An Ax-Schanuel theorem for the modular curve and the j-function

The classical Ax-Schanuel theorem states that, in a differential field, any algebraic relations involving the exponential function must arise in a 'trivial'

manner. It turns out that one can formulate natural analogues of this theorem in the context of uniformization maps arising from Shimura varieties, the simplest case of which is the j-function. Besides their inherent appeal, such analogues have applications to the Zilber-Pink conjecture in number theory; a far reaching generalization of Andre-Oort.

We will explain these analogues and sketch a proof in the case of the j-function. This is joint work with J.Pila.

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