Mochizuki's "phenomena in number theory" outside the scope of Langlands

Question

(Crossposted from math.stackexchange by suggestion)

On page 12 of Shinichi Mochizuki's "On the Verification of Inter-universal Teichmuller Theory: A Progress Repor", he writes

"The representation-theoretic approach exemplified by the Langlands program does indeed constitute one major current of research in modern number theory. On the other hand, my understanding is that the idea that every essential phenomenon in number theory may in fact be incorporated into, or somehow regarded as a special case of, this representation-theoretic approach is simply not consistent with the actual content of various important phenomena in number theory."

Does anyone know what these 'various important phenomena' are that he's referring to?

Answer 1

As noted by Lucia, large parts of number theory are completely "beyond the scope of the Langlands program": most of analytic number theory, obviously, is, but also many important, active and beautiful subfields of algebraic number theory -- for a list of examples, see for instance the list of publications of Bjorn Poonen, which cover a large scope of subjects in algebraic number theory, but touch the Langlands program only tangentially.

To make sense of the proposed quote of Mochizuki, it is therefore necessary to give a much more restrictive interpretation of "number theory", which I imagine would be in this context the set of number-theoretic questions that can be solved or at least attacked by understanding the Galois groups of number fields, together with all its attached tractors of decomposition groups, inertia subgroups, and Frobenius elements. This set contains all the reciprocity laws, proved or conjectured, and many diophantine equations and problems, from Fermat's last theorem to Mordell's conjecture and its generalizations, through Birch and Swinnerton-Dyer.

My guess is that Mochizuki means that even in this relatively restricted sense, many phenomenons of number theory lie beyond the scope of Langlands. In saying so, he is following an idea that Grothendieck tried to disseminate in the 80's ("avec force", as Deligne said), which I will try to summarize as follows. The Langlands program tries to understand Galois groups by looking at their representations, that is, their action over vector spaces over fields or slightly more generally over modules over commutative rings. At any rate, these objects form an abelian category (even Tannakian), a linear object. Grothendieck's theory of Motives should be attached to this general program, because the motives also are "linear objects", and as Langlands himself famously argued in his famous 1979 paper "Automorphic Representations, Shimura Varieties, and Motives. Ein Märchen".

But, Grothendieck argues, there are many other kind of objects on which we may let Galois group act in order to study them in a different direction than with representations: sets, for instance, as in the version of Galois theory developed in SGA 1 (1960), or non-abelian (profinite) groups, as the étale fundamental groups of various variety over $\mathbb Q$: to this aspect belong all the theory of Grothendieck-Teichmüller, the Dessins d'enfants, the anabelian geometry. That is of course a subject in which Mochizuki has himself proved some remarkable results, and which he claims to have enormously developed up to the point of deducing the ABC conjecture.

An example of a number theoretic (in the restricted sense mentioned above) problem that arguably lies beyond the Langlands program but that the practicians of anabelian geometry plans to study successfully is Mordell's conjecture (proved by Faltings, it is true, with methods not so far from the Langlands program, at least in that they study "linear objects", namely abelian varieties) and all its generalizations (still open). For a very interesting, if speculative, discussion of these questions, see "Galois Theory and Diophantine Geometry" by Minhyong Kim.

Answer 2

The basic idea is that sometimes arithmetic objects aren't "nice enough" for the tools that support the Langlands program (representation theory, noncommutative harmonic analysis, etc) to work.

They work fine, pretty much by definition, as long as the arithemtic object is essentially linear ($GL_n(\mathbb{A})$, cohomologies, Galois groups...).

But when you consider the kind of highly non-linear arithmetic objects (étale fundamental groups, moduli spaces...) that anabelian geometry studies, those theories just aren't meant to apply.

I would recomend section 1.3.3 of "Survey of the Hodge-Arakelov Theory of Elliptic Curves I" (pages 17 and 18), called "The Meaning of Nonlinearity", for a concrete (yet philosophical) example of this kind of reasoning in Mochizuki's own words.

For a pre-anabelian example of the perceived limitations of the Langlands tools and techniques, notice that after all these years, no natural proof (from the point of view of the theory of automorphic representations) exists of the Ramanujan conjecture for classical modular forms.