Quadratic forms, orthogonal groups, and modular forms

Or in the comfort of you home if you do not insist on a live performance.

However, you are

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$\bullet$ 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

$\bullet$ 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.

Davide Accadia

Hughes Bellemare

Niccolo Bosio

Marcel Goh

Jad Hamdan

Hazem Hassan

Arihant Jain

Marti Roset Julia

Sun Kai Leung

Reginald Lybbert

Paul-Antoine Seitz

Course recording.

Organisational remarks. Overview of the course.

Course recording.

Quadratic forms. Basic definitions. General algebraic theory. Orthogonal decompositions. Witt's theorem.

The Wednesday lecture will

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Quadratic spaces over finite fields. Statement of the Hasse-Minkowski theorem. Quadratic spaces over

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Quadratic spaces and orthogonal groups over

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

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Classification of quadratic forms over $\mathbb Q_p$.

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Classification of quadratic forms over $\mathbb Q$. The Hasse-Minkwoski theorem. Sums of three squares.

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Quadratic modules over $\mathbb Z$. Minkowski's theorem. Unimodular lattices and their classificaiton.

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Classification of indefinite unimodular lattices.

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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.

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A quick introduction to modular forms. Eisenstein series and their fourier expansions. Structure of the graded ring of modular forms.

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Fourier coefficients of cusp forms. Genera of quadratic forms and adelic quotients. Automorphic forms.

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

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

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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).

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The Weil representation and the theta correspondence. Relation with the Siegel-Weil formula.

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Hecke operators. Abstract Hecke operators.

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

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Hecke operators on orthogonal groups. The notion of a p-neigbour. The Heisenberg group and its representation theory.

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Review of representations of finite groups. Uniqueness of the Heisenberg representation (over a finite field).

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Construction of the Heisenberg representation. The Weil representation.

Course Recording.

Definition and construction of the Weil representation. The metaplectic group. Explicit formula for the Weil representation.

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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.

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Hecke's indefinite theta series. Symmetric spaces attached to indefinite orthogonal groups.

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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 $(1,1)$.

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