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The universal family of semistable $$p$$-adic Galois representations. (English) Zbl 07244791
Let $$K$$ be a finite field extension of $$\mathbb{Q}_p$$ and let $$\mathscr{G}_k$$ be its absolute Galois group. We construct the universal family of filtered $$(\varphi, N)$$-modules, or (more generally) the universal family of $$(\varphi, N)$$-modules with a Hodge-Pink lattice, and study its geometric properties. Building on this, we construct the universal family of semistable $$\mathscr{G}_k$$-representations in $$p$$-algebras. All these universal families are parametrized by moduli spaces which are Artin stacks in schemes or in adic spaces locally of finite type over $$\mathbb{Q}_p$$ in the sense of Huber. This has conjectural applications to the $$p$$-adic local Langlands program.
The study of such families was begun in [M. Kisin, Prog. Math. 253, 457–496 (2006; Zbl 1184.11052); J. Am. Math. Soc. 21, No. 2, 513–546 (2008; Zbl 1205.11060); G. Pappas and M. Rapoport, Mosc. Math. J. 9, No. 3, 625–663 (2009; Zbl 1194.14032)] and in [E. Hellmann, J. Inst. Math. Jussieu 12, No. 4, 677–726 (2013; Zbl 1355.11065)], where a universal family of filtered $$\varphi$$-modules was constructed and, building on this, a universal family of crystalline representations with Hodge-Tate weights in $${0, 1}$$. The approach is based on Kisin’s integral $$p$$-adic Hodge theory cf. [M. Kisin, Prog. Math. 253, 457–496 (2006; Zbl 1184.11052)].
In this article, these results are generalized in two directions. First, the authors consider more general families of $$p$$-adic Hodge-structure, namely families of $$(\varphi, N)$$-modules together with a so called Hodge-Pink lattice. The inspiration to work with Hodge-Pink lattices instead of filtrations is taken from the analogous theory over function fields; see [R. Pink, “Hodge structures over function fields”, Preprint, http://www.math.ethz.ch/~pinkri/ftp/HS.pdf; A. Genestier and V. Lafforgue, Ann. Sci. Éc. Norm. Supér. (4) 44, No. 2, 263–360 (2011; Zbl 1277.14036); U. Hartl, Ann. Math. (2) 173, No. 3, 1241–1358 (2011; Zbl 1304.11050)]. It was already applied to Kisin’s integral $$p$$-adic Hodge theory by A. Genestier and V. Lafforgue [Compos. Math. 148, No. 3, 751–789 (2012; Zbl 1328.11112)] in the absolute case for $$\varphi$$-modules over $$\mathbb{Q}_p$$. Second, the authors generalize [E. Hellmann, J. Inst. Math. Jussieu 12, No. 4, 677–726 (2013; Zbl 1355.11065)] to the case of semistable representations.
##### MSC:
 11S20 Galois theory 11F80 Galois representations 13A35 Characteristic $$p$$ methods (Frobenius endomorphism) and reduction to characteristic $$p$$; tight closure
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