# Math 726: Topics in Number Theory Modular forms and the theory of complex multiplication

Instructor: Henri Darmon

Time and venue. MW 1:00-2:30, in ENGTR (Trottier Building) 1100.
Or anywhere you want if you do not inisist on a live performance.
A private zoom link will be sent to registered participants by email.

Health and Wellness Resources at McGill.

Student well-being is a priority for the University. All of our health and wellness resources have been integrated into a single Student Wellness Hub, your one-stop shop for everything related to your physical and mental health. If you need to access services or get more information, visit the Virtual Hub or drop by the Brown Student Services Building (downtown) or Centennial Centre (Macdonald campus). Within your faculty, you can also connect with your Local Wellness Advisor (to make an appointment, visit mcgill.ca/lwa).

The Committee on Equity, Outreach, and Student Well-Being in the Mathematics Department has recently revamped its its website to make information more accessible for students.

How do I register?

$\bullet$ If you just want to audit the course and do not need a grade (notably, if you are not a student), just send me an email and I will put you on the mailing list. Of course, signing up as an auditeur libre entails no commitment in terms of attendance or participation.

$\bullet$ If you are a graduate student and want to take the class for credit, write to Raffaella Bruno (raffaella.bruno AT mcgill DOT ca) and she will walk you through a byzantine process which involves filling out forms and securing multiple authorisations.

Syllabus.

The course will touch on various topics growing out of the theory of complex multiplication, including (but not necessarily in that order):

$\bullet$ Singular moduli and their factorisations, following Gross and Zagier.

$\bullet$ p-adic variants based on the Cerednik-Drinfeld uniformisation.

$\bullet$ Extensions to real quadratic fields.

$\bullet$ The theorem of Gross-Kohnen-Zagier.

$\bullet$ The ABC conjecture and Siegel zeroes, following Granville-Stark.

$\bullet$ Traces of singular moduli, and modular forms of half integral weight.

$\bullet$ Generalisations via Borcherds' theory of singular theta lifts.

Logistics.

The course will be taught in hybrid format. Access to the lecture room (the teaching pod, as such things are now called for some reason) is limited to 15 participants, and priority for access will be given to the McGill students who are taking the course for credit.

McGill's oft proclaimed devotion to inclusiveness does not extend to those who want to audit classes. In particular, no mecanism seems available for giving CRM-ISM postdocs, or graduate student trainees, access to materials on Minerva and MyCourses. For this reason, all recordings and other course materials will be available on this public web site, and nowhere else.

Our other means of communicating with the class will be via an old-fashioned email list. If ever you want to write to the whole class, just use "reply to all". But don't do it frivolously, of course. If you are interested in being added to this list please just send me an email.

The student grade will be based on in class participation. I will reserve some of the class time (typically, 30 minutes or so in most of the lectures) for a student presentation of certain more basic or ancillary topics. Anyone who wants course credit should volunteer to give at least one such presentation during the semester. (Ideally, two, but since there will be 26 lectures and we already have 11 registered students this might be a bit tight, barring some attrition.)

Registered student Participants

Cédric Dion
Antoine Giard
Ting-Han Huang
Arihant Jain
Dhruva Rasesh Kelkar
Reginald Lybbert
Siva Sankar Nair
Marti Roset Julia
Subham Roy
Jhan-Cyuan Syu
Christian Tafula

Auditing student Participants

Francesc Gispert-Sanchez
David Lilienfeldt
Isabella Negrini
James Rickards
Ricardo Toso
Ju-Feng Wu
Peter Xu

Non-student participants (auditeurs libres)

Lea Beneish
Antonio Cauchi
Mathilde Gerbelli-Gauthier
Lennart Gehrmann
Eyal Goren
Alice Pozzi
Raul Alonso Rodriguez

Francesc's notes: Francesc Gispert-Sanchez has kindly volunteered to make his class notes available. Any mistakes or obscurities of exposition they might contain can safely be blamed on the lecturer; do let me know if you spot any.

Schedule:

Wednesday, September 9. (At McGill).
The course recording. is now available. I was mortified to see that the zoom recording displays me along with around 20 of the more photogenic participants throughout the lecture, with the result that the blackboard appears in a small portion of the screen and is quite illegible. Many apologies for this; I will figure out how to fix this for next time.
Thanks to Francesc Gispert who has kindly offered his texed notes of the first lecture to make up for the illegible material on the blackboard.
Summary of the lecture. Organisational remarks. The main statements of the theory, and a quick overview of their applications, to such questions as explicit class field theory, the ABC conjecture and class numbers of quadratic imaginary fields, $S$-unit equations, the Iwasawa main conjecture for imaginary quadratic fields, and the Birch and Swinnerton-Dyer conjecture. Most of these topics will not be explored further in the class, and this first presentation just aimed to give you a sense of the broad sweep of the subject and its wide applicability.

Monday, September 14. (At the CRM).
The recording of this lecture is now available.
A more detailed overview of the course syllabus, outlining the three main topics that will be touched on during the term:
1. Basic theory, and applications of CM theory to the study of class numbers of imaginary quadratic fields, following Heegner-Stark, Granville-Stark,Gross-Zagier, etc.
2. p-adic analytic formulation of the theory of complex multiplication via CM points on Shimura curves and the Cerednik-Drinfeld uniformisation.
3. The transposition of the description in 2. to the (far less well understood) setting of real quadratic fields, based on the notion of rigid meromorphic cocycles."

Wednesday, September 16. (At McGill).
Course recording.
The moduli of elliptic curves and the algebraic theory of modular forms.
Jhan Cyuan Syu (live from Taiwan): How Riemann Roch is used to understand the moduli of (framed) elliptic curves.

Monday, September 21. (At CRM).
Course Recording.
The Tate curve, and q-expansions of modular forms.
Elliptic curves with complex multiplication.
Marti Roset Julià. (live from Barcelona). The ring of modular forms.

Wednesday, September 23 (At McGill).
Course recording.
Elliptic curves with complex multiplication, cont'd. The commiting actions of the class group and of the Galois group on the set of isomorphism classes of elliptic curves with complex multiplication. Proof that the j-invariants of CM elliptic curves whose endomorphism ring is an imaginary quadratic order are defined over an abelian extension of the associated imaginary quadratic field.

Monday, September 28.
Class cancelled because of Yom Kippur.

Wednesday, September 30. (At McGill).
Lecture recording.
The notes from Dhruva's presentation on the class number one problem.
A discussion of CM elliptic curves over finite fields. Reduction of endomorphisms. Ordinary and supersingular reduction.
Dhruva Kelkar. (live from Essen). The class number one problem, following Heegner, Stark, Kenku, Serre, Siegel, Chen, ...

Monday, October 5. (At CRM).
Course Recording.
The Galois action on torsion points of CM elliptic curves. Integrality of the j-invariant. The class number one problem, revisited.
Cédric Dion. (from Quebec City). Elliptic curves over finite fields, their p-torsion subgroups and their endomorphism rings.

Wednesday, October 7. (At McGill).
Course Recording
CM points on modular curves. Other applications.
Antoine Giard (from ... Montreal!)
Kronecker's solution to Pell's equation.

Monday, October 12. (At CRM).
There will be no lecture that day because of the Canadian Thanksgiving.

Wednesday, October 14. (At McGill).
Course recording.
Factorisations of differences of singular moduli, d'après Gross-Zagier. Proof that the primes that divide $J(D_1,D_2)$ are non-split in both imaginary quadratic fields ${\mathbb Q}(\sqrt{D_1})$ and ${\mathbb Q}(\sqrt{D_2})$, and divide an integer of the form $(D_1D_2-x^2)/4$.
Christian Tafula (live from Sao Paulo). The work of Granville-Stark on ABC and Siegel zeroes.
Slides of the presentation.

On the week of October 19-23, there will be no course because of the workshop onArithmetic quotients of locally symmetric spaces and their cohomology which will be held at the CRM, and which you are all encouraged to attend.

For those who feel an acute need for their weekly ration of CM theory, I recommend watching last week's lecture by Philippe Michel in the number theory web seminar, which touches on many notions that were also covered in class, and develops the theme of equidistribution of CM points which we also encountered in Christian's lecture.

The week before, we discussed modular curves and the special importance of non-split Cartan modular curves in studying the class number one problem, in particular. These curves are discussed in this week's lecture by Pedro Lemos in the members' seminar; this is a good chance to catch up on it if you missed the live presentation.

After this week of break, we will switch to somewhat more advanced and less standard topics involving the p-adic uniformisation of Shimura curves and the theory of real multiplication" based on rigid meromorphic cocycles. Some of the student presentations will remain focused on deepening some of the topics covered in the first part of the course, however.

Monday, October 26. (On zoom).
Course recording.
Shimura curves, and their archimedean and non-archimedean uniformisations. CM points on Shimura curves.
Arihant Jain. Factorisation of singular moduli; statements of the precise theorem of Gross-Zagier. Calculations and examples. Slides of Arihant's presentation.

Wednesday, October 28. (On zoom).
Course recording.
Shimura curves and CM points on them, cont'd.
Isabella Negrini. The eficient evaluation of p-adic theta functions and the p-adic uniformisation of Mumford-Schottky curves. (Isabella's notes.)

Monday, November 2. (On zoom).
Course recording.
The modular cross-ratio. $p$-adic meromorphic functions on the $p$-adic upper half plane. p-adic theta functions and invariant differentials.
Instructor's course notes.

Wednesday, November 4. (On zoom).
Lecture recording.
p-adic theta functions, cont'd. Motivation for $SL_2({\mathbb Z}[1/p])$.
Siva Sankar Nair. The arithmetic of quaternion algebras over ${\mathbb Q}$ and their maximal orders.

Monday, November 9. (On zoom).
Lecture recording.
$p$-adic integration on $\Gamma\backslash {\cal H}_p$.
Ting-Han Huang. Coleman integration on curves.

Wednesday, November 11. (On zoom).
Lecture recording
$p$-adic integration on the upper half-plane, cont'd.
Sofia Giampietro. A Gross-Zagier style factorisation for singular moduli attached to Shimura curves.

Monday, November 16. (On zoom).
Lecture recording.
First example of a rigid analytic theta-cocycle: the Dedekind-Rademacher cocycle. Gross-Stark units in ring class fields of real quadratic fields.
Reginald Lybbert. Heegner points and the Gross-Zagier formula. Calculations and examples.

Wednesday, November 18. (On zoom).
Lecture recording.
Construction of the Dedekind-Rademacher cocyle. Siegel units and the Siegel distribution.
Subham Roy. The Chowla-Selberg formula.
Subham's slides.

Monday, November 23. (At the CRM, and on zoom).
Lecture recording.
End of construction of the Dedekind-Rademacher cocyle.

Wednesday, November 25. (At the CRM, and on zoom).
Lecture recording.
Other rigid analytic theta cocyles: elliptic modular cocycles; the winding cocycle; rigid meromorphic cocycles. Modular generating series.

Monday, November 30. (At the CRM, and on zoom).
Lecture recording.
Stark-Heegner points as lifting obstructions. Rigid meromorphic cocycles and theta-cocycles. Real quadratic singular moduli.

Wednesday, December 2. (At the CRM, and on zoom).
Lecture recording.
Summary of the course. p-adic analytic families of Hilbert Eisenstein series and their diagonal restrictions. Conclusion.