Wenhan Dai

Papers and Notes

Papers

1. A geometric Jacquet-Langlands correspondence for unitary Shimura varieties mod ramified p, undergraduate thesis (supervised by Liang Xiao), Peking University, 2023.

Expository Writings and Notes

(Disclaimer: The following scattered materials are written for studying goals only, and the originality is thus hardly claimed. These notes possibly obtain many major or minor gaps, typos, and logical mistakes due to the note-taker, as well as a lack of references.)

Geometry of Shimura varieties:

• The Hodge-Tate period map on perfectoid Shimura varieties (lectures by Yihang Zhu).
  Live-texed notes for a lecture series at YMSC, based on Scholze’s torsion paper.
• Integral models of Shimura varieties of Hodge type (lectures by Xu Shen).
  Following Mark Kisin’s work on smooth integral models after classification of Breuil-Kisin modules of p-divisible groups.
• Integral models of Shimura varieties of Abelian type (lectures by Mark Kisin).
  Scribed from Kisin’s lectures at Clay Mathematical Institute; including description of special fibres.

Relative Langlands duality:

• Hyperspherical Hamiltonian varieties (with Xiangqian Yang).
  Notes for two talks on relative Langlands duality.

Modular forms and modularity:

• Chow groups and L-derivatives of automorphic motives for unitary groups.
  Based on Chao Li’s talk at Columbia in February, 2021; we survey the background of the joint work by Chao Li and Yifeng Liu on Beilinson–Bloch conjecture for unitary Shimura varieties.
• Modularity lifting (lectures by Patrick Allen).
  The crucial ingredient in understanding Andrew Wiles’ proof of Fermat’s last theorem; expanded from Allen’s McGill lectures.
• Hida theory on p-adic modular forms (lectures by Bin Zhao).
  About Haruzo Hida’s construction of analytic families of ordinary p-adic modular forms and their associated Galois representations.

Representation theory and group theory:

• On the mod p Jacquet-Langlands correspondence for GL(2) (lectures by Yongquan Hu).
  Including basic preliminaries and recent works, such as topics on Serre weight, Gelfand-Kirillov dimension, and Scholze’s functor.
• On representations of Lie algebras (lectures by Jinpeng An).
  Notes for a basic course in routine.

General algebraic geometry:

• On Lusztig’s middle extension of perverse sheaves.
  A survey adapted from a seminar talk about Springer theory.
• Abelian varieties.
  Following David Mumford’s book closely.

Others:

• Complex analysis.
  Studying notes for the PhD Qualification.
• Notes on Ratner’s measure classification theorem.
  Survey context on homogeneous dynamics and unipotent flows; including Oppenheim conjecture but lack of precision.