We present an algorithm to perform arithmetic operations over small extension field via numerical routines. The idea is to convert the X-adic representation of modular polynomials, with X an indeterminate, to a q-adic representation where q is a prime power larger than the field characteristic. With some control on the different involved sizes it is then possible to perform some of the q-adic a...

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...

Throughout this paper we use the following notations. By Zp we denote the ring of p-adic rational integers, Q denotes the field of rational numbers, Qp denotes the field of p-adic rational numbers, C denotes the complex number field, 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-exten...

The T -adic deformation of p-adic exponential sums is defined. It interpolates all classical p-power order exponential sums over a finite field. Its generating L-function is a T -adic entire function. We give a lower bound for its T -adic Newton polygon and show that the lower bound is often sharp. We also study the variation of this L-function in an algebraic family, in particular, the T -adic...

We initiate and examine integer powers of the (possibly unbounded) diagonal operators on the so-called p-adic Hilbert space Eω (see [10], [11], and [3]). For that, we first give and recall the required background on author’s recent work related to the formalism of unbounded linear operators in the p-adic setting [5]. Next, we shall be dealing with integer powers of the diagonal operators, their...

The Čebotarev Density Theorem, generalizing Dirichlet’s theorem on primes in arithmetic progression, gives us a notion of the density of prime ideals in a number field. We motivate our exposition of the theorem by giving a brief introduction to `-adic representations, as can be found in [Hus04]. We give a brief introduction to class field theory, drawing from [Cox89], and present Deuring’s [Deu...

where ((x)) = x − [x]G − 1 2 , if x / ∈ Z, ((x)) = 0, x ∈ Z, where [x]G is the largest integer ≤ x cf. ([1], [5], [9], [11], [12], [13]). In this paper, Zp, Qp, Cp, C and Z, respectively, denote the ring of p-adic integers, the field of p-adic rational numbers, the p-adic completion of the algebraic closure of Qp normalized by |p|p = p −1, and the complex field and integer numbers. Let q be an ...

How often is the p-adic λ-invariant of an imaginary quadratic field equal to m? We model this by the statistics of random p-adic matrices, and test these predictions numerically.

The notion of `-adic Tate modules, for primes ` away from the characteristic of the ground field, is incredibly useful. The analogous notion at the prime p is that of Dieudonné modules. At finite level, Dieudonné modules classify commutative finite group schemes of p-power order over a field of characteristic p. Dieudonné modules can be used to determine the local Brauer invariant of the endomo...

Let p be a fixed odd prime number. 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. Let vp be the normalized exponential valuation of Cp with |p|p = p −vp(p) = 1 p . When one talks of q-extension, q is variously considered as and in...