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