|p = 1 p . Let q be an indeterminate in Cp such that |1− q|p < p −1 p−1 . The q-extension of number x be defined as [x]q = 1−qx 1−q . Note that limq→1[...

Motivated by an application to LDPC (low density parity check) algebraic geometry codes described by Voloch and Zarzar, we describe a computational procedure for establishing an upper bound on the arithmetic or geometric Picard number of a smooth projective surface over a finite field, by computing the Frobenius action on p-adic cohomology to a small degree of p-adic accuracy. We have implement...

Throughout this paper Z,Zp,Qp and Cp will be denoted by the ring of rational integers, the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of Qp, respectively, cf. [7, 8, 9, 10]. Let νp be the normalized exponential valuation of Cp with |p|p = p −νp(p) = p. When one talks of qextension, q is variously considered as an indeterminate, a comple...

Adelic quantum mechanics is form invariant under an interchange of real and p-adic number fields as well as rings of p-adic integers. We also show that in adelic quantum mechanics Feynman’s path integrals for quadratic actions with rational coefficients are invariant under changes of their entries within nonzero rational numbers.