This is a learning seminar with the main goal being to understand the key ideas of the paper Heegner points and derivatives of L-series by Gross and Zagier (reference [GZ] below). It will also serve as an introduction to some foundations regarding modular curves, local heights, arithmetic surfaces, L-series of modular forms, etc. If we get through the paper, we will continue on with the generalizations and extensions that have arisen since then.
Meeting time: Wednesdays, 11:00am ET (Zoom)
Here are my notes for the first several weeks of the seminar. Notes provided by other speakers may be found below.| Date | Topic | Speaker | Notes | References |
|---|---|---|---|---|
| June 15 | Modular Curves I | Vijay Srinivasan | ** | [GZ §I, II.1], [DR §I-III] |
| June 22 | Modular Curves II | Mikayel Mkrtchyan | [KM §13], [GZ §III.1], [Con2] | |
| June 29 | Heegner points and L-series | Vijay Srinivasan | ** | [Gro1 §I], [Gro2 §I], [DS], [Sil] |
| July 6 | Local heights and arithmetic surfaces | Niven Achenjang | PDF |
[Lan1 §3], [Gro1], [Lan2 §11], [Sil §IV] |
| July 13 | Nonarchimedean local heights on X0(N) I | Vijay Srinivasan | ** | [GZ §III.2-6], [Con1] |
| July 20 | No meeting | – | – | – |
| July 27 | Nonarchimedean local heights on X0(N) II | Vijay Srinivasan | ** | [GZ §III.7-9], [Con1] |
| Aug 3 | Archimedean local heights on X0(N) | Ryan Chen | PDF |
[GZ §II] |
| Aug 9* | Rankin–Selberg method | Mikayel Mkrtchyan | [GZ §IV.1-3] | |
| Aug 17 | No meeting | – | – | – |
| Aug 24 | End of proof and applications | Vijay Srinivasan | [GZ §V] | |
| – | p-adic Gross–Zagier formulae | – |
[Con1] B. Conrad, Gross–Zagier revisited
[Con2] B. Conrad, Arithmetic moduli of generalized elliptic curves
[DR] P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques
[DS] F. Diamond and J. Shurman, A First Course in Modular Forms
[Gro1] B. Gross, Local heights on curves
[Gro2] B. Gross, Heegner points on X0(N)
[GZ] B. Gross and D. Zagier, Heegner points and derivatives of L-series
[KM] N. Katz and B. Mazur, Arithmetic moduli of elliptic curves
[Lan1] S. Lang, Introduction to Arakelov Theory
[Lan2] S. Lang, Fundamentals of Diophantine Geometry
[Ser] J.P. Serre, Lectures on the Mordell–Weil Theorem
[Sil] J. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves