RÉSUMÉ

We shall consider the problem, given a curve with rational coefficients, of how one tries to find all of its rational points. We shall discuss various techniques, and explain how to solve a wide range of naturally occurring problems relating to: cycles of polynomials, Fermat quartics (including the recent solution of a challenge curve of Serre), a riddle of Diophantus, and part of the classification problem for Q-derived polynomials (a polynomial is Q-derived if it and all of its derivatives have all of their roots in Q, and so is ideal for setting students as a calculus problem). Some of the techniques require the use of the Jacobian of a curve, but this will be explained from first principles, and the talk should be accessible to a general Pure Mathematics audience.

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