Lecture notes

During my studies I have written several mathematical notes with $\LaTeX$ for my own use. Even so, sometimes others have asked me to share them. Below I make them available for whoever might find them interesting. Please let me know if you find any mistakes.

Courses

In autumn 2019, after years contemplating the option without daring to take the step, I tried to take class notes directly with $\LaTeX$ as the professor wrote on the blackboard and with little edition afterwards. The experiment was quite a success and the following year I did it again, even with the added difficulty of lectures being online.

Next I list the courses for which I have typed notes. Unfortunately, the notes of the courses that I followed previously are all handwritten. On the other hand, Adrian Iovita explicitly requested that the notes of his courses not end up in any website. If you are interested in them for personal use, send me an email promising that you will share them nowhere and with no one else; then I will send them to you.

Modularity lifting

Professor(s)

Patrick Allen

Term

Autumn 2020

Notes available

Here

Comments

Introduction to the Taylor–Wiles method, mainly for $\mathrm{GL}_2(\mathbb{Q})$. The first half was deformation theory. The focus of the second half was patching. Some results for $\mathrm{GL}_2(F)$ where $F$ is a totally real or a CM field were discussed at the end.

Complex multiplication

Professor(s)

Henri Darmon

Term

Autumn 2020

Notes available

Here

Comments

The first third was the classical theory of complex multiplication. The second third was about generalizations for Shimura curves using p–adic methods. The last part was an introduction to the real multiplication theory, a work in progress of Henri and collaborators.

Galois representations

Professor(s)

Adrian Iovita and Giovanni Rosso

Term

Autumn 2020

Notes available

In private

Comments

Course about p–adic Galois representations. The first quarter was about global representations. The second quarter was about local representations away from p. The last half was an introduction to p–adic Hodge theory. I dropped the course midway.

Sheaf cohomology

Professor(s)

Adrian Iovita

Term

Autumn 2019

Notes available

In private

Comments

The first half was the standard theory of sheaf cohomology. The second half was an introduction to Clausen–Scholze’s recent theory of condensed mathematics. The original plan was too ambitious and it ended up being an extended version of Scholze’s first few lectures on the topic with more details on the background. I omitted some review that happened in class.

Seminars

I have also typed notes of most of the talks I have given in seminars. These are summaries of different topics in number theory and usually focus on the main ideas rather than the technical details. All documents but one are in English.

Algebraic de Rham cohomology and the Gauss–Manin connection

Definition of algebraic de Rham cohomology, computations using Čech cohomology and spectral sequences and definition of the Gauss–Manin connection.

Notes of a talk in a seminar on Lawrence–Venkatesh’s new proof of the Mordell conjecture at the University of Montreal.

Modular forms modulo p and p–adic modular forms

Reminder of the theories of modular forms modulo a fixed prime number p, of p–adic modular forms (after Serre) and of compact operators on p–adic Banach spaces.

Notes of an introductory talk in a seminar on Coleman theory at the University of Montreal.

Representations of semisimple Lie algebras

Reminder of the classical theory of representations of semisimple complex Lie algebras and study of $\mathfrak{sl}_2$ as a basic example.

Notes of a talk in a seminar on the Bernstein–Gelfand–Gelfand complex at the University of Montreal.

Rational points of bounded size on the base of an abelian-by-finite family

Explanation of the technical core of Lawrence–Venkatesh’s proof of the Mordell conjecture, which is the finiteness of the set of rational points of bounded size on the base of an abelian-by-finite family satisfying certain properties.

Notes of a talk in a seminar on Lawrence–Venkatesh’s new proof of the Mordell conjecture at McGill University in Montreal.

Modular forms modulo p

Structure of the algebra of modular forms (of level 1) modulo a fixed prime number p in terms of their q–expansions.

Notes of a talk in a graduate students seminar at Concordia University in Montreal.

Localizing classes

Definition of localizing classes of morphisms and construction of the localization of a category with respect to such a class by means of roof diagrams.

Notes of a talk in a seminar on derived categories at Concordia University in Montreal.

(Classical) modular forms and modular symbols

Introduction for non-specialist mathematicians to a simple case of the (classical) theory of modular forms, Hecke operators and modular symbols.

Notes of a talk in a graduate students seminar at Concordia University in Montreal.

Integrality of the j–invariant of elliptic curves with complex multiplication

Proof that the j–invariant of an elliptic curve with CM is integral by studying its $\ell$–torsion inside a p–adic uniformization.

Presentation slides of a talk in a course on elliptic curves with complex multiplication at McGill University in Montreal.

Classical and overconvergent modular symbols and p–adic L–functions attached to eigenforms

Introduction to the theory of classical and overconvergent modular symbols, the control theorem and application to the construction of p–adic L–functions attached to eigenforms.

Notes of a talk in a seminar on Greenberg–Stevens’s proof of the Mazur–Tate–Teitelbaum conjecture at McGill University in Montreal.

Étale morphisms

Equivalent definitions of étale morphism, basic properties of these morphisms and their sections, review of henselian rings and first results about Galois finite étale coverings.

Notes of a talk in a seminar on Katz’s correspondence at the University of Padua.

The class number formula and Dirichlet’s prime number theorem

Conclusion of the proof of the class number formula (from the properties of the completed Dedekind zeta function) and proof of the equidistribution (with respect to Dirichlet’s density) of prime numbers in congruence classes.

Notes of a talk in a seminar on analytic number theory at the University of Regensburg.

The theorem of the cube

Proof of the theorem of the cube for line bundles over complete varieties and some consequences of this result, among which the theorem of the square for abelian varieties.

Notes of a talk in a seminar on abelian varieties at the University of Regensburg.

Abelian varieties: definition and first properties

Definition of abelian variety as a group variety (i.e., as a group object of the category of algebraic varieties over a field) and basic properties that these varieties and their morphisms satisfy.

Notes of a talk in a seminar on abelian varieties at the University of Regensburg.

Residues and duality

Proof of the formula for the sum of residues on a function field following Tate’s approach and explanation of the duality between differential forms and Weil differentials on a geometric function field.

Notes of a talk in a seminar on algebraic function fields at the University of Regensburg.

The Brauer group

Proof that tensor products induce an abelian group structure on the Brauer group of a field and explanation of the behaviour of this group with respect to base changes.

Notes of two talks in a seminar on central simple algebras at the University of Regensburg.

Classical modular forms and symbols

Reminder of the classical theory of modular forms and introduction to classical modular symbols as a theoretical tool to describe the space of cusp forms $S_2(\Gamma_0(N))$.

Notes (in Catalan) of two introductory talks in a seminar on overconvergent modular symbols at the University of Barcelona.