Solvability in Polynomials of Pell Equations in a Pencil and a Conjecture of Pink

Solvability in Polynomials of Pell Equations in a Pencil and a Conjecture of Pink - Umberto Zannier

Umberto Zannier
Scuola Normale Superiore de Pisa, Italy
April 10, 2013

The classical Pell equation $X^2-DY^2=1$, to be solved in integers $X,Y\neq 0$, has a variant for function fields (studied already by Abel), where now $D=D(t)$ is a complex polynomial of even degree and we seek solutions in nonzero complex polynomials $X(t),Y(t)$. In this context solvability is no longer ensured by simple conditions on $D$ and may be considered `exceptional'.
In the talk we shall mainly let $D(t)=D_\lambda(t)$ vary in a pencil. When $D_\lambda(t)$ has degree $\le 4$, it may be seen that for infinitely many $\lambda\in\C$ there are nontrivial solutions. On the other hand, it is not so when $D_\lambda$ has degree $ 6$ (provided natural assumptions are verified).
Such finiteness result, obtained jointly with D. Masser, represents a conjecture of Pink for `Unlikely Intersections' in a simple abelian scheme over a curve. We shall survey on the whole context and also comment on related problems.