## STUDY GROUP ON EXPLICIT CLASS FIELD THEORY OVER QUADRATIC FIELDS

In Autumn 2021, we are running a study group on CM and RM theory, coordinated by Chris Lazda, Chris Williams and Óscar Rivero. Here you may find the schedule, some useful links, as well as some notes written by the speakers.

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Main references:

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Darmon's course on CM theory (Autumn 2020). The webpage is available here: https://www.math.mcgill.ca/darmon/courses/20-21/cm/cm.html

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In particular, you will find the set of lecture notes (by Francesc Gispert) of the course.

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Darmon's book Rational points on modular elliptic curves.

Available here: https://www.math.mcgill.ca/darmon/pub/Articles/Research/36.NSF-CBMS/chapter.pdf

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Silverman's book Advanced topics in the arithmetic of elliptic curves (chapter 2 and 5).

Auxiliary references:

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The article of Darmon and Vonk on the construction of rigid meromorphic cocycles:

https://www.math.mcgill.ca/darmon/pub/Articles/Research/69.DV1/paper.pdf

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Darmon's original article on the construction of Stark--Heegner points:

https://www.math.mcgill.ca/darmon/pub/Articles/Research/28.Integration/paper.pdf

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Complementary texts.

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The paper of Gross and Zagier on the difference of singular moduli:

https://wstein.org/papers/bib/gross-zagier-on_singular_moduli.pdf

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Chris prepared a very detailed schedule with the topics covered in each lecture, that may be found here.

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Session 0 (Oscar, October 8th). Introduction and general overview. Here you can find the slides.

Main references.

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Session 1 (Kat, October 15th). CM elliptic curves. Here you can read the detailed notes she wrote for her talk.

Main references. Darmon's course, Sections 3.1-3.3.

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Session 2 (James, October 22nd). Explicit class field theory for imaginary quadratic fields.

Main references. Silverman, Sections 2.3-2.5

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Session 3 (Arshay, October 29th). Heegner points. Here you can see the slides of Arshay's presentation.

Main references. Darmon's book, Chapter 3.

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Session 4 (David, November 5th). Singular moduli and the class number one problem.

Main references. Darmon's course, Sections 3.9-3.10 (it could be interesting to take a look to the Gross--Zagier paper linked above).

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Session 5 (Chris L., November 12th). Tate uniformisation elliptic curves.

Main references. Silverman, Chapter 5 (see also Chris Williams' master thesis).

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Session 6 (Elvira, November 19th). The Bruhat--Tits tree. Here you can read Elvira's notes.

Main references. Darmon's book Chapter 5.

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Session 7 (Muhammad, November 26th). Stark--Heegner points I.

Main references. Darmon's book, Sections 9.1-9.2.

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Session 8 (Oscar, December 3rd). Stark--Heegner points II. Here you can read Oscar's notes.

Main references. Darmon's book, Sections 9.3-9.5

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Session 9 (Chris W., December 10th). Real quadratic singular moduli

Main references. The paper of Darmon and Vonk.