Some new cases of Zilber-Pink in $Y(1)^{3}$
Appears in collection : Diophantine approximation and transcendence 2025 / Approximation diophantienne et transcendance 2025
I will discuss joint work with M. Orr (Manchester) and G. Papas (IAS) on the Zilber-Pink conjecture for $Y(1)^{3}$. This is known for so-called asymmetric curves by the 2012 work of Habegger-Pila. More recently, an approach known as the G-function method, has yielded further cases, namely, curves intersecting (∞, ∞, ∞) (D-Orr) and curves intersecting a special point in the boundary (Papas). In this work, we extend the method to deal with curves intersecting a boundary modular curve, and to give an unconditional result for points with few places of supersingular reduction.
