# Contemporary methods for solving Diophantine equations (12ss131)

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

## Organizers

Michael Bennett (University of British Columbia)

Nils Bruin (Simon Fraser University)

Yann Bugeaud (Université de Strasbourg)

Bjorn Poonen (Massachusetts Institute of Technology)

Samir Siksek (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.

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.