The Digit Principle

Abstract

A number of constructions in function field arithmetic involve extensions from linear objects using digit expansions. This technique is described here as a method of constructing orthonormal bases in spaces of continuous functions. We illustrate several examples of orthonormal bases from this viewpoint, and we also obtain a concrete model for the continuous functions on the integers of a local field as a quotient of a Tate algebra in countably many variables. Academic Press

Functional identities for L-series values in positive characteristic

“…Now, the claim follows from (8). By Proposition 10 we have, with k as above, that the following expression…”

“…A careful investigation of the polynomials V 1,s and an application of the digit principle ( [8]) to the function ω will allow us to show that, for s ≥ 2 congruent to one modulo q − 1,…”

Normal bases of rings of continuous functions constructed with the (qn)-digit principle

“…Nevertheless, there is another version that holds in any characteristic: Proposition 1.5 (Digit principle in any characteristic [6,Theorem 3]). Let (e n ) n≥0 be a sequence of elements of C(V, V ) such that the reductions e i ∈ C(V, k) are constant on cosets modulo M i+1 and the map…”

“…Let (E, · ) be an ultrametric Banach space over K. Definition 1.1. A sequence (e n ) n≥0 of elements of E is called a normal basis of E (orthonormal basis in [6]) if (1) each x ∈ E has a representation as x = n≥0 x n e n where x n ∈ K and lim n→∞ x n = 0, (2) in the representation x = n≥0 x n e n , we have x = sup n |x n |. Definition 1.3.…”

Digit derivatives and application to zeta measures

“…Let A v be the completion of A at v, k v be the fraction field of A v , and let C(A v , k v ) be the k v -Banach space of continuous functions from A v into k v equipped with the usual sup norm. It is then well known (see [W,Go1,Co,Sn,J2]) that the space C(A v , k v ) has two sets of orthonormal bases consisting of the Carlitz polynomials and digit derivatives. Now the zeta measure is defined as a 1-parameter family of measures for all x such that |x| ∞ < 1.…”