INTER-UNIVERSAL TEICHMÜLLER THEORY: A PROGRESS REPORT

Shinichi Mochizuki
October 2010

The analogy between number ﬁelds and function ﬁelds of curves (e.g., hyperbolic curves)
over ﬁnite ﬁelds is quite classical. In the present talk, we survey work in progress concerninga theory developed by the lecturer during the last decade — in the spirit of this analogy —whose goal is to construct an analogue for number ﬁelds “equipped with an elliptic curve” of thep-adic Teichmüller theory developed by the lecturer during the early 1990’s for hyperbolic
curves over a ﬁnite ﬁeld “equipped with a nilpotent ordinary indigenous bundle”. From an evenmore classical point of view, one may think of this theory as a sort of analogue for number ﬁeldsof classical complex Teichmüller theory, in which canonical deformations of the holomorphic
structure of a hyperbolic Riemann surface of ﬁnite type are constructed by dilating one of thetwo underlying real dimensions of the Riemann surface, while leaving the other dimension ﬁxed
(i.e., “undeformed”).
In the case of number ﬁelds equipped with an elliptic curve, one thinks of the ring structure of the number ﬁeld as a sort of “arithmetic holomorphic structure”. One then constructs
canonical deformations of this arithmetic holomorphic structure — i.e., analogues of the canonical liftings of p-adic Teichmüller theory — by applying the general theory of Frobenioids, aswell as the theory of the Frobenioid-theoretic theta function (developed in earlier papers by thelecturer). At a more concrete level, if one thinks of the ring structure (i.e., “arithmetic holomorphic structure”) of the given number ﬁeld as consisting of “two underlying combinatorialdimensions” corresponding to addition and multiplication, then working with Frobenioids
corresponds, roughly speaking, to performing operations with the multiplicative monoids involved (i.e., multiplicative portions of the rings involved) — in a fashion motivated by the theoryof log structures; in particular, such operations are not necessarily compatible with the additive
portions of the ring structures involved. Alternatively, if one thinks of the ring structure (i.e.,“arithmetic holomorphic structure”) of the various local ﬁelds that arise as localizations of thegiven number ﬁeld as consisting of “two underlying combinatorial dimensions” corresponding
to the group of units and the value group, then one may think of these canonical deformationsof the arithmetic holomorphic structure as deformations in which the value groups are (canonically!) dilated — by means of the theta function — while the units are left undeformed.
Typeset by AMS-TEX2SHINICHI MOCHIZUKI

Since such “arithmetic Teichmüller dilations” are manifestly incompatible with the ring structure of the given number ﬁeld, it follows that they are not compatible, in general, with various
classical scheme-theoretic constructions performed over the number ﬁeld which depend on thisring structure. In particular, these arithmetic Teichmüller dilations fail to be compatible withthe various basepoints of arithmetic fundamental groups involved (e.g., Galois groups) which
are deﬁned by considering scheme-theoretic geometric points. The resulting incompatibilityof (conventional scheme-theoretic) basepoints on either side of the “arithmetic Teichmüllerdilation” gives rise to numerous indeterminacies; these indeterminacies lead naturally to theintroduction of tools from anabelian geometry. It is this fundamental aspect of the theory thatis referred to by the term “inter-universal”.
The (expected) main theorem of inter-universal Teichmüller theory consists of a fairly explicit computation, up to certain relatively mild indetermacies, of the “arithmetic Teichmüller
deformations of a number ﬁeld equipped with an elliptic curve” discussed above by applying
various results obtained in previous papers by the lecturer concerning local and global absolute
anabelian geometry, tempered anabelian geometry, and the étale theta function. This passage
from the Frobenioid-theoretic deﬁnition of the arithmetic deformations involved to a more
explicit Galois-theoretic description may be thought of as asort of global arithmetic ana2∞
logue of the classical computation of the Gaussian integral (i.e., −∞ e−x dx) by means of thepassage from cartesian to polar coordinates. Inequalities of interest in diophantine geometry
may then be obtained as (expected) corollaries of this (expected) main theorem.