Wir müssen wissen, Wir werden wissen

                                     —— David Hilbert

I have been interested in Algebraic Number Theory and Arithmetic Geometry since 1995, when I was a graduate student in Tsinghua University, China.  Right now I have been focusing on  p-adic Galois representations from algebraic geometry, more specifically, integral p-adic Hodge theory.  


Roughly speaking, integral p-adic Hodge theory is the study of Galois stable Z_p lattices in semi-stable p-adic representations together with their links with the various integral p-adic cohomologies of proper smooth scheme over base fields F. It gives rise to p-torsion phenomena which makes the theory very complicated. For the introduction of integral p-adic Hodge theory, please see Breuil's survey paper " Integral p-adic Hodge Theory".     




Preprints and Publications: 

[23] Potentially GL_2-type Galois representations associated to noncongruence modular forms, (with Wen-Ching Winnie Li, Ling Long), preprint, (arXiv)

[22] Breuil-Kisin modules via crystalline cohomology, (with Bryden Cais), preprint, appear at Transactions of AMS, (PDF)

[21] Loose crystalline lifts and overconvergence of etale (φ, τ)-modules, (with Hui Gao), preprint, (arXiv)

[20] Potentially crystalline lifts of certain prescribed types, (with Florian Herzig, Toby Gee and David Savitt), Documenta Mathematica, Vol. 22 (2017), 397-422 (arXiv)

[19] On Automorphy of certain Galois representations of GO_4 -type, (with Jiu-Kang Yu, with Liang Xiao's appendix "Tensor being crystalline implies each factor being crystalline up to twist"), Jounal of Number Theory (special issue for Prof. Winnie Li) 161 (2016) 49--74 (PDF)

[18] On F-crystalline representation, (with Bryden Cais), Documenta Mathematica, 21 (2016) 223--270 (arXiv)

[17] The weight part of Serre's conjecture for GL(2), (with Toby Gee and David Savitt), Forum of Mathematics, π 3 (2015), e2 (52 pages) (PDF)

[16] Compatibility of Kisin modules for different uniformizers, preprint, appear at Journal für die reine und angewandte Mathematik (pdf)

[15] A Note on Potential Diagonalizability of Crystalline Representations, (with Hui Gao), Mathematische Annalen, (2014) 360:481--487 (pdf)

[14] The Buzzard-Diamond-Jarvis conjecture for unitary groups, (with Toby Gee and David Savitt), J. Amer. Math. Soc. 27 (2014), no. 2, 389--435. (arXiv)

[13] Crystalline extensions and the weight part of Serre's conjecture, (with Toby Gee and David Savitt), Algebra & Number Theory, Vol. 6 (2012), No. 7, 1537--1559 (arXiv)

[12] Filtration associated to torsion semi-stable representations, Annales de l'Institut Fourier, 65 no. 5 (2015), p. 1999-2035 (pdf)

[11] The correspondence between Barsotti-Tate groups and Kisin modules when p=2 , Journal de Théroie des Nombres de Bordeaux, 25 (2013), no. 3, 661--676.(pdf)

[10] Galois Representations with Quaternion Multiplications Associated to Noncongruence Modular Forms , (with A.O.L. Atkin, Wen-Ching Winnie Li, Ling Long), Transactions of of AMS, 365 (2013), no 12, 6217--6242. (pdf)

[9] Lattices in filtered (φ, N)-modules, J. Inst. Math. Jussieu 2, Volume 11, Issue 03, 2012, pp 659-693 (pdf) ((Errata in the end of [16]) )

[8] Some Bounds for ramification of p^n-torsion semi-stable representations , (with Xavier Caruso), Journal of Algebra, 325, 2011, Issue 1, 70-96 (arXiv)

[7] A note on lattices in semi-stable representations, Mathematische Annalen, 346, (2010), No. 1, 117--138. (PDF)

[6] Quasi-semi-stable representations, (with Xavier Caruso), Bull. Soc. math. de France, 137 (2), 2009, p. 185-223 (PDF)

[5] Torsion p-adic Galois representation and a conjecture of Fontaine, Ann. Scient. de l'E.N.S., Volume 40, Issue 4, July-August 2007, Pages 633-674. (PDF) (Errata in the end of [7])

[4] On lattices in semi-stable representations: a proof of a conjecture of Breuil , Compositio Mathematica, 144, 2008, No. 1, 61--88, (PDF)

[3] Potentially Good Reduction of Barsotti-Tate Groups. Jounal of number theory, 126 (2007), no. 2, 155-184. (PDF)  

[2] Steinitz class of Mordell-Weil groups of elliptic curves with complex  multiplication.  (with Xiake Zhang), Pacific J. Math.193 (2000), no. 2, 371--379.(DVI)

[1] Congruences for the class numbers of real cyclic sextic number fields.    Sci. China Ser. A 42 (1999), no. 10, 1009--1018.

Drafts (papers that not 100% finished, use very carefully): 

Notes (Please be aware of typoes and mistakes, use very carefully): 


[1] CM Seminar Notes,  Classification of Abelian Varieties with CM over C and Reduction of Abelian Varieties. (PDF
  The link to 2004-05 VIGRE Number Theory Working Group
[2] Mini course in Morningside center on an introduction to p-adic Hodge theory:
Note1, Note2


[3] Modularity of compatible family of p-adic Galois representations PDF

Slices for presentations: 

[1] F-crystalline representation and Kisin module (talk in Chicago AMS meeting 2015).

[2]Introduction to Modern Number Theory and Arithmetic Geometry (talk in Qcean University of China 2017)


Seminar on Integral p-adic Hodge Theory

Seminar on Perfectoid Space

Seminar on Rigid Analytic Space and Berkovich Space

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