Bounded height problems and applications – Francesco Amoroso (Université de Caen)


Department of Mathematics, Aula Magna.


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.

Further information is available on the event page on the Indico platform.

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