We shall report on some recent joint works with D. Masser and U. Zannier.
Let $C$ be a curve defined over $\mathbb{Q}$. In 1999, Bombieri, Masser, and Zannier proved a result which may be rephrased as a toric analogue of Silverman’s Specialization Theorem:
Let $\Gamma \subset \mathbb{G}_m(C)$ be a finitely generated subgroup of non-zero rational functions on $C$ which does not contain non-trivial constant functions. Then the set of $P\in C(\overline{\mathbb{Q}})$ such that the restriction of the specialization map $\sigma_P\colon \mathbb{G}_m(C) \to \mathbb{G}_m(\overline{\mathbb{Q}})$, $x \mapsto x(P)$ to $\Gamma$ is not injective is a set of bounded height.
Some years ago we prove, under some technical assumptions on $\Gamma$, the following generalisation:
Let $V$ be an algebraic subvariety of $\mathbb{G}^r_m(C)$ and let $\sigma_P \colon \mathbb{G}^r_m(C)\to \mathbb{G}^r_m(\overline{\mathbb{Q}})$ be the specialization map. Then the set of $P\in C(\overline{\mathbb{Q}})$ such that for some $x\in \Gamma^r\setminus V$ we have $\sigma_P(x)\in \sigma_P(V)$ is a set of bounded height.
As a corollary, we obtain a bounded height result for some degenerate un-likely intersections. Moreover, our specialisation result allows us to develop a new approach to treat families of norm form equations. We prove that, under suitable assumptions, all solutions of a norm form diophantine equation over an algebraic function field come from specialisation of functional equations. For instance for Thomas cubic equation we get:
All diophantine solutions $(t, x, y)\in \mathbb{Z}^3$ of Thomas cubic equation $X(X −A_1(T)Y)(X −A_2(T)Y)+Y^3= 1$ (with $A_1, A_2\in \mathbb{Z}[T], 0 < \mathsf{deg}(A_1) < \mathsf{deg}(A_2)$) for $t\in \mathbb{N}$ (effectively) large enough, are specialisations of a functional solution $(T, X, Y)$.
A simple but significant example (already known by Beukers) of our specialization result is given by the family of equations $x^n+ (1 − x)^n= 1$ with $n$ an integral parameter. In this simple case, the height of the solutions is bounded by $\log(216)$. Very recently we make a start on the problem of generalising to rational exponents, which corresponds to the step from groups that are finitely generated to groups of finite rank. We discover some un- expected obstacles in principle. The proofs are partly based on our earlier work but there are also new considerations about successive minima over function fields.