Chao Li's homepage

Info

• I am an Associate Professor at Columbia University. I graduated from Harvard under the supervision of Professor Benedict Gross in 2015.
• My research interests: Number Theory, Arithmetic Geometry, Automorphic Forms.
• Address:
  • Math Building 614, MC4430
  • Department of Mathematics, Columbia University
  • 2990 Broadway
  • New York, NY 10027
• Email:
• I am on sabbatical leave in 2024.

Research

• Arithmetic Fundamental Lemma for the spherical Hecke algebra (with Michael Rapoport, Wei Zhang), preprint
• Geometric and arithmetic theta correspondences
  Lectures at the IHES 2022 Summer School on the Langlands Program
  Proc. Symp. Pure. Math, to appear
• Degrees of unitary Deligne-Lusztig varieties
  Contribution to the Kudla 70 volume
  Fields Inst. Commun., to appear
• A proof of Kudla-Rapoport conjecture for Kramer models (with Qiao He, Yousheng Shi, Tonghai Yang)
  Invent. Math., 234 (2023), no. 2, 721-817. doi:10.1007/s00222-023-01209-1
• A note on Tate's conjectures for abelian varieties (with Wei Zhang)
  Essent. Number Theory, 1 (2022), no. 1, 41-50. doi:10.2140/ent.2022.1.41
• From sum of two squares to arithmetic Siegel-Weil formulas (survey)
  Bull. Amer. Math. Soc., 60 (2023), no. 3, 327-370. doi:10.1090/bull/1786
• On the arithmetic Siegel-Weil formula for GSpin Shimura varieties (with Wei Zhang)
  Invent. Math., 228 (2022), no. 3, 1353-1460. doi:10.1007/s00222-022-01106-z
• Chow groups and L-derivatives of automorphic motives for unitary groups, II (with Yifeng Liu)
  Forum Math. Pi, 10 (2022), e5, 77pp. doi:10.1017/fmp.2022.2
• Chow groups and L-derivatives of automorphic motives for unitary groups (with Yifeng Liu)
  Ann. of Math., 194 (2021), no. 3, 817-901. doi:10.4007/annals.2021.194.3.6
• Kudla-Rapoport cycles and derivatives of local densities (with Wei Zhang)
  J. Amer. Math. Soc., 35 (2022), no. 3, 705-797. doi:/10.1090/jams/988
• Proof of (Fourier-Jacobi) arithmetic fundamental lemma for the minuscule case (with Yihang Zhu)
  An appendix to Yifeng Liu, Fourier-Jacobi cycles and arithmetic relative trace formula
  Camb. J. Math., 9 (2021), No. 1, 90-95. doi:/10.4310/CJM.2021.v9.n1.a1
• Fine Deligne-Lusztig varieties and arithmetic fundamental lemmas (with Xuhua He and Yihang Zhu)
  Forum Math. Sigma, 7 (2019), e47, 55pp. doi:/10.1017/fms.2019.45
• On the 2-part of the BSD conjecture for quadratic twists of elliptic curves (with Li Cai and Shuai Zhai)
  J. Lond. Math. Soc., 101 (2020), No. 2, 714-734. doi:10.1112/jlms.12284
• Recent developments on quadratic twists of elliptic curves (survey)
  Proceedings of the ICCM 2017, First Annual Meeting, 381-400. ISBN 9781571463920
• Prime twists of elliptic curves (with Daniel Kriz)
  Math. Res. Lett., 26 (2019), No. 4, 1187-1195. doi:10.4310/MRL.2019.v26.n4.a10
• Goldfeld's conjecture and congruences between Heegner points (with Daniel Kriz)
  Forum Math. Sigma, 7 (2019), e15, 80pp. doi:10.1017/fms.2019.9
• Arithmetic intersection on GSpin Rapoport-Zink spaces (with Yihang Zhu)
  Compos. Math., 154 (2018), No. 7, 1407-1440, doi:10.1112/S0010437X18007108
• Remarks on the arithmetic fundamental lemma (with Yihang Zhu)
  Algebra Number Theory, 11 (2017), No. 10, 2425-2445, doi:10.2140/ant.2017.11.2425
• 2-Selmer groups, 2-class groups and rational points on elliptic curves
  Trans. Amer. Math. Soc., 371 (2019), 4631-4653, doi:10.1090/tran/7373
• Level raising mod 2 and obstruction to rank lowering
  Int. Math. Res. Not. IMRN (2019), no. 8, 2332-2355. doi:10.1093/imrn/rnx188
• Level raising mod 2 and arbitrary 2-Selmer ranks (With Bao V. Le Hung)
  Compos. Math. 152 (2016), no. 8, 1576-1608, doi:10.1112/S0010437X16007454

Teaching

• I taught Math 4044 (Representations of finite groups), Fall 2022.
• I taught a minicourse at Elliptic Curves 2022 Summer School, Baskerville Hall.
  • Lecture 1 BSD Conjecture, Modularity Theorem, Gross-Zagier-Kolyvagin Theorem
  • Lecture 2 Gross-Zagier formula for X0(N), applications to the BSD conjecture and the Gauss class number problem
  • Lecture 3 Gross-Zagier formula for Shimura curves, Waldspurger's formula, a proof outline of the Gross-Zagier formula
• I taught Math 6657 (Class field theory), Spring 2022
• I taught Math 4044 (Representations of finite groups), Fall 2021
• I taught Math 6657 (Class field theory), Spring 2021.
• I taught Math 4044 (Representations of finite groups), Fall 2020
• I taught Math 6657 (Class field theory), Spring 2020.
• I taught Math 1101 (Calculus I), Fall 2019.
• I taught Math 6657 (Class field theory), Spring 2019.
• I taught Math 8674 (Arithmetic of L-functions), Fall 2018. Course Notes by Pak-Hin Lee.
• I taught Math 1101 (Calculus I), Fall 2017 and Spring 2018.
• I taught Math 6657 (Class field theory), Spring 2017.
• I taught Math 1101 (Calculus I) and Math 4043 (Algebraic number theory), Fall 2016.
• I taught Math 1101 (Calculus I), Fall 2015 and Spring 2016. (Columbia Departmental Teaching Award)
• I CAed Math 123 (Theory of rings and fields) and Math 223b (Class field theory 2), Spring 2015. (Harvard Certificate of Distinction in Teaching)
• I taught Math 21a (Multivariable calculus), Fall 2013 and Fall 2014. (Harvard Certificate of Distinction in Teaching)
• I CAed Math 223b (Class field theory 2), Spring 2013. (Harvard Certificate of Distinction in Teaching)
• I CAed Math 223a (Class field theory), Fall 2012. (Harvard Certificate of Distinction in Teaching)
• I taught Knots and Primes, Summer 2012.
• I taught Math 21a (Multivariable calculus), Fall 2011.

Expository Notes

• By clicking the orange links below, you solemnly swear to send me any mistakes you find.

• Moonshine and the BSD conjecture (a Student Mathematics Colloquium talk)
• Basic functions and the arc space of L-monoids
• Quadratic polynomials and modular forms (a UMS talk)
• Higher Gross-Zagier formulas: Cohomological spectral decomposition and finish the proof
• Kato's explicit reciprocity laws
• The Hodge-Tate period map and the cohomology of Shimura varieties
• Vector bundles on the Fargues-Fontaine curve

• Lusztig's classification of irreducible representations of reductive groups over finite fields
• K2 and L-functions of elliptic curves (a Trivial Notions talk)
• Mixed Tate motives, algebraic K-theory and multiple zeta values
• Applications of Deligne's Weil II
• Kolyvagin's conjecture
• Almost mathematics
• Deligne-Lusztig curves (a Trivial Notions talk)
• What is the Tate conjecture?
• Basic Dieudonne theory
• Eisenstein descent and rational points on modular curves
• Mumford curves (a Trivial Notions talk)
• What is the Birch and Swinnerton-Dyer conjecture?
• Elliptic Surfaces and Mordell-Weil Lattices
• Heights and Finiteness theorems of abelian varieties
• Enriques classification of complex algebraic surfaces
• Stable vector bundles on curves
• Shimura Curves (a Trivial Notions talk)
• Endomorphism rings of elliptic curves and singular moduli — my minor thesis:
  • Chapter I: Endomorphism rings of elliptic curves
  • Chapter II: Supersingular elliptic curves
  • Chapter III: Complex multiplication and singular moduli

Course Notes

• Relative trace formulae and the Gan-Gross-Prasad conjectures
• Geometric aspects of p-adic Hodge theory
• Bloch-Kato conjectures for some polarized motives
• Langlands correspondence for general reductive groups over function fields
• Perverse sheaves and fundamental lemmas

• Intersection theory in algebraic geometry
• Galois Representations
• Pure Motives and Rigid Local Systems
• Topics in Automorphic Forms
• Class field theory: proofs
• Class field theory
• Abelian varieties
• Representation theory and number theory
• Perverse sheaves in representation theory
• Discrete subgroups of Lie groups and discrete transformation groups
• Arithmetic surfaces and successive minima
• Teichmuller Theory