Toward the $p$-adic Hodge parameters in the potentially crystalline representations of ${\mathrm{GL}}_n$

Abstract

Let $p$ be a prime number, $n\ge 2$, and let $L$ be a finite extension of ${\mathbf{Q}}_p$. Let ${\rho }_L$ be an $n$-dimensional non-critical generic potentially crystalline $p$-adic representation of the absolute Galois group of $L$ with regular Hodge--Tate weights. Building on Ding's results and strategy in the crystabelline case, and on the recent work of Breuil--Ding in the critical crystalline case, we construct an explicit locally analytic representation ${\pi }_1({\rho }_L)$ and describe explicitly the Hodge-filtration data of ${\rho }_L$ that it determines. When ${\rho }_L$ arises from a patched $p$-adic automorphic representation, we show, under mild hypotheses, that ${\pi }_1({\rho }_L)$ is a subrepresentation of the ${\mathrm{GL}}_n(L)$-representation globally associated with ${\rho }_L$ by using the framework of Bernstein eigenvarieties that were developed by Breuil-Ding.

arXiv comment39 pages