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According to Mochizuki himself (here), the essential prerequisites for the IUTeich papers are:

• Semi-graphs of Anabelioids (sections 1 to 6)

• The Geometry of Frobenioids I: The General Theory (complete)

• The Geometry of Frobenioids II: Poly-Frobenioids (sections 1 to 3)

• The Etale Theta Function and its Frobenioid-theoretic Manifestations (complete)

• Topics in Absolute Anabelian Geometry I: Generalities (sections 1 and 4)

• Topics in Absolute Anabelian Geometry II: Decomposition Groups and Endomorphisms (section 3)

• Topics in Absolute Anabelian Geometry III: Global Reconstruction Algorithms (sections 1 to 5)

• Arithmetic Elliptic Curves in General Position (complete)

While other sources also recomend:

• The Hodge-Arakelov Theory of Elliptic Curves: Global Discretization of Local Hodge Theories

• The Galois-Theoretic Kodaira-Spencer Morphism of an Elliptic Curve

• A Survey of the Hodge-Arakelov Theory of Elliptic Curves I

• A Survey of the Hodge-Arakelov Theory of Elliptic Curves II

Particularly interesting are Fesenko recent extended remarks on IUT (and learning IUT):

• Fesenko, Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions

As for how the (considerable) gap between Hartshorne and Mochizuki's work, the references on each paper are quite concrete and helpful (see for example the ones on Topics in Absolute Anabelian Geometry I for a good sample).