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.
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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
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.
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.
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.
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.
Marti Roset Julia
Sun Kai Leung
Wednesday, September 1.
Organisational remarks. Overview of the course.
Monday, September 6.
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.
Quadratic spaces over finite fields.
Statement of the Hasse-Minkowski theorem.
Quadratic spaces over R. Orthogonal groups.
Examples in low dimension.
Wednesday, September 15.
Quadratic spaces and orthogonal groups over R. Examples in low dimension.
Quadratic spaces over $\mathbb Q_p$. The Hilbert symbol.