## Current students

- 2017-present:
**Arthur Razafindrasoanaivolala**(co-supervision with Jean-Marie De Koninck) - 2016-present:
**Jesse Larone**(co-supervision with Omar Kihel and Claude Levesque) - 2014-present:
**Gautier Ponsinet**

## Former students

- 2015-2016:
**Wei Lu** - 2016:
**José Ibrahim Villanueva Gutierrez**(exchange student from Bordeaux 1)

## Summer project students

- 2017:
**Cédric Dion** - 2015, 2016:
**David Ayotte** - 2015:
**Jean-Christophe Rondy-Turcotte**

## Research environment at Laval

If you study Algebraic Number Theory as a graduate student at University Laval, you will be part of:

- Algebra and Number Theory research group of the university, which consists of four professors and ten graduate students currently;
- Centre interuniversitaire en calcul mathématiques algébriques, which organizes research activities for researchers and students from five universities across eastern Canada. There are regular seminars that take place in Montreal, which you will receive financial support to attend;
- Maine-Quebec Number Theory Conference, which takes place every year. You will have the opportunity to attend talks given by international experts in the field as well as give talks on your research topics.

## Scholarships and salaries

Several scholarships are available to garduate students doing a master or a PhD at the Mathematics departement of Laval University. Students also have the opportunity to carry teaching duties for an extra salary. In addition, international PhD students receive an extra scholarship which allows them to effectively pay the same tuition fees as students from the province of Quebec.

PhD scholarships under an agreement between China Scholarship Council and Laval University are also available to qualified students from China. For more information about this program, write to SponsoredStudent.China@br.ulaval.ca.

## Language policy

While courses are given in French, students have the option to write their thesis in either English or French. Students might also take reading courses in English. The university also provides French classes to students.

Research Projects

If you are interested in working with me as a graduate students, you can expect to work on one of the following projets.

### 1. Factorization of p-adic L-functions

An elliptic curves is a cubic curve that is equipped with a group structure. They are used heavily in modern-day cryptography. In this project you will study the p-adic L-function of an elliptic curve, which is a power series defined over p-adic numbers. It tells us about arithmetic properties of the curve over extensions of the rational numbers that are given by p-power roots of unity. You will investigate how the coefficients of these functions behave. By analysing values of these functions evaluated at a set of special points, you will study whether they factor into functions that are easier to understand. This will result in new arithmetic information on the elliptic curve.

### 2. Iwasawa theory of modular forms

A modular form is an analytic function that satisfies invariant properties under certain Mobius transforms. In order to prove Fermat’s Last Theorem, Andrew Wiles showed that an elliptic curve is in fact a special case of modular forms. The analytic structure of modular forms allow us to obtain new information about elliptic curves. In this project, you will study certain representations of the Galois group of the rational numbers that are described by the Fourier coefficients of a modular form. More precisely, you will look at the behaviour of these representations as you restrict them to a tower of extensions and understand their cohomology groups.

### 3. Construction of Euler systems

An Euler system is a family of cohomology classes of a representation that satisfy certain compatibility condition. It is a very powerful tool in Iwasawa Theory because they give us a link between algebraic objects and to analytic objects. In this project, you will study the construction of Euler systems for a certain family of representations and apply them to study asymptotic behaviour of these representations. The construction will rely on understanding the geometry of modular curves and modular surfaces.