# When does a p-adic function have a Mahler expansion?

Let $f: \mathbb{Z}_p \rightarrow \mathbb{C}_p$ be any continuous function. Then Mahler showed there are coefficients $a_n \in \mathbb{C}_p$ with

$$f(x) = \sum^{\infty}_{n=0} a_n {x \choose n}.$$

This is known as the Mahler expansion of $f$. Here, to make sense of $x \choose n$ for $x \not\in \mathbb{Z}$ define

$${x\choose n} = \frac{x(x-1)\ldots(x-n+1)}{n!}.$$

There's an important application to number theory: expressing a function in terms of its Mahler expansion is one step in translating the old fashioned interpolation-based language of $p$-adic $L$-functions into the more modern measure-theoretic language. This application is explained in Coates and Sujatha's book Cyclotomic Fields and Zeta Values.

However, when the $p$-adic $L$-function is more complicated than Kubota-Leopoldt's, it seems to me that this "translation" really requires one to be able to write down a Mahler expansion of a function $f$ with much larger domain, e.g. the ring of integers of the completion of the maximal unramified extension of $\mathbb{Q}_p$ or some finitely ramified extension. (See, for instance, line (8), p. 19 of de Shalit's Iwasawa Theory of Elliptic Curves with Complex Multiplication).

I can't find a published reference that these functions really have Mahler expansions. Mahler's paper uses in an essential way that the positive integers are dense in $\mathbb{Z}_p$, so it doesn't instantly generalize.

So is it true or false that for a ring of integers $\mathcal{O}$ in a finitely ramified complete extension of $\mathbb{Q}_p$, and a function $f: \mathcal{O} \rightarrow \mathbb{C}_p$, there is a Mahler expansion as above?

• Just a remark that p-adic L-functions are analytic, which is much stronger than just continuous, so power series expansions for them are in a sense much easier to come by, even if the source is much more general. – Kevin Buzzard Feb 13 '10 at 21:55

It is false for the valuation ring in any nontrivial finite extension of $\mathbb{Q}_p$. The coefficients of the Mahler expansion of a continuous function $\mathcal{O} \to \mathbb{C}_p$ are determined by its restriction to $\mathbb{Z}_p$ (they are given as $n$-th differences of the sequence of values on nonnegative integers, in fact). But there are different continuous functions $\mathcal{O} \to \mathbb{C}_p$ with the same restriction to $\mathbb{Z}_p$.

Even worse, the Mahler expansions need not even converge because if $x$ is not in $\mathbb{Z}_p$, the binomial coefficient values may have negative valuation.

EDIT: As Kevin Buzzard and dke suggest, one can give a positive answer if your question is interpreted differently. The point of this edit is to make a few explicit remarks in these two directions.

1) If it is known in advance that $f \colon \mathcal{O} \to \mathbb{C}_p$ is represented by a single convergent power series, then the Mahler expansion of $f|_{\mathbb{Z}_p}$ converges to $f$ on all of $\mathcal{O}$. This can be deduced from the theorem that a continuous function $\mathbb{Z}_p \to \mathbb{C}_p$ is analytic if and only if the Mahler expansion coefficients $a_n$ satisfy $a_n/n! \to 0$ (see Theorem 54.4 in Ultrametric calculus: an introduction to $p$-adic analysis by W. H. Schikhof).

2) If one chooses a $\mathbb{Z}_p$-basis of $\mathcal{O}$, then $f$ can be interpreted as a continuous function $\mathbb{Z}_p^r \to \mathbb{C}_p$, and any such function has a multivariable Mahler expansion $$\sum a_n \binom{x_1}{n_1} \cdots \binom{x_r}{n_r},$$ where the sum is over tuples $n=(n_1,\ldots,n_r)$ with $n_i \in \mathbb{Z}_{\ge 0}$, and $a_n \to 0$ $p$-adically.

Ehud de Shalit has a preprint called "Mahler's theorem for local fields" which does what you want.

As Bjorn says in his answer, the set of binomial coefficient functions just isn't sufficient in general. However, plenty has been written about analogues of Mahler expansions, i.e. finding nice bases for various spaces of continuous functions, going back to Amice in the 1960's for finite extensions of ${\mathbb Q}_p$, as well as positive characteristic versions. This is all very nicely explained in Keith Conrad's The Digit Principle, J. Number Theory 84 (2000), no. 2, 230--257. arXiv version

S. Evrard has recently extended some of these results to cases with infinite residue field Normal bases of rings of continuous functions constructed with the $(q_n)$-digit principle. Acta Arith. 135 (2008), no. 3, 219--230.

Edited to add: probably of more relevance to your question though would be the theory of Mahler-type expansions developed in p-Adic Fourier Theory by Schneider and Teitelbaum.

You can look up the nice paper by Manjul Bhargava and Kiran Kedlaya titled "Continuous functions on compact sets of local fields". The results in this paper are weaker than the papers proposed by other people but nonetheless the paper is really easy and a pleasure to read.

Let $R$ be a complete discrete valuation ring with finite residue field, and let $K$ be its fraction field. Let $C(R,K)$ be the (ultrametric Banach) space of continuous functions $f: R \rightarrow K$. This 1999 paper of K. Tateyama gives an analogue of Mahler's Theorem in $C(R,K)$.

Namely: let $\operatorname{Int}(R,R)$ be the ring of "integer-valued polynomials," i.e. the set of $P \in K[t]$ such that $P(R) \subseteq R$. Let $\{\psi_n(t)\}_{n=0}^{\infty}$ be a regular basis of $\operatorname{Int}(R,R)$: that is, it is a basis for $\operatorname{Int}(R,R)$ as an $R$-module and $\deg \psi_n = n$ for all $n \in \mathbb{Z}^+$.

That regular bases always exist in this context is well known (e.g. a very special case of work of Bhargava). In fact Tateyama constructs an explicit regular basis using algebraic combinations of the "Fermat quotient polynomials" $\frac{t^q-t}{\pi}$, where $q$ is the residue cardinality and $\pi$ is a uniformizer. Then he shows:

(i) For any sequence $\{a_n\}$ in $K$ [or any complete extension thereof] with $a_n \rightarrow 0$, the series $x \mapsto \sum_{n=0}^{\infty} a_n \psi_n(x)$ is uniformly convergent and thus defines an element of $C(R,K)$.

(ii) [Thm. 3.3] For any $f \in C(R,K)$, there is a unique sequence $\{a_n\}_{n=0}^{\infty}$ in $K$ such that $a_n \rightarrow 0$ and $\sum_{n=0}^{\infty} a_n \psi_n(x)$ converges uniformly to $f$.

At the end of his paper, Tateyama mentions a connection with Lubin-Tate formal groups and explains how this generalizes an earlier construction of Carlitz-Wagner in the case $R = \mathbb{F}_q[[t]]$. It seems to me at the moment that Tateyama's paper contains much of the more elementary material of de Shalit's recent paper (which leans more heavily on Lubin-Tate formal groups) referred to in Laurent Berger's answer. (Please correct me if I'm wrong; I am interested in this material of late but do not claim to have fully absorbed it.) de Shalit then goes on to discuss deeper material related to the Schneider-Teitelbaum $p$-adic Fourier theory. It is a very nice paper.

See also Chapter 3 of the 1997 book Integer-Valued Polynomials of Cahen and Chabert for a more systematic treatment of these sort of expansions from the perspective of Stone-Weierstrass theory.

The answer to another interpretation of the question is: any continuous function from the p-adic integers to a Banach space over the p-adic numbers has a Mahler expansion, where the coefficients of the Newton polynomials are elements of the Banach space, computed by Newton's interpolation sum.

Carl