Overview
This research details the establishment of a ramification bound for mod $p$ Galois representations. These representations are specifically those associated with Wach modules of a bounded height, denoted as at most $i$. The context for this work is an absolutely unramified $p$-adic field, referred to as $K$.
A key aspect of this methodology involves the application of $q$-crystalline cohomology. This approach is specifically implemented in its prismatic form. The overall effect of this combination of Wach modules and $q$-crystalline cohomology is the derivation of improved bounds on ramification. These improved bounds pertain to $\mathrm{H}^{i}_{et}(X_{\mathbb{C}_K}, \mathbb{Z}/p\mathbb{Z})$ for smooth proper $p$-adic formal schemes over $\mathcal{O}_K$, applicable for arbitrarily large degrees $i$.
Research Context
The study operates within the theoretical framework of $p$-adic fields and formal schemes. The field $K$ is characterized as an absolutely unramified $p$-adic field. The representations under consideration are mod $p$ Galois representations. These representations are linked to Wach modules, a specific mathematical construct, with a height constraint of at most $i$. The ramification bound established depends exclusively on the prime $p$ and the integer $i$.
Approach
The primary approach involved two central components:
- Establishing a ramification bound for mod $p$ Galois representations associated with Wach modules. This bound is contingent solely on the prime $p$ and the integer $i$.
- Utilizing an instance of $q$-crystalline cohomology, specifically in its prismatic formulation. This application contributed to obtaining updated ramification bounds.
Findings
- A ramification bound has been established for mod $p$ Galois representations. This bound is defined by the given prime $p$ and an integer $i$. These representations are those associated with Wach modules having a height of at most $i$.
- Through the use of $q$-crystalline cohomology in its prismatic form, improved bounds have been obtained. These bounds relate to the ramification of $\mathrm{H}^{i}_{et}(X_{\mathbb{C}_K}, \mathbb{Z}/p\mathbb{Z})$. This improvement is noted for smooth proper $p$-adic formal schemes $X$ over $\mathcal{O}_K$, for degrees $i$ of arbitrary size.