BIRS Summer School on Contemporary Methods for Solving Diophantine Equations

June 10-17th, 2012 : Banff International Research Station, Banff, Alberta, Canada



Goals of the Summer School

The main objective of the summer school will be to familiarize the participants with each of the following approaches to Diophantine problems:

  • 1. The modular approach: Frey curves, newforms, level-lowering, recipes for various families of ternary equations, the 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. 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. 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.

  • 4. Other techniques: Brauer-Manin and descent obstructions, the hypergeometric method of Thue-Siegel, miscellaneous approaches to Diophantine equations.

    Format: We intend to offer a lecture series on each of these subjects. Each lecture will include a number of linked exercises, ranging from the routine to research level. 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

    Information for Prospective Participants

    The summer school is aimed at graduate students, though other applications will be considered. To express interest in attending, send an email by December 1st, 2011 to describing your background and indicating your interest in the school, and arrange to have your adviser send a short follow-up email (to the same address). Applications received after this date will be considered, space permitting.

    All meals and housing will be provided free of charge for participants, but travel expenses are the student's responsibility. More information (including a tentative schedule) will be made available as soon as possible.


    The summer school is scheduled to immediately precede the 12th meeting of the Canadian Number Theory Association, to be held from June 17th until June 22nd at the University of Lethbridge in Alberta, Canada. Participants interested in this, the largest regularly scheduled meeting of number theorists in North America, are directed to the CNTA XII website.