Lubin-Tate representations over nontrivial finite Galois extensions of $\mathbb{Q}_{p}$ are not Aut-intrinsically Hodge-Tate

Abstract

In the present paper, we show that, for an odd prime number $p$ and a nontrivial finite Galois extension $k$ of $\mathbb{Q}_{p}$, the $p$-adic representation of the absolute Galois group of $k$ determined by a Lubin-Tate formal group over the ring of integers of $k$ is not Aut-intrinsically Hodge-Tate [in the sense of Hoshi]. This settles the odd-degree cases left open in the previous works of Hoshi and the author and, together with the known even-degree case, completes the picture for finite Galois extensions of $\mathbb{Q}_{p}$ in the case where $p$ is odd. This exhibits a sharp contrast, from the viewpoint of anabelian geometry, between the $p$-adic cyclotomic character and other $p$-adic Lubin-Tate characters.