Supersingular reduction and strongly special intersections in powers of the modular curve

Abstract

We show that Lang--Trotter-type sparsity for simultaneous supersingular reduction of pairs of elliptic curves provides a new arithmetic input for unlikely intersections in powers of the modular curve. Assuming such a sparsity statement, we prove two Zilber--Pink-type finiteness results for Hodge generic curves in $Y(1)^n$. The proof proceeds through height bounds obtained by applying the $G$-function method of Yves André.