Describing the Gamma-transform explicitly in terms of power series

Question

The Gamma transform of a measure is defined as follows. If $\alpha$ is a $\mathbf{Z}_p$-valued measure on $\mathbf{Z}_p$, then the Gamma transform of $\alpha$ is: $$\Gamma_{\alpha}(s) = \int_{\mathbf{Z}_p^{\times}} \langle x \rangle^s \, d\alpha(x). $$ So the Gamma transform takes as input a measure $\alpha$, and returns an analytic function of the variable $s$, which we call $\Gamma_{\alpha}(s)$. But we can also think of the Gamma transform in a different way: as taking as input a power series and returning as output a power series. Namely:

• as input, the Gamma transform takes in the power series $F_{\alpha}(T)$ corresponding to the measure $\alpha$
• as output, the Gamma transform returns the power series $G$ such that $G((1+p)^s - 1) = \Gamma_{\alpha}(s)$.

My question is: can one explicitly describe the Gamma transform as a map from power series to power series? That is, given a power series $f(T) = \sum a_nT^n$, is there an explicit formula for the power series expansion of $\Gamma_F(T)$ in terms of the power series expansion of $F(T)$?

Here is my motivation for asking this. Washington has a very nice article, "On Sinnott's Proof of the Vanishing of the lwasawa Invariant $\mu_p$", where he gives a different proof of the Ferrero–Washington Theorem, inspired by a proof of Sinnott. On page 3, Washington does a few calculations with power series and says that "this is essentially the Gamma-transform". But I don't know what the Gamma transform looks like as a map from power series to power series, so I don't understand what that comment means. It seems to lie at the heart of the proof, so I want to ask this question to make sense of that step in the paper.

Answer 1

This is a hard problem (and one which is easily overlooked by the unwary)! Just to be clear, I'll summarize (how I think about) the problem: as a relation between additive and multiplicative Fourier transforms. We start with a measure $\mu$, i.e. a linear map

$\mu$: (continuous functions on $\mathbf{Z}_p$) $\to \mathbf{Z}_p$.

For any $t$ with $|t| < 1$, there is a unique character $\kappa_t$ of $\mathbf{Z}_p$ as an additive group which sends 1 to $1 + t$; and we have $\mu(\kappa_T) = F_\mu(T)$ for some power series $F_\mu(T) \in \mathbf{Z}_p[[T]]$: the additive Fourier transform of $\mu$.

On the other hand, for each $u$ with $|u| < 1$, there is also a character $\chi_u$ of $\mathbf{Z}_p^\times$ as a multiplicative group, which is trivial on roots of unity and sends $1 + p$ to $1 + u$. This is given by $\mu(\chi_u) = G_\mu(u)$ for some power series $G_\mu \in \mathbf{Z}_p[[U]]$: the multiplicative Fourier transform of $\mu$. (I am deliberately using a different name for the variable here!)

Your question is, then, how $F_\mu$ and $G_\mu$ are related. The answer is: "not in any straightforward way". For a given $t$, the functions $\chi_u$ and $\kappa_u$ are totally different as functions on $\mathbf{Z}_p$, so the values of $F_\mu$ and $G_\mu$ at a specific $u$ do not determine each other. On the other hand, if $\mu$ is supported in $1 + p\mathbf{Z}_p$ (so we lose nothing by restricting), then the two series $F_\mu$ and $G_\mu$ do determine each other: we get a bijection (the "Mellin transform") $$\mathfrak{M}: (1 + T) \cdot \mathbf{Z}_p[[(1+T)^p - 1]] \longleftrightarrow \mathbf{Z}_p[[U]]$$ which is a bijection of additive groups mapping $F_\mu(T)$ to $G_\mu(U)$ (but not respecting multiplication on either side). As an example, if $\mu$ is the Dirac measure sending a function to its value at $a$, for some $a \in 1 + p\mathbf{Z}_p$, then $F_\mu(T) = (1 + T)^a$, and $G_\mu(U) = (1 + U)^{log_p(a) / log_p(1 + p)}$. (This formula actually determines $\mathfrak{M}$ uniquely, since the Dirac measures are dense.)

One can pull various information about the power series through this isomorphism -- for instance, the Iwasawa $\lambda$ and $\mu$-invariants of the two series are the same, and the Newton polygons coincide beyond a certain point (Sarah Zerbes and I proved this in an old paper of ours). But there is no simple formula that would allow you to read off the coefficients of $F_\mu$ from those of $G_\mu$ or vice versa.