Chol Park

Let F / Q be a CM field where p splits completely and let r ¯ : Gal ( Q ¯ / F ) → GL 3 ( F ¯ p ) be a Galois representation whose restriction to Gal ( Q ¯ p / F w ) is ordinary and strongly generic for all places w above p. In this paper, we specify the set of Serre weights in which r ¯ can be modular. To this aim, we develop a technique in integral p-adic Hodge theory to describe extensions of rank-one Breuil modules.

Let p be a prime number and r a positive even integer less than p???1. In this paper, we find a Galois stable lattice in each two-dimensional semistable noncrystalline representation of GQp with Hodge???Tate weights (0,r) by constructing the corresponding strongly divisible module. We also compute the Breuil modules corresponding to the mod p reductions of these strongly divisible modules, and determine the semisimplification of the mod p reduction of the original representations. We use these r

Let F / Q be a CM field where p splits completely and r ¯ : Gal ( Q ¯ / F ) → GL 3 ( F ¯ p ) a continuous modular Galois representation. Assume that r ¯ is non-ordinary and non-split reducible (niveau 2) at a place w above p. We show that the isomorphism class of r ¯ | Gal ( F ¯ w / F w ) is determined by the GL 3 ( F w ) -action on the space of mod p algebraic automorphic forms using the refined Hecke action of Herzig, Le and Morra [Compos. Math. 153 (2017) 2215–2286]. We also give a nearly opt

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 3"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p&gt;3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a prime number and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Subscrip

Let [Formula: see text] be a prime number and [Formula: see text] a positive even integer less than [Formula: see text]. In this paper, we find the strongly divisible modules corresponding to the Galois stable lattices in each 2-dimensional semi-stable non-crystalline representation of [Formula: see text] with Hodge–Tate weights [Formula: see text] whose mod-[Formula: see text] reductions are corresponding to nontrivial extensions of two distinct characters. We use these results to construct the

Let $F/F^+$ be a CM field and let $\widetilde{v}$ be a finite unramified place of $F$ above the prime $p$. Let $\overline{r}: \mathrm{Gal}(\overline{\mathbb{Q}}/F)\rightarrow \mathrm{GL}_n(\overline{\mathbb{F}}_p)$ be a continuous representation which we assume to be modular for a unitary group over $F^+$ which is compact at all real places. We prove, under Taylor--Wiles hypotheses, that the smooth $\mathrm{GL}_n(F_{\widetilde{v}})$-action on the corresponding Hecke isotypical part of the mod-$p

Let $p$ be a prime number, $n&gt;2$ an integer, and $F$ a CM field in which $p$ splits completely. Assume that a continuous automorphic Galois representation $\overline{r}:\mathrm{Gal}(\overline{\mathbf{Q}}/F)\rightarrow\mathrm{GL}_n(\overline{\mathbf{F}}_p)$ is upper-triangular and satisfies certain genericity conditions at a place $w$ above $p$, and that every subquotient of $\overline{r}|_{\mathrm{Gal}(\overline{\mathbf{Q}}_p/F_w)}$ of dimension $&gt;2$ is Fontaine--Laffaille generic. In this

Let $p&gt;3$ be a prime number and let $G_{\mathbb{Q}_p}$ be the absolute Galois group of $\mathbb{Q}_p$. In this paper, we find Galois stable lattices in the irreducible $3$-dimensional semi-stable and non-crystalline representations of $G_{\mathbb{Q}_p}$ with Hodge--Tate weights $(0,1,2)$ by constructing their strongly divisible modules. We also compute the Breuil modules corresponding to the mod $p$ reductions of the strongly divisible modules, and determine which of the semi-stable represent

In this dissertation, we study semi-stable representations of G(Q(p)) and their mod p-reductions, which is a part of the problem in which we construct deformation spaces whose characteristic 0 closed points are the semi-stable lifts with Hodge-Tate weights (0, 1, 2) of a fixed absolutely irreducible residual representation ρ : G(Q(p)) → GL₃(F(p)). We first classify the isomorphism classes of semi-stable representations of G(Q(p)) with regular Hodge-Tate weights, by classifying admissible filtere

Let K be a finite extension of Qp. It is believed that one can attach a smooth Fp-representation of GLn(K) (or a packet of such representations) to a continuous Galois representation of Gal(K/Qp) with coefficients in GLn(Fp) in a natural way, that is called mod p Langlands program for GLn(K). This is known only for GL2(Qp): one of the main difficulties is that there is no classification of such smooth representations of GLn(K) unless K = Qp and n = 2. However, for a given continuous Galois repre

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K slash double-struck upper Q Subscript p"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">K/\mathbb {Q}_p</mml:annotation

We classify 3-dimensional semi-stable representations of the Galois group of Q_p with coefficients and regular Hodge--Tate weights, by determining the isomorphism classes of admissible filtered (phi,N)-modules of Hodge type (0,r,s) with 0 &lt; r &lt; s.

Let $p$ be an odd prime, and $\mathbf{Q}_{p^f}$ the unramified extension of $\mathbf{Q}_p$ of degree $f$. In this paper, we reduce the problem of constructing strongly divisible modules for $2$-dimensional semi-stable non-crystalline representations of $\mathrm{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_{p^f})$ with Hodge--Tate weights in the Fontaine--Laffaille range to solving systems of linear equations and inequalities. We also determine the Breuil modules corresponding to the mod-$p$ reduction