2012-01-23
・Let me report on my progress in writing up the series of papers on
IUTeich. The task of writing up and making a final check on the
contents of the series of papers is proceeding smoothly, and it
appears that the four papers in the series will come to a total of
about 500 pages. Previously (cf. the entry made on 2009-10-15) I
stated that I hope to finish this project by the summer of 2012;
although it is not clear whether or not I will be able to meet
this summer deadline, I do hope to finish by the latter half of the
year 2012. (Nevertheless, of course, I am not able to guarantee
anything at the present time.)
2011-06-20
・Let me report on my progress in writing up the series of papers on
IUTeich. I finished writing a first draft of IUTeich III (except
for the Introduction) and am currently in the midst of writing
IUTeich IV. After thinking over again the contents of IUTeich III
and IV, I decided to change the title of IUTeich IV to
IUTeich IV: Log-volume Computations and Set-theoretic Foundations.
By contrast to the quite abstract content of IUTeich I〜III
involving arguments concerning the ``reconstruction of the ring
structure'' (in situations in which this ring structure does not
exist) cast in the framework of ``inter-universal geometry'', which
may be thought of as a sort of generalization of anabelian geometry
and, in particular, ``absolute p-adic anabelian geometry'', the
``volume computations'' of IUTeich IV consist of extremely elementary
arguments (of a similar flavor to the arguments of ``Arithmetic
Elliptic Curves...'') --- a sort of ``shift of gears'' to which it
is taking some time for me to become accustomed.
2011-05-02
・I realized that the valuation-theoretic problem left pending in my
comments of 2011-03-03 may be resolved quite easily. In particular,
one may derive the main theorem obtained in Pop-Stix' work on the
p-adic Section Conjecture as a consequence of an entirely elementary
argument in graph theory and commutative algebra (i.e., without
resorting to the use of difficult arithmetic results such as
Tamagawa's ``resolution of nonsingularities''). Click here for more
details.
2011-03-03
・As I explained in a previous note, one may apply the work of
Pop-Stix on the p-adic Section Conjection to reduce the Profinite
p-adic Section Conjecture to its tempered counterpart (a remark
due to Y. Andre). On the other hand, I recently noticed that this
reduction may also be obtained as a consequence of an entirely
elementary graph-theoretic argument (i.e., without resorting to
the use of difficult arithmetic results such as Tamagawa's
``resolution of nonsingularities''). Click here for more details.
2010-11-18
・Let me report on my progress in writing up the series of papers on
IUTeich. I have currently finished writing up a bit more than
two-thirds of the entire series of four papers and am in the midst
of preparing to start writing up the final section of IUTeich III.
In this final section, I plan to formulate and prove the ``Main
Theorem'' of IUTeich (cf. the OHP sheets and Abstract of my
lecture last month for more on this ``Main Theorem''). If
everything proceeds smoothly, I should be able to begin writing up
the ``volume computations'' of IUTeich IV around April or May 2011.
2010-11-11
・I made slight modifications to my comments the other day
concerning the the theorem of Pop-Stix. Click here for more
details.
2010-11-02
・I realized, with regard to the relationship between the
``combinatorial section conjecture'' and the work of Pop-Stix on
the p-adic Section Conjection, that a similar argument holds over
more general fields and for more general sets of primes Σ. This
lead me to modify the argument I gave on 2010-10-29. Click here
for more details.
2010-10-29
・I recently realized that the work of Pop-Stix on the p-adic
Section Conjecture is related to the ``combinatorial section
conjecture'' that I proved in my paper ``Semi-graphs of
anabelioids'' a few years ago. Click here for more details.
2010-04-12
・Let me report on my progress in writing up the series of papers on
IUTeich. Recently, I finished writing up IUTeich II (except for
the Introduction), thus completing roughly half of this project
(i.e., of writing up the series of papers on IUTeich), which I
began in July of 2008. In fact, I expect that IUTeich III and IV
will be shorter than IUTeich I and II. Thus, I have written up
slightly more than half of the series of papers in slightly less
than two years. On the other hand, recently, I am burdened with
more non-research duties than before, so I am afraid that the
remaining portion, i.e., consisting of slightly less than half of
the series of papers, will require slightly more than two years to
complete. In particular, my current expectation concerning the
approximate time of completion of this project remains unchanged,
i.e., roughly around the summer of 2012.
From the point of view of the analogy with pTeich (i.e., p-adic
Teichmuller theory), the theory of IUTeich I and II corresponds to
the construction of ``canonical liftings of curves'' and
``associated Galois representations''. Of the papers that I have
already made public, the two papers that I apply most essentially
in this theory are the papers on ``Semi-graphs of anabelioids''
and the ``Etale theta function''. As is suggested by the analogy
with pTeich, the portion of the theory that I have finished
writing up so far already constitutes a quite substantial portion
of the entire theory. On the other hand, the remaining portion of
the theory, constituted by IUTeich III and IV, is, of course, by
no means ``trivial'' and will still require a substantial amount of
time to complete. Relative to the analogy with pTeich, IUTeich
III corresponds to the construction of ``Frobenius liftings'',
while IUTeich IV corresponds to the computation of the ``Hasse
invariant'' (i.e., the derivative of the Frobenius lifting).
Thus, IUTeich I, II, and III may be thought of as constituting a
single ``natural unit'' in the sense that they concern the
construction of a certain apparatus from the ``parts'' that I
prepared in the various papers that I have already made public.
That is to say, I have already completed more than two-thirds of
the construction of this apparatus. By contrast, IUTeich IV
consists of a certain computation concerning this apparatus.
2009-10-15
・Let me report on my progress in writing up my ideas on IUTeich.
In my last report (in February 2009), I stated that I was
currently in the process of writing up the theory in a series of
three papers, but subsequent to this, the series has grown and now
consists of four papers, as follows.
IUTeich I: Construction of Hodge Theaters
IUTeich II: Hodge-Arakelov-theoretic Evaluation
IUTeich III: Canonical Splittings of the Log-Theta-Lattice
IUTeich IV: An Analogue of the Hasse Invariant
This is not the result of an increase in the mathematical content
of the theory. Indeed, on the contrary, by formulating the theory
in precise terms, my understanding of the theory has become
somewhat more refined, and this has in fact allowed me to make
a number of simplifications in the theory. On the other hand,
I found that in order to expose the theory in a sufficiently
leisurely fashion while adhering to the restriction of limiting
each of the papers in the series to be roughly of the order of 100
pages in length, it would be necessary to divide the series into
four papers, as described above.
At the present time, I have written about half of IUTeich II.
Thus, if one thinks of the entire project as being of length
``4.0'', then my current position is ``1.5''. Although I had
provisionally completed a version of IUTeich I around February,
I subsequently decided to rewrite certain parts of it and to add
a new section. This had the effect of diminishing my pace a bit.
At the present time, my writing pace is roughly of the order of
one paper a year, so if I am able to continue writing at my
present pace, I should be able to finish around the summer of 2012
(although I am certainly not able to guarantee this!).
I have arrived at a more refined understanding concerning quite a
number of topics in the theory, but perhaps the most notable
development in this regard is that I have succeeded in completely
eliminating the complicated theory of limits
that existed previously in the final portion of the theory.
Relative to the analogy with p-adic Teichmuller theory, this
amounts to the realization that instead of working with
``canonical liftings to a p-adic field'', it suffices to work with
``canonical liftings mod p^2''. This simplification has resulted
in a more transparent logical structure in the theory overall, and,
in particular, has rendered more explicit the quite central and
essential nature within IUTeich --- as I had asserted previously!
--- of the ``anabelian theory'' of my previous papers ``Etale
Theta'' and ``Absolute Topics III''.
2009-02-11
・I would like to report on my recent progress, since July of last
year, in writing up my ideas on IUTeich. First of all, although I
stated in my report of 2008-03-25 (cf. Report on Past and Current
Research) that I was planning to write up this theory in two
papers, during the last half a year or so, I decided to divide
the theory up into three papers (so that the length of each paper
would remain roughly of the order of 100 pages). At the present
time, the titles of these papers are as follows:
IUTeich I: Construction of Hodge Theaters
IUTeich II: Hodge-Arakelov-theoretic Evaluation
IUTeich III: Canonical Splittings
Of these, I have essentially finished writing IUTeich I (except
for the Introduction), and I have started writing IUTeich II. If
I can continue writing at my current pace, then I expect to finish
writing roughly by the end of the year 2010 (i.e., as scheduled in
my report of 2008-03-25), but of course, at the present time, I am
unable to guarantee anything with regard to finishing dates.
In IUTeich I, in addition to
(a) the theory of Frobenioids I, II,
I make use, in an essential way, of a nontrivial, though
relatively superficial, portion of
(b) the theory of the Etale Theta Function
as well as of
(c) the theory of Absolute Topics III.
On the other hand, in IUTeich II, I plan to make use of the
deeper portion of (b), while in IUTeich III, I intend to apply
the deeper portion of (c).
2008-06-11
・I finished writing my paper (cf. Papers) on combinatorial
cuspidalizations (cf. my last report on April 9). In this paper,
in the case of proper hyperbolic curves, injectivity as the
dimension of the configuration space is lowered from two to one
is not proven, but as a result of subsequent joint research with
Yuichiro Hoshi, it now appears that this injectivity can be
proven. Thus, once this joint research is completed, one obtains
a generalization of Matsumoto's theorem to the proper case. One
interesting aspect of this development is the fact that in sharp
contrast to the apparently intractable nature of the proper case
if one confines oneself to the bounds of scheme theory, by
applying a theory of a combinatorial nature that, while patterned
after scheme theory, lies outside the realm of scheme theory, this
proper case follows practically effortlessly. That is to say,
this development may be thought of as a good example of the
efficacy of the ``spirit of inter-universal geometry''.
In my paper on combinatorial cuspidalizations, the symmetry of the
GT (=Grothendieck-Teichmuller group) plays an important role in
the proof of surjectivity (i.e., of the outer automorphism groups
of the geometric fundamental groups of configuration spaces when
the dimension is lowered). Recently, I noticed that by applying
this ``GT-symmetry'', one can prove a pro-p GC (=Grothendieck
Conjecture) type result --- the first of its kind --- in the
context of absolute anabelian geometry over p-adic local fields.
The argument is quite simple, but since I would like to return to
writing up IUTeich, it is not clear when I will get around to
writing up this argument.
2008-04-09
・It appears that I should be able to prove a ``combinatorial
version'' of a famous ``injectivity theorem'' of Makoto Matsumoto
(i.e., Theorem 2.2 of his 1996 paper in Crelle) by applying the
``combGC'' (i.e., a combinatorial version of the Grothendieck
Conjecture --- cf. my 2007 paper). This strikes me as a fascina-
ting development for the following two reasons. First of all,
it is interesting to obtain an *application* of a ``Grothendieck
Conjecture-type'' theorem. Second of all, in the proof that I
envision, one applies the ``combGC'' to construct a sort of
*``canonical splitting''*; this is reminiscent of the way that
one applies anabelian geometry to construct a ``canonical split-
ting'' in IUTeich (cf. my recent ``Report on Past and Current
Research''). The resemblance with IUTeich may also be observed
in the fact that this ``canonical splitting'' takes the place of
the property of ``arising from scheme theory'' in Matsumoto's
argument. Finally, I might remark that it appears that the proto-
type of this phenomenon of ``something like the GC giving rise to
splittings/semi-simplicity'' may be seen in the fact that ``if
G is a center-free group and H an arbitrary group, then to give
an extension of H by G is equivalent to giving an outer action of
H on G''.
2006-10-31
・Erratum concerning ``symmetrized configuration spaces'': After
Tamagawa's lecture (concerning the algebraic and anabelian
geometry of configuration spaces) given at the Galois theory
conference held at RIMS the other day, a question was posed (by
Professors Y. Ihara and T. Szamuely) concerning the issue of
whether or not the results obtained in the joint paper by Tamagawa
and myself may be extended to the case of ``symmetrized
configuration spaces'' (i.e., spaces obtained by forming the
quotient of a configuration space by the action of the symmetric
group). At that point, I jumped to a hasty conclusion and
responded that such a generalization is possible for one of the
two types of theorem obtained in our joint paper, but this was the
result of a misunderstanding on my part, for which I wish to
express my apologies.
Upon returning home from the lecture and rethinking the problem,
I reached the following conclusion: The desired generalization to
symmetrized configuration spaces is possible for both types of
theorem under the condition ``pro-l, where l > the dimension of
the configuration space'' (i.e., simply by considering maximal
pro-l subgroups), but does not appear to be possible as an
immediate consequence of the results obtained in our joint paper
for any other cases (e.g., the general profinite case). In
particular, the general extension to symmetrized configuration
spaces appears, at the present time, to be an interesting
**unsolved problem**. Moreover, the issue of whether or not the
decomposition groups associated to the various diagonal divisors
are preserved by isomorphisms of the fundamental groups of
configuration spaces (again a question due to Prof. Ihara) also
remains **unresolved** at the present time.
2006-08-25
・More on the pro-(p,l) abs PGC: Although I still believe the
argument used to prove ``geometricity from point-theoreticity''
[whose existence I announced on 06-24] is correct, I have found
a gap in the argument that I intended to use to prove ``point-
theoreticity'' which I do see any immediate way to fix.
2006-06-24
・More on the pro-(p,l) abs PGC: As of 05-17, in fact, there was
still a portion (involving Green's trivializations) that remained
to be proven, but I now believe that I have solved this problem.
Nevertheless, the ``pro-(p,l) abs PGC'' that I believe that I can
prove is under the assumption of ``point-theoreticity'' (cf. my
paper on ``Cuspidalizations'' for an explanation of this term).
On the other hand, I believe that point-theoreticity should follow
in the not so distant future from joint work that I am currently
carrying out with my student Y. Hoshi. Thus, in my current opinion,
the ``most important unsolved problem in p-adic anabelian
geometry'' is the ``pro-p abs pGC''. Given what I have done so far,
this should essentially amount to the problem of reconstructing the
geometry of the special fiber of the curve (i.e., closed points and
irreducible comonents) in a ``pro-p absolute p-adic setting''.
・Joint research with A. Tamagawa concerning the anabelian geometry
of configuration spaces of hyperbolic curves is progressing
smoothly, and I expect to be able to release our paper on this topic
in the not so distant future.
2006-05-17
・Although I have yet to check the details, it seems as though
I should be able to show a pro-(p,l) absolute version of the
Grothendieck Conjecture (abs pGC) over p-adic local fields. As a
result, I feel more than ever that showing that pro-p Green's
trivializations are preserved in this sort of abs pGC context is
the most important unsolved problem in p-adic anabelian geometry.
(For more on Green's trivializations, cf. my paper on ``Cuspidal-
zations''.)
