Third Italian Number Theory Meeting

On the "almost'' Pell's equation over polynomial rings

speaker: Laura Capuano (Indam - Scuola Normale Superiore di Pisa)

abstract: It is a classical result that, for any positive integer d not a perfect square, there exist integers x and \(y \neq 0\) such that \(x^2-dy^2=1\). The analogous assertion for \(D,X,Y \neq 0\) in \(\mathbb C[t]\) with \(X^2-DY^2=1\) clearly requires that the degree of D should be even and in general solvability is no longer guaranteed if we only ask that D is not a perfect square in \(\mathbb C[t]\). In the talk we shall 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 complex \(\lambda\) there are nontrival solutions. On the other hand, this is not so when \(D_{\lambda}(t)\) has degree 6. In this context, as an application of some results about Unlikely Intersections for certain families of abelian surfaces, Masser and Zannier proved that, if \(D_{\lambda}(t)=t^6+t+\lambda\), the Pell's equation is solvable nontrivially only for finitely many \(\lambda\in \mathbb C\). Here we consider a variant of this, namely the ``almost'' Pell equation \(X^2~-~DY^2=~P\), where \(P(t)\in \overline{\mathbb Q}[t]\) is a polynomial of degree at most 2. When \(D_{\lambda}(t)\) varies in the previous pencil, we have another finiteness result. This is a consequence of theorems on Unlikely Intersections for points on Jacobians of genus two curves.

timetable:
Tue 22 Sep, 15:00 - 15:25, Aula Contini
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