Seminar

We will begin with a discussion of some recent results obtained with David Masser. These results are simultaneously "relative" cases of the Manin-Mumford conjecture (a theorem of Raynaud) and special cases of the so-called Pink Conjecture.

We will continue with a presentation of applications of our results to the solvability of the Pell equation X^2-DY^2 = 1 in nonconstant polynomials X(t),Y (t) when D = D(t) is also a polynomial.

This analogue of Pell's equation for integers was studied already by Abel. In the polynomial situation, solvability is no longer ensured by simple conditions on D, and in fact may be considered "exceptional." In our treatment, we let D(t) = DL(t) vary over a pencil. In the particular case where DL(t) = t^6 + t + L, our results imply that the Pell equation is solvable with nonzero Y(t) only for finitely many complex values of L.