Diophantine problems of avoiding unlikely intersections

Thesis

Masser and Zannier have proved in 2019 that “most” abelian varieties, defined over Qbar, are not isogenous to any jacobian of a curve. In other terms, “most” points of A_g(Qbar) “avoid” the countably many “isogenous images” of the Torelli locus.

We generalise their statement, also in the context of powers of modular curves, in a number of ways, including allowing for countable families “to be avoided” rather than the fixed Torelli locus and for a smaller source of “avoiding points”, rather than the whole of A_g, to be chosen.

Our second main topic is “double unlikely intersections”: we generalise a result by Marché-Maurin related to curves in powers of the multiplicative group and we prove an analogous version in the context of powers of modular curves.

Authors: Ballini, F

• DOI: 10.5287/ora-ovyz7y7nq
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: elliptic curves; Pila-Wilkie; unlikely intersections; modular curves; heights; diophantine geometry
• Subjects: Number theory; Mathematics; Logic