Relating two conjectures in $p$-adic Hodge theory

Abstract: Let $K$ be a finite extension of $\mathbf{Q}_p$ and let $\mathcal{G}_K = \mathrm{Gal}(\overline{\mathbf{Q}_p}/K)$. Fontaine has constructed a useful classification of $p$-adic representations of $\mathcal{G}_K$ in terms of cyclotomic $(\varphi,\Gamma)$-modules. Lately, interest has risen around a generalization of the theory of $(\varphi,\Gamma)$-modules, replacing the cyclotomic extension with an arbitrary infinitely ramified $p$-adic Lie extension. Computations from Berger suggest that locally analytic vectors should provide such a generalization for any arbitrary infinitely ramified $p$-adic Lie extension, and this has been conjectured by Kedlaya. In this paper, we focus on the case of $\mathbf{Z}_p$-extensions, using recent work of Berger-Rozensztajn and Porat on an integral version of locally analytic vectors and explain what is the structure of the locally analytic vectors in the higher rings of periods $\widetilde{\mathbf{A}}^{\dagger}$ in this setting. We then use this result to construct, in the anticyclotomic setting and assuming that Kedlaya's conjecture holds, an element in the field of fractions of the Robba ring which ``should not exist'' according to a conjecture of Berger. As a consequence, we prove that this conjecture of Berger on substitution maps on the Robba ring is incompatible with Kedlaya's conjecture. Should Berger's conjecture hold, this would provide an example of an extension for which there is no overconvergent lift of its field of norms and for which there exist nontrivial higher locally analytic vectors.