(click on each title to show descriptions of the research)
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Geometry of Shimura varieties
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| I primarily focus on understanding the geometry of the special fibers of the Shimura varieties, in particular, global descriptions of the Newton strata. The first result along this line was in a joint work with Yichao Tian, in which we gave a global description of the Goren-Oort strata of the Hilbert modular variety. Using this, we gave a cohomology proof of the generalization of Coleman's classicality of overconvergent modular forms to the Hilbert case. More interestingly, this leads to studying the Tate conjecture of the special fiber of the Hilbert modular varieties, or more general Shimura varietes. This now consists of several on-going projects with David Helm, Yichao Tian, and Xinwen Zhu. |
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p-adic automorphic forms
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| The theme here is to study the p-adic variation of automorphic forms, either in rigid analytic family, or considered as torsion forms. My research uses more geometry of Shimura varieties than many other approaches. |
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Slopes of Newton polygons
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| Instead of working with general questions on slopes of Newton polygons, my research interest comes from one interesting computation of Buzzard and Kilford (addressing a question of Coleman and Mazur): the eigencurve for level 2 and p=2 over the boundary of the weight space is an infinite disjoint union of connected components each finite and flat over the weight space, and the slopes on each component approaches to 0 as the point moves towards the boundary. We generalized this to arbitary prime number p and arbitrary level for the definite quaternion algebra over Q. This is proved in a recent joint work with Daqing Wan. This work is built on an idea of Coleman explained in his private notes and some ideas of the two previous papers in joint works with Chris Davis, Daqing Wan, and Jun Zhang. It would be interesting to extend this result to general eigenvarieties. |
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p-adic Hodge theory and (φ, Γ)-modules
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| This series of research concerns arithmetic applications of the theory of (φ, Γ)-modules. To start, in a joint paper with Jay Pottharst and Kiran Kedlaya, we proved the finiteness of cohomology of an arithmetic family of (φ, Γ)-modules. This has immediate applications to generalizing many results in the ordinary case to that of the finite-slope case. For example, in a joint on-going project with Jay Pottharst, we use this to prove parity conjecture for Hilbert modular forms under mild hypothesis. |
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Ramification theory
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| There are basically two approaches to the ramification theory for a complete discrete valuation field with imperfect residue field: l-adic and p-adic. My method falls into the second category; it makes use the theory of p-adic differential equations. The most successful achievement is to prove the Hasse-Arf theorem for the Swan conductors defined by Abbes and Saito. I can also prove certain generalization of Grothendieck-Ogg-Shavarevich formula to the higher dimensional case, in the set up of vector bundles with flat connections. |