Some arithmetic unlikely intersection problems in $Y(1)^2$

Contact: Pieter Moree (moree@mpim-bonn.mpg.de) 

A result of Burgeaud, Corvaja and Zannier shows that in the multiplicative group G_m^{^2} over the integers, the size of the arithmetic intersection between a non-special section and the kernel of the raising-to-the-nth-power morphism is “small”. In this talk, we study the modular analogue in Y(1)^{^2} of this problem. We show that a direct generalization of the aforementioned result does not hold in general due to the presence of supersingular primes. The discussion will naturally lead to consider the modular version of the so-called “support problem”, first formulated by Erdős in the multiplicative case. Everything I will discuss is joint work with Gabriel Dill.