Department of Mathematics, Aula Magna.<\/p>\n
We shall report on some recent joint works with D. Masser and U. Zannier.<\/p>\n
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\u2019s Specialization Theorem:<\/p>\n
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. <\/i><\/p>\n
Some years ago we prove, under some technical assumptions on $\\Gamma$, the following generalisation:<\/p>\n
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.<\/i><\/p>\n
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:<\/p>\n
All diophantine solutions $(t, x, y)\\in \\mathbb{Z}^3$ of Thomas cubic equation $X(X \u2212A_1(T)Y)(X \u2212A_2(T)Y)+Y^3= 1$ (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 $(T, X, Y)$.<\/i><\/p>\n
A simple but significant example (already known by Beukers) of our specialization result is given by the family of equations $x^n+ (1 \u2212 x)^n= 1$ 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.<\/p>\n
Further information is available on the event page<\/a> on the Indico platform.<\/p>\n","protected":false},"excerpt":{"rendered":" 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\u2019s Specialization…<\/p>\n