# 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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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).

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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 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.

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

Hughes Bellemare
Niccolo Bosio
Marcel Goh
Hazem Hassan
Arihant Jain
Marti Roset Julia
Sun Kai Leung
Reginald Lybbert
Paul-Antoine Seitz

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.

Monday, September 20.
Course recording.
Quadratic spaces and orthogonal groups over $\mathbb Q_p$. The Hilbert symbol, and the Hasse-Witt invariant.

Wednesday, September 22.
Course recording.
Classification of quadratic forms over $\mathbb Q_p$.

Monday, September 27.
Course recording.
Classification of quadratic forms over $\mathbb Q$. The Hasse-Minkwoski theorem. Sums of three squares.

Wednesday, September 29.
Course recording.
Quadratic modules over $\mathbb Z$. Minkowski's theorem. Unimodular lattices and their classificaiton.

Monday, October 4.
Course recording.
Classification of indefinite unimodular lattices.

Wednesday, October 6.
Course recording.
Definite unimodular lattices. Theta series of lattices. Poisson summattion formula. Modularity of theta series.

Note that the week of October 11-13 is a study break", meaning that there are no classes on those days. On the other hand, the normally sacro-sanct (for number theorists) Thursday has been turned into a Monday, so we will be meeting on that day at the usual time and usual place.

Having proved that the theta series of (even, unimodular) lattices are modular forms on ${\mathbb SL}_2(\mathbb Z)$, one then feels compelled to say something about modular forms, which is what I plan to do on Thursday. However, I will try to cover that material very quickly. Those who have never seen modular forms before are advised to go through the relevant chapter in Serre's book, in order to come with some prior exposure. There is also an excellent set of on-line lectures given by Richard Borcherds, which covers the material in Serre in a very elegant and efficient way. It is made up of 13 segments of 15 to 25 minutes each, and the novices among you are urged to take in at least the first half of this very nice mini-course before next Thursday.

Thursday, October 14.
Course recording.
A quick introduction to modular forms. Eisenstein series and their fourier expansions. Structure of the graded ring of modular forms.

Monday, October 18.
Course recording..
Fourier coefficients of cusp forms. Genera of quadratic forms and adelic quotients. Automorphic forms.

Wednesday, October 20.
Course recording.
Adelic quotients and automorphic forms on $G$ for various $G$. (Additive group, multiplicative group, orthogonal group, ${\bf SL}_2$).

Monday, October 25.
Course recording.
Fourier analysis on the adèles and on the adèlic points of a quadratic space. Fourier series, fourier inversion, and the Poisson summation formula.

Wednesday, October 27.
Course recording.
Fourier analysis on the adeles and on the adeles modulo $\mathbb Q$. Adelic poisson summation formula. Proof that all unimodular lattices of given parity and rank lie in the same genus (Marti).

Monday, November 1.
Course recording
The Weil representation and the theta correspondence. Relation with the Siegel-Weil formula.

Wednesday, November 3.
Course recording
Hecke operators. Abstract Hecke operators.

Monday, November 8.
Lecture recording. Hecke operators, continued. Gelfand's criterion. Hecke algebras for the orthogonal group.

Wednesday, November 10.
Course recording.
Hecke operators on orthogonal groups. The notion of a p-neigbour. The Heisenberg group and its representation theory.

Monday, November 15.
Course recording.
Review of representations of finite groups. Uniqueness of the Heisenberg representation (over a finite field).

Wednesday, November 17.
Course recording
Construction of the Heisenberg representation. The Weil representation.

Monday, November 22.
Course Recording.
Definition and construction of the Weil representation. The metaplectic group. Explicit formula for the Weil representation.

Wednesday, November 24.
Course Recording.
Quick review of the theta correspondence for orthogonal groups. Hecke's theta series for indefinite quadratic spaces of rank two. Automorphic forms on indefinite orthogonal groups. Adelic quotients and archimedean symmetric spaces.

Monday, November 29.
Course recording.
Remark. The recording for this lecture leaves much to be desired. The owl was not working that day, and I forgot to start the recording until 20 minutes into the class.
Hecke's indefinite theta series. Symmetric spaces attached to indefinite orthogonal groups.

Wednesday, December 1.
Course recording.
Classical and Hilbert modular forms, as forms on $O(2,1)$ and $O(2,2)$. Proof of modularity of Hecke's indefinite theta series of quadratic spaces of signature $(1,1)$.

Monday, December 6.
Course recording.
Proof of modularity of Hecke's indefinite theta series of quadratic spaces of signature $(1,1)$, cont'd. Quick summary of the course.