ETD

• algebraic number theory
• diophantine geometry
• elliptic curves
• number theory
• Pila-Wilkie
• singular moduli
• unlikely intersections

The aim of the thesis is to explain a result of Pila and Tsimerman stating that for every positive integer n there exist finitely many multiplicatively dependent n-tuples of singular moduli, excluding trivial cases with dependent subsets. The main ingredient of the proof is the theorem of Pila-Wilkie. In particular, one defines a particular set X of quadratic points, corresponding to the dependent n-tuples of singular moduli, then with some arithmetic estimates finds a lower bound for the number of points with bounded height, depending on the height. The theorem of Pila-Wilkie shows that the number of quadratic points in the transcendental part of X grows less than the lower bound, then they must be contained in the algebraic part of X. With some mixed Ax-Schanuel results we show that an algebraic subvariety of X containing a singular-dependent n-tuple must be atypical and with some other work we show that such a subvariety cannot exist. This concludes the proof.