$p$-Adic Fourier Theory of Differentiable Functions

Nagel, Enno

Documenta mathematica, Tome 23 (2018), pp. 939-967.

Résumé

Let $\bold K$ be a finite extension of $\Bbb{Q}_p$ of degree $d$ and $\Cal{O}_{\bold{K}}$ its ring of integers; let $\Bbb{C}_p$ be the completed algebraic closure of $\Bbb{Q}_p$. The Fourier polynomials $P_n:\Cal{O}_{\bold{K}}\to\Bbb{C}_p$ show that the topological algebra of all locally analytic distributions $\mu:\Cal{C}^{\mathrm{la}}(\Cal{O}_{\bold{K}},\Bbb{C}_p)\to\Bbb{C}_p$ is, by $\mu\mapsto\sum\mu(P_n) X^n$, isomorphic to that of all power series in $\Bbb{C}_p[[X]]$ that converge on the open unit disc of $\Bbb{C}_p$. Given a real number $r\geq d$, we determine the power series that correspond under this isomorphism to all distributions $\mu:\Cal{C}^r(\Cal{O}_{\bold{K}},\Bbb{C}_p)\to\Bbb{C}_p$ that extend to all $r$-times differentiable functions (as arisen in the $p$-adic Langlands program): A function $f:\Cal{O}_{\bold{K}}\to\Cal{C}_p$ is $r$-times differentiable if and only if $f(x)=\Sigma a_nP_n(x)$ with $|a_n|n^{r/d}\to 0$ as $n\to\infty$.

Détail

Classification : 11S80, 11S31, 14G22, 12J25, 32P05, 46S10
Keywords: Fourier transform, Mahler basis, Amice transform, Lubin-Tate formal group, Taylor polynomials