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''.)

・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''.)