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In particular, when one computes the height of a rational point of the projective line
minus three points as a suitable weighted sum of the valuations of the q-parameters of
the corresponding elliptic curve, one may ignore, up to bounded discrepancies, contributions to the height that arise, say, from the archimedean valuations or from the
nonarchimedean valuations that lie over some “exceptional” prime number such as 2.

§ 2.2. Arithmetic degrees as global integrals

§ 2.7. The apparatus and terminology of mono-anabelian transport
Example 2.6.1 is exceptionally rich in structural similarities to inter-universal
Teichm¨uller theory, which we proceed to explain in detail as follows. One way to understand these structural similarities is by considering the quite substantial portion of
terminology of inter-universal Teichm¨uller theory that was, in essence, inspired by
Example 2.6.1:
(i) Links between “mutually alien” copies of scheme theory: One central
aspect of inter-universal Teichm¨uller theory is the study of certain “walls”, or “filters”
− which are often referred to as “links” − that separate two “mutually alien”
copies of conventional scheme theory [cf. the discussions of [IUTchII], Remark
3.6.2; [IUTchIV], Remark 3.6.1]. The main example of such a link in inter-universal
Teichm¨uller theory is constituted by [various versions of] the Θ-link. The log-link also
plays an important role in inter-universal Teichm¨uller theory. The main motivating
example for these links which play a central role in inter-universal Teichm¨uller theory
is the Frobenius morphism ΦηX of Example 2.6.1. From the point of view of the
discussion of §1.4, §1.5, §2.2, §2.3, §2.4, and §2.5, such a link corresponds to a change of coordinates.

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