**
“****”**

**
—— David Hilbert**

I have been interested in
Algebraic Number Theory and Arithmetic Geometry since 1995, when I was a
graduate student in

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)
((

[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)** **

**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)

**Seminars: **