Sen Operators and Lie Algebras arising from Galois Representations over $p$-adic Varieties
Tongmu He
TL;DR
The paper constructs a canonical family of Sen operators for representations of the fundamental group of a $p$-adic variety with a semi-stable chart, defining a Lie algebra action via the canonical Faltings extension and the Hyodo (Hyodo-ring) structure arising from the $p$-adic Simpson correspondence of Tsuji. This canonical action yields Sen operators that recover Brinon–Ohkubo in valuation-field cases and relate to inertia subgroups at height-1 primes, while extending to infinite-dimensional representations through completion and descent techniques. A key achievement is showing that geometric Sen operators annihilate locally analytic vectors, generalizing Pan, and providing a robust relative framework (via quasi-adequate algebras, Kummer towers, and adequate charts) for descent, decompletion, and analytic continuation of Sen operators. The work thereby integrates $p$-adic analytic group theory, logarithmic geometry, and $p$-adic Hodge-theoretic tools to advance a relative Sen theory with practical implications for understanding Galois actions on $p$-adic varieties. Its machinery offers a pathway to systematic control of infinitesimal Galois actions in families and across non-perfect residue fields, with implications for the study of $p$-adic representations in arithmetic geometry.
Abstract
Any finite-dimensional $p$-adic representation of the absolute Galois group of a $p$-adic local field with imperfect residue field is characterized by its arithmetic and geometric Sen operators defined by Sen and Brinon. We generalize their construction to the fundamental group of a $p$-adic affine variety with a semi-stable chart, and prove that the module of Sen operators is canonically defined, independently of the choice of the chart. Our construction relies on a descent theorem in the $p$-adic Simpson correspondence developed by Tsuji. When the representation comes from a $\mathbb{Q}_p$-representation of a $p$-adic analytic group quotient of the fundamental group, we describe its Lie algebra action in terms of the Sen operators, which is a generalization of a result of Sen and Ohkubo. These Sen operators can be extended continuously to certain infinite-dimensional representations. As an application, we prove that the geometric Sen operators annihilate locally analytic vectors, generalizing a result of Pan.