The universal family of semistable \(p\)-adic Galois representations.

*(English)*Zbl 07244791Let \(K\) 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.

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.

Reviewer: Mouad Moutaoukil (Fès)

##### MSC:

11S20 | Galois theory |

11F80 | Galois representations |

13A35 | Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure |