 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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