On families of strongly divisible modules of rank 2

TL;DR

The paper develops a framework to study the mod-$p$ reductions of 2-dimensional semi-stable non-crystalline Galois representations of ${ m Gal}(ar{f Q}_p/f Q_{p^f})$ by translating the problem into constructing rank-2 strongly divisible modules. It introduces pseudo-strongly divisible modules as tractable isotypic components parameterized by $(oldsymbol{ extLambda},oldsymbol{ extTheta},oldsymbol{ extOmega},x)$ and additional delta-data, and reduces the lifting problem to explicit linear systems and inequalities (A,B) that govern when these pseudo-objects arise from genuine strongly divisible modules. The approach yields a correspondence with admissible filtered $(oldsymbol{ extphi},N)$-modules $D(oldsymbol{ extlambda},oldsymbol{rak L})$, and, after reduction, to Breuil modules that determine the mod-$p$ representations. The results recover known $f=1$ cases, provide concrete $f=2$ examples, and conjecturally produce at least one Galois-stable lattice in every case, with the two-lattice phenomenon occurring when the mod-$p$ reduction is a two-character extension; this enriches the understanding of mod-$p$ reductions and offers a concrete computational framework for further cases.

Abstract

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 of the strongly divisible modules. We expect our method to produce at least one Galois-stable lattice in each such representation for general $f$. Moreover, when the mod-$p$ reduction is an extension of distinct characters, we further expect our method to provide the two non-homothetic lattices. As applications, we show that our approach recovers previously known results for $f=1$ and determine the mod-$p$ reduction of the semi-stable representations with some small Hodge--Tate weights when $f=2$.

Figures (5)

• Figure 1: $R(\vec{r};\vec{k}')$ for $\vec{r}=(2,2)$ and $\vec{k}'\in J(\vec{r})$
• Figure 2: Partition by $S_1,\cdots,S_9$, displayed on $v_p(\vec{\xi})$-plane.
• Figure 3: $R(\vec{r};\vec{k}')$ for $\vec{r}=(1,5)$ and $\vec{k}'\in J(\vec{r})$
• Figure 4: Partition of $E^2$ by $S_1,\cdots,S_7,S_8\cup S_9\cup S_{10}, S_{11}\cup S_{12}\cup S_{13}$.
• Figure 5: Partitions of $S_8\cup S_9\cup S_{10}$ (left) and of $S_{11}\cup S_{12}\cup S_{13}$ (right)

Theorems & Definitions (142)

• Theorem 1.1.4: Theorem \ref{['theo: main 2']}
• Conjecture 1.1.5
• Example 1.1.6
• Definition 2.1.1
• Theorem 2.1.2: CF
• Example 2.1.3
• Lemma 2.1.5
• proof
• Example 2.2.1
• Definition 2.2.2