Abstract
We study the intersection of an algebraic curve C lying in a multiplicative torus over \({\bar{\mathbb{Q}}}\) with the union of all algebraic subgroups of codimension 2. Finiteness of this set has already been proved by Bombieri, Masser and Zannier under the assumption that C is not contained in a translate of a proper subtorus. Following this result, the question of the minimal hypothesis implying finiteness has been raised by these authors, giving rise to the conjecture~: finiteness holds precisely when C is not contained in a proper subgroup. We prove here this statement which is also a special case of more general conjectures stated independently by Zilber and Pink. Our proof takes its inspiration from an article by Rémond and Viada concerning the Zilber-Pink conjecture for curves lying in a power of an elliptic curve. Hence, it relies on a uniform version of the Vojta inequality proven via the generalized Vojta inequality of Rémond. The main task is to establish a lower bound for some intersection numbers, here on a whole family of surfaces obtained by blowing up a compactification of C × C.
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Maurin, G. Courbes algébriques et équations multiplicatives. Math. Ann. 341, 789–824 (2008). https://doi.org/10.1007/s00208-008-0212-9
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DOI: https://doi.org/10.1007/s00208-008-0212-9