# Contemporary methods for solving Diophantine equations (12ss131)

Arriving in Banff, Alberta Sunday, June 10 and departing Sunday June 17, 2012

## Organizers

(University of British Columbia)

(Simon Fraser University)

(Université de Strasbourg)

(Massachusetts Institute of Technology)

(University of Warwick)

## Objectives

textbf{Objectives}. By the end of the workshop the participants
should be familiar with each of the following approaches
to Diophantine equations:

1. For the modular approach: Frey curves, newforms, level-lowering, recipes for
various families of ternary equations (e.g. $A x^p+B y^p=Cz^p$, $A x^p+B y^p=C z^2$, etc.), proof of Fermat's Last Theorem from the recipes, Serre's method of
bounding the exponent, Kraus' method for eliminating one exponent at a time,
predicting the exponents of constants.

2. For Chabauty and the Mordell-Weil sieve: The Mordell-Weil group.
Global $1$-forms. Basics of $p$-adic integration. Chabauty's method.
Mordell-Weil sieve (examples where this aids Chabauty and others
where it proves the non-existence of rational points).

3. For linear forms in logarithms. Statements of standard bounds for
linear forms in logarithms (e.g. Matveev). Using these bounds
to obtain bounds for exponential equations. Descent arguments on
Thue equations, hyperelliptic equations, superelliptic equations etc.
to reduce to exponential equations and obtain bounds for variables.

textbf{Format}. We intend to achieve the objectives by offering a lecture
series on each of the three subjects. Each lecture will also suggest some
exercises for that day. The students will work on the exercises in groups and
present their solutions in a daily wrap-up session. We have ensured that each of
the areas of expertise is represented by at least one of the organizers:

Michael Bennett - Modular approach, linear forms in logarithms, hypergeometric methods

Nils Bruin - Descent, Mordell-Weil Sieving, Chabauty methods

Samir Siksek - Modular approach, Mordell-Weil Sieving, Chabauty methods

We expect the program to have the following form:

textbf{Morning}. Three hour-long lectures.

textbf{Afternoon}. The participants will work on the exercises in groups. We
expect that the groups will be able to work in the lecture hall, the break-out
rooms and the lounge.

textbf{Late afternoon}. The participants will gather for presentation
and discussion of solutions.

textbf{Participants}. We think that the summer school would be most successful if we allow people to apply for participation and select the most promising candidates from that pool. We have included a list of people who would definitely want to attend this summer school if it were to be held now, to give an indication that there will be no problem finding more than enough participants for this event.

textbf{Timeliness, Importance and Relevance}. The most exciting
future progress in the subject of Diophantine equations is likely
to come from applying combinations of available tools to solve
outstanding problems. However, the tools which we have outlined
above belong to separate branches of number theory, each with a separate
history, and very few number theorists have mastered more than one of
these tools. Moreover, the modular approach poses a particular challenge
to non-experts as most of the literature is high-brow and there is
a definite lack of accessible textbooks.

The summer school will give the participants an introduction to each of the
above subjects using a hands-on approach geared towards solving explicit
problems. Unlike most training events, we will emphasize the interconnects
between the methods and the judgement in choosing which method is most
appropriate for a given problem.

Our target audience is graduate students and postdoctoral researchers in
number theory. The prerequistes are modest: Algebraic Number Theory (up to
class groups and unit groups) and a basic acquientence with curves
(particularly elliptic curves)---no more than what is contained in the first
few chapters of Silverman's book "Arithmetic of Elliptic Curves".

We also expect that the course notes from the lecture series will form the firm basis
for a much-needed accessible graduate text book on the various contemporary methods for solving diophantine equations. Indeed, a secondary goal is develop these notes further and put them in book form.