# Graduate Student Seminar

**Professors:** Chantal David and Henri Darmon

**Time: ** Wednesday 3:00-4:00.

**Room: ** Concordia 5th floor.

**Syllabus**: The graduate student seminar is an
informal seminar which will meet
once a week.
Everyone is welcome to attend,
but the number theory graduate students
and those who are attending the graduate
course 189-726A on algebraic
number fields
are strongly encouraged to do so! It is expected
that they will give most of the lectures.
Topics that we can expect to touch upon may include:
- Elliptic curves, L-functions, and the Birch Swinnerton-Dyer
conjecture.

- Modular forms.

- Arithmetic of modular curves.

- Shimura curves and the theory of Cerednik-Drinfeld.

- Iwasawa theory and p-adic L-functions.

- Galois representations attached to modular forms and Fermat's Last Theorem.

**Grading Scheme**: There will be no assignments. The grade will be
based on in-class participations, and on the lectures given
by the students. It is expected that every student who is registered
in Math 189-726A
will give at least one, and hopefully two, lectures.

**Schedule of talks: **
- Sept 16:
** Ignazio Longhi**, Introduction to p-adic numbers.
- Sept 23:
** Ignazio Longhi**, Introduction to p-adic numbers, cont'd.
- Sept 30:
** Malcolm Harper**, The Euclidean algorithm in number rings.
- Oct 7:
** Jack Fearnley **, Introduction to elliptic curves.
- Oct 21:
** Nicolas Arsenault**, Class groups of GL(2,p)-extensions.
- Oct 28:
** Lassina Dembele**, The Hasse-Minkowski theorem.
- Nov. 4:
** Jessica Bridson**, Units in real quadratic fields.
- Nov. 11:
** Yu-Ru Liu**, Cubic reciprocity.
- Nov. 18:
** Francesco Sica**, Riemann's Memoir.
- Nov. 25:
** Wei-Dong Tian**, The Kronecker-Weber theorem
- Dec. 2:
** Dominic Lemelin**, The proof of Fermat's Last Theorem (Overview)
- Dec. 9:
** Dominic Lemelin**,
The proof of Fermat's Last Theorem (Cont'd)

- Jan. 22:
** Jack Fearnley**, Curves of genus 1 and Weierstrass
equations
- Jan. 29:
** Dominic Lemelin**, Elliptic curves over C
- Feb. 5:
** Ignazio Longhi**, Tate curves
- Feb. 12:
** Steve Thiboutot**, Galois representations attached to
elliptic curves: an introduction.
- Feb. 19:
** Nicolas Arsenault**, The Weil Pairing.
- Feb. 26:
** WEEK OFF **
- March 5:
**Odette Barutciski**,
Rational torsion points on elliptic curves.
- March 12:
**Francesco Sica**,
Hecke L-functions.
- March 19:
**Lassina Dembele**,
Local-global principles for curves of genus one.
- March 26:
** Dominic Lemelin**,
Elliptic curves with complex multiplication.
- April 2:
** Ignazio Longhi**,
Class field theory for function fields of curves over finite fields
- April 9:
** Nicolas Arsenault**,
The Chebotarev density theorem.
- April 16:
** TBA**, TBA
- April 23:
** TBA**, TBA
- April 30:
** Yu-Ru Liu**, The Hasse principle for certain
non-homogenous ternary equations.
- Coming attraction:
** Steve Thiboutot**,
Elliptic curves, Galois representations and the
equation x^2+y^3=z^5.