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The idea of the "field with one element", or $\mathbb{F}_{1}$, is supposed to allow us to do for number fields what we can do for function fields. Hence this idea often comes up regarding problems where a function field analogue has been solved, but not a number field one, e.g. the Riemann hypothesis and the abc conjecture.

In Mochizuki's early approach to the Szpiro conjecture (which is equivalent to the abc conjecture), involving "Hodge-Arakelov Theory", he aims to follow an older proof of the function field conjecture, due to Szpiro himself, which involves a family of elliptic curves and differentiation on the base. In the number field case this would be something in the line of the philosophy of $\mathbb{F}_{1}$. From what I can follow from Minhyong Kim's answer to this question, it involves replacing the de Rham cohomology with etale cohomology, and the Gauss-Manin connection with the Galois action (from there several complications then arise and modifications have to be made).

However, Hodge-Arakelov Theory is eventually not what Mochizuki used in his claimed proof of the abc conjecture. Instead he developed what he calls "Inter-Universal Teichmuller Theory". In papers related to Inter-Universal Teichmuller Theory, such as this recent one by Go Yamashita, there are some mentions of $\mathbb{F}_{1}$, but I find it hard to understand in what sense this philosophy is fulfilled.

Intuitively, how is Inter-Universal Teichmuller Theory related to $\mathbb{F}_{1}$?

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    $\begingroup$ The arithmetic Kodaira-Spencer morphism is codified in the Theta-link. So it's implicitly used there. The Theta-link is kind of like a Frobenius, so it gives a descent data over F_1 $\endgroup$ – user40276 Sep 23 '17 at 2:14

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