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Math 596+726: Topics in Number Theory

Quadratic forms, orthogonal groups, and modular forms





Instructor: Henri Darmon

Time and venue. MW 10:00-11:30, in BH 1214.
Or in the comfort of you home if you do not insist on a live performance.
However, you are strongly encouraged to attend the live lectures if it is at all possible for you, and to avail yourself of the opportunity of interacting with the other students, and learning from them.



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, 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 getting various people to sign them.



Syllabus.

The course will touch on the classical theory of quadratic forms and their associated orthogonal groups, as well as relations with the theory of modular form Topics will include the classification of quadratic forms, the representations of integers by quadratic forms, theta functions, the Weil representation and the theta correspondence, as well as Hilbert and Siegel modular forms as forms on orthogonal groups.



Logistics.

The course will be taught in hybrid format, to accomodate the graduate students who did not yet make it into the country, and to keep on our toes and stay ready for the next lockdown. While it is still possible, you are encouraged to attend the lectures in person and engage with your fellow students.

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.



Student Grade.

I will reserve some of the class time (typically, 15 minutes or so in most of the lectures) for a student presentation of the solutions of problems which will be mentionned during lectures and assigned to participants on a rotating basis.

These questions are recorded here..

The student grade will be based on class participation, and on a final exam.

The final exam will be a three hour in class affair, but, to avoid bad surprises, questions will be selected from the master list that will have been accumulated over the term.



Registered Participants

Davide Accadia
Niccolo Bosio
Arnab Chakraborthy
Max Chemtov
Jad Hamdan
Arihant Jain
Marti Roset Julia
Sun Kai Leung



Schedule:

Wednesday, September 1.
Course recording.
Organisational remarks. Overview of the course.



Monday, September 6.
Course recording.
Quadratic forms. Basic definitions. General algebraic theory. Orthogonal decompositions. Witt's theorem.



Wednesday, September 8.
The Wednesday lecture will not be given in person because of a conflict with a Jewish holiday but is available in recorded form under the "Monday September 6" heading.



Monday, September 13.
Course recording.
Quadratic spaces over finite fields. Statement of the Hasse-Minkowski theorem. Quadratic spaces over R. Orthogonal groups. Examples in low dimension.



Wednesday, September 15.
Course recording.
Quadratic spaces and orthogonal groups over R. Examples in low dimension. Quadratic spaces over $\mathbb Q_p$. The Hilbert symbol.