# 189-666A/667B. Graduate Student Seminar

**Organiser:** Eyal Goren, Adrian Iovita and Henri Darmon

**Time: ** Thursdays, 4:00-5:30 PM.

**Room: ** Concordia Seminar room.

**Regular Participants:**

Dylan Attwell-Duval

Luca Candelori

Francesc Castella

Amy Cheung

Vitoria de Quehen

Yara Elias

Andrew Fiori

Francesca Gala

Clément Gomez

Alexandre Paquin

Juan Ignacio Restrepo

Francois Seguin

Bahare Mirza Hossein

Luiz Takei

This seminar is aimed at all graduate students in the number theory group.
The more advanced participants will lecture on
topics that are closely related to, or provide background for, their
thesis problems. We will also choose a theme that (many, but not
necessarily all)
lectures in the seminar shall be devoted
to understanding better. This year's theme is Minhyong Kim's approach to
Diophantine geometry exploiting the pro-unipotent completion of the
fundamental group of
an algebraic curve. The talks will be at a basic level, and aim to prepare
the students who will attend this May's Barbados workshop on
Minhyong Kim's work.

Here are a few rules of the seminar:

1. Each week will be devoted to lectures by one student.

2. This is a working seminar, aimed at students with varying backgrounds.
It is important that lectures be accessible to all participants.

3. Speakers should allow, in fact, welcome, questions,
interruptions, and constructive comments from the audience.

4. Participants are encouraged to ask questions during the presentations, at
any time, and to put in their two cents' worth.

5. One of the themes of this year's seminar is the work of Minhyong Kim on the
fundamental group of
P1-{0,1,infty}.
Only the students who invest some time and effort in following
the relevant lectures, and giving some talks of their own,
will be funded to attend the conference on Bellairs in which
Kim will lecture about his work.

Here is the schedule (to be made up as we go along).

**Thursday, September 22, 4:00-5:30 PM. **

A lecture by **Luiz Takei** on his generalisation of a theorem
of Hecke.

**Thursday, September 29, 4:00-5:30 PM. **

* No seminar, since the organisers will all be absent and dispersed
in three different countries.*

**Thursday, October 6. **

Organisational meeting; general comments on the topic of the seminar.

**Thursday, October 13. **

A lecture by **Bahare Mirza** on integral
models of Shimura curves over totally real fields,
related to her ongoing thesis project.

**Thursday, October 20. **

A lecture by **Francesca Gala**:
Proof that Faltings' theorem (née the Mordell conjecture) implies Siegel's
theorem. (This proof illustrates the role of unramified coverings in the
study of questions of this sort concerning integral points on curves.)

**Thursday, October 27.**

**Juan Restrepo**
Proof of the Mordell-Weil theorem for
elliptic curves, emphasising the role of the Siegel-Weil theorem and
the parallel with the proof that Francesca explained in her lecture.

**Thursday, November 3. **

**Adrian Iovita**

Overview lecture for the seminar, explaining some of the topics that will be covered, including:

Introduction to the pro-unipotent fundamental group of
a projective curve.

Introduction to (mixed) Hodge structures. The mixed Hodge structure on
the complex de Rham realisation of
J/J^{3}.

Chen's iterated integrals.

The *p*-adic étale and de Rham realisations of
J/J^{3}. *p*-adic iterated integrals.

Kim's new proof of Siegel's theorem.

*This overview lecture will be followed by a session of vigorous arm-twisting to
recruit volunteers to speak on the various topics.*

**Thursday, November 10.**

**Luiz Takei**: Proof of Belyi's theorem (in 30 minutes or less).
Followed by:

**Yara Elias**: Elkies' proof that ABC implies Faltings.

**Thursday, November 17.**

In the last weeks, we were treated to a glimpse of how the study of etale covers
of curves leads to diophantine applications.

**Francesca** explained how to prove that Faltings implies Siegel's
theorem on the S-unit equation (i.e., finiteness of *integral points*
on P^{1}-{0,1,infty}) by resorting to a suitable unramified covering
of this curve, having genus at least two.

**Juan** showed us how a similar idea occurs in the proof of the weak
Mordell-Weil theorem, where one analyses the rational points on an
elliptic curve E by studying the unramified morphism from E to E
given by the multipication by
n map. This leads to the proof of the weak Mordell Weil theorem, that
E(Q)/nE(Q) is finite.

** Luiz** explained that the curve P1-{0,1,infty}, rather than being
just an isolated single example, might in fact be ubiquitous in the study
of curves of genus at least 2, by proving *Belyi's theorem*,
the striking assertion tht *every* curve over the algebraic
closure of Q occurs as an unramified covering of P1-{0,1,infty}.

** Yara** started to explain how this insight might be parlayed into
a proof that the ABC conjecture (a statement about how heights and conductors
of rational points on P1-{0,1,infty} behave) leads to a proof
of (effective) Mordell (the statement, proved by Faltings
without an effective bound on the heights of rational
solutions, that a curve of genus at least two over a number field has finitely many rational points.)
There are still a number of interesting issues to be explained, which
Yara could come back to next semester eventually, but in the interests of
moving forward we will put this particular topic on the back burner for a while.

This week, ** Adrian** will give a lecture on

*The pro-p etale fundamental group of a hyoerbolic curve*

This is also a good time to volunteer to give a lecture in the seminar.

** Winter Semester Schedule **

**Thursday, February 2**.

**Juan Restrepo**: The method of Chabauty-Coleman, following the
expository article by McCallum and Poonen.

**Thursday, February 9**.

**Francesca Gala and/or Clément Gomez**: Selmer Groups I. The general notion of Selmer
group attached to a p-adic representation of Q. Local and global duality
and their use in estimating the size of a Selmer group.

**Thursday, February 16**.

**Francesca Gala and/or Clément Gomez**: Selmer Groups II.
The general theory of the previous lecture, applied to the simplest examples.
(Tate twists, Galois representations of elliptic curves or tensor powers
of CM elliptic curves.)

**Thursday, February 23**.

No lecture (because of Spring Break).

**Thursday, March 1**.

** Clément Gomez**. Selmer groups, cont'd.

*(Time permitting)*
** Luiz Takei**. Non abelian Galois cohomology (mainly H^{1})
and its
relation to torsors. The notion of a ``Selmer pointed set".

**Thursday, March 8**.

** Luca Candelori**. Basic facts about the structure of the pro-p unipotent
fundamental group. Its filtration by the lower central series, action of
Galois on the graded pieces of the etale realisation.
A few basic examples. (P1 minus three points, Elliptic curve minus a point,
Curves with CM Jacobians...).

Here are the notes of Luca's lecture.

**Thursday, March 15**.

* There will be no seminar since most of the participants
will be fooling around in Toronto that week.*.

**Thursday, March 22**.

** Luca Candelori**, cont'd. Basic facts about the
structure of the pro-p unipotent
fundamental group. Its filtration by the lower central series, action of
Galois on the graded pieces of the etale realisation.
A few basic examples. (P1 minus three points, Elliptic curve minus a point,
Curves with CM Jacobians...)

**Thursday, March 29**.

** Francesc Castella**.
The "Selmer variety" classifying torsors under (certain finite-dimensional quotients of) the pro-p fundamental group.
Estimates on the dimensions of Selmer varieties.
A few basic examples. (P1 minus three points, Elliptic curve minus a point,
Curves with CM Jacobians...)

**Thursday, April 5**.

** As-yet undetermined victim**.
The ``local points" of the Selmer variety, the etale-de Rham comparison theorem,
p-adic iterated integrals and functions on the local points of the Selmer variety.

**Thursday, April 12<**.

** As-yet undetermined victim**.
Proof of Minhyong Kim's pro-unipotent generalisation of Chabauty's theorem.
Application to the proof of Faltings' theorem for Fermat curves and their
twists over Q, following the article by Coates and Kim.