What question did this study set out to answer?

This research aims to construct Wach modules of rank three and to compute the mod-p reductions of the associated crystalline representations.

May 24, 2026Open Access

Construction of Some Wach Modules of Rank Three

Key Points

This research aims to construct Wach modules of rank three and to compute the mod-p reductions of the associated crystalline representations.
Utilized approximation methods inspired by Yamashita and Yasuda's previous work on rank two Wach modules.
Conducted calculations over OE[[π]] to establish the conditions necessary for the construction of rank three modules.
Investigated the p-adic and π-adic approximations to simplify the construction process.
Successfully constructed several Wach modules of rank three over OE[[π]].
Calculated mod-p reductions that correspond to the crystalline representations of Gal(Qp/Qp) over E.

Abstract

Fix a prime p≥3 and let E be an algebraic extension of Qp. We construct some Wach modules over OE[π] of rank three and calculate mod-p reductions of the corresponding crystalline representations of Gal(Qp/Qp) over E. The main idea in the construction of some Wach modules is to approximate p-adically and π-adically the conditions of Wach modules by simpler conditions. Yamashita and Yasuda (in preparation) have constructed some Wach modules of rank two by employing some approximation methods. We apply their method to construct some Wach modules of rank three.

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