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BIMSA-YMSC Tsinghua Number Theory Seminar
BIMSA-YMSC Tsinghua Number Theory Seminar
On $\varphi$-modules without $\varphi$-stable lattices
On $\varphi$-modules without $\varphi$-stable lattices
Organizers
Hansheng Diao
, Heng Du
, Yueke Hu
, Bin Xu
, Yihang Zhu
Speaker
Zhongyipan Lin
Time
Monday, April 13, 2026 10:00 AM - 11:00 AM
Venue
Shuangqing-B654
Abstract
In (integral) $p$-adic Hodge theory, an important tool is various lattices (for example, Wach lattices and Breuil-Kisin lattices) inside étale $\varphi$-modules. The moduli of such lattices is crucial for Kisin to construct and analyze potentially semistable deformation rings, and for Emerton and Gee to construct the moduli of $(\varphi, \Gamma)$-modules.
However, when a given Galois representation has wildly ramified Weil-Deligne type, or when the reductive group in consideration is wildly ramified, the relevant $\varphi$-modules will not possess lattices that are $\varphi$-stable. As a consequence, very little is known about the structure of wildly potentially semistable deformation rings.
In this talk, I will explain how the study of certain $\varphi$-unstable lattices of étale $(\varphi, \Gamma)$-modules allow us to construct the Emerton-Gee stacks for wildly ramified reductive groups $G$ -- indeed, for any embedding ${}^L G \to GL(V)$, the corresponding morphism of Emerton-Gee stacks is relatively representable by algebraic spaces of finite presentation.
If time permits, I will say a few words about the expectation of the correct notion of “Breuil-Kisin lattices” for a wildly potentially semistable Galois representation (part of a joint work in progress with Y. Min and S. Morra), when its Weil-Deligne type is cyclically ramified. Such lattices are different but closely related to the lattices introduced earlier in the talk.