Welcome to Chao's Math 8674 Topics in number theory!

Time and place: MW 4:10pm-5:25pm in Room 507. First meeting on Sep. 10.

Our topic is Arithmetic of L-functions. More specifically, we will discuss the conjecture of Birch and Swinnerton-Dyer, which predicts deep connections between the L-function of an elliptic curve and its arithmetic, and the vast conjectural generalizations for motives due to Beilinson, Bloch and Kato. In the first half, we will provide necessary background and explain a proof of the BSD conjecture in the rank 0 or 1 case. We will emphasize new tools which generalize to higher dimensional motives. In the second half, we will study recent results on the Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives.

Prerequisite: first graduate courses in algebraic number theory, modular forms, and algebraic geometry.

Some tentative topics include:

  • L-functions of elliptic curves and modular forms
  • BSD and BBK conjectures
  • The Waldspurger formula and the Gross-Zagier formula
  • Geometry of modular and Shimura curves
  • Level raising congruences for modular forms
  • Rapoport-Zink weight spectral sequences
  • Bertolini-Darmon explicit reciprocity laws
  • Kolyvagin's method
  • Rankin-Selberg L-functions and Rankin-Selberg motives
  • Gan-Gross-Prasad conjectures and Zhang's formula
  • Geometry of unitary Shimura varieties
  • Tate's conjecture via the geometric Satake correspondence
  • Explicit reciprocity laws for unitary Shimura varieties


Lecture Topic Reference
09/10/2018 Overview and the statement of main theorems
09/12/2018 L-functions of elliptic curves and modular forms [S1] V, [S2] I.11, [DS] 5.9
09/17/2018 Rank part of BSD conjecture, computing the leading term, periods [BSD], [Cre] 2.13, [S1] III.5
09/19/2018 Rationality of L-values, heights, full BSD conjecture [Cre] 2.1, [S1] VIII.9, [Gro] 2.4, [Coa] 5
09/24/2018 Tate-Shafarevich groups, Tamagawa number conjecture [S1] X.4, X.6, [Blo]
09/26/2018 Bloch's reformulation of BSD, Selmer groups [Blo], [Zha13], [S1] X.4, [BK]
10/08/2018 Bloch-Kato reformulation, Tate's local duality [BK], [Tat], [Mil], [PR]
10/10/2018 Kolyvagin's method (simplest case), Heegner points [Gro2], [Zha13]
10/15/2018 the Gross-Zagier formula and consequences [GZ], [Zha13]
10/17/2018 the Yuan-Zhang-Zhang formula, the Waldspurger formula [Zha13], [Gro3]
10/22/2018 Weight 3/2 modular forms, Level raising congruences [Gro3], [Rib]
10/24/2018 Ihara's lemma, geometry of modular curves [DT], [LHL] 5.5, [DS] 8
10/29/2018 Geometry of Shimura curves [BD] 5
10/31/2018 Weight spectral sequences [Ill], [Sai], [Liu]
11/07/2018 Examples of weight spectral sequences [Liu]
11/12/2018 First explicit reciprocity law, proof of BSD/BK in rank 0 [BD] 8, 3, 4
11/14/2018 Second explicit reciprocity law, proof of BSD/BK in rank 1 [BD] 9, 3, 4
11/19/2018 Rankin-Selberg L-functions [JPSS], [Cog]
11/26/2018 Gan-Gross-Prasad conjectures, the Ichino-Ikeda-Zhang formula [GGP], [Zha14], [Zha17]
11/28/2018 Bloch-Kato Selmer groups, Bloch-Kato conjecture [Bel]
12/03/2018 BBK for Rankin-Selberg motives, unitary Shimura varieties, GGP cycles [CH], [Shi], [Bei], [Nek]
12/05/2018 Geometry of unitary Shimura varieties, Tate conjecture for the special fiber, explicit reciprocity laws for GGP cycles [VW], [XZ]
12/10/2018 Tate conjecture via geometric Satake [XZ], [Zhu]


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  • [LTXZZ] Y. Liu, Y. Tian, L. Xiao, W. Zhang, X. Zhu, On the Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives, in preparation.
  • A plan for MCM summer school organized by Ye Tian, L. Xiao and myself.
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  • [LHL] Le Hung, Bao V.; Li, Chao Level raising mod 2 and arbitrary 2-Selmer ranks. Compos. Math. 152 (2016), no. 8, 1576-1608.
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  • [Liu] Yifeng Liu, Bounding cubic-triple product Selmer groups of elliptic curves, Journal of the European Mathematical Society, to appear
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  • [Zha14] Zhang, Wei Automorphic period and the central value of Rankin-Selberg L-function. J. Amer. Math. Soc. 27 (2014), no. 2, 541-612.
  • [Zha17] Zhang, Wei Periods, cycles, and L-functions: a relative trace formula approach, arXiv:1712.08844
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  • [CH] Chenevier, Gaetan; Harris, Michael Construction of automorphic Galois representations, II. Camb. J. Math. 1 (2013), no. 1, 573.
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  • [Z] Xinwen Zhu, Geometric Satake, categorical traces, and arithmetic of Shimura varieties, arXiv:1810.07375