Dear Colleague:

    The purpose of this letter is to give an informal and intuitive explanation 
of Dwork's unit root conjecture and some new "results" I have recently obtained. 
We shall try to put them in a wider perspective. 

    Dwork's conjecture can be viewed as the starting point of a truly p-adic 
extension of the Weil conjectures from a single variety over a finite field 
to a family of varieties over a finite field. It provides a possible ground 
to extend the Gouvea-Mazur conjecture about modular forms (the ordinary family 
of elliptic curves) to an ordinary family of algebraic varieties over a finite 
field. A few other closely related conjectures will be mentioned along the way. 
It is my hope that this mini-expose gives an useful indication about many of 
the open p-adic questions related to L-functions of algebraic varieties over 
finite fields. If there is anything here that is misleading or contradictory, 
it is most likely caused by my misunderstanding and naive imagination. 
Thus, your comments, corrections, suggestions and updated information for 
improvements are warmly welcome and will be greatly appreciated. 


1. Geometric case of Dwork's conjecture. 
   (Normalized period matrices II, Ann. Math., 98(1973), 1-57)
   
 
Let Fq be a finite field of q elements of characteristic p>0. Let X be a 
variety over Fq (connected smooth affine would be suffice for us). Dwork's 
conjecture says that if H is a continuous p-adic representation of the 
arithmetic fundamental group of X which "arises" from algebraic geometry, 
then the L-function L(H, T) is p-adic meromorphic. 

The conjecture can be reformulated more precisely in terms of p-adic etale 
cohomology. Let f: Y->X be a family of algebraic varieties defined over Fq. 
Let H be one of the relative p-adic etale cohomology groups with compact 
support of the family f: Y->X. Dwork's conjecture says that the L-function 
L(H,T) attached to the p-adic etale sheaf H on X is p-adic meromorphic. 

In the case Y=X and f is the identity map, then L(H,T) gives the zeta 
function of X which is rational (by Weil conjecture) and hence p-adic 
meromorphic. In fact, the key step of Dwork's rationality proof is to 
show that the zeta function is p-adic meromorphic. 

If f: Y->X is a finite map, then L(H,T) gives the L-function of a finite 
covering which is again a rational function by Grothendieck's l-adic theory 
and hence p-adic meromorphic. More generally, if H is any constructible 
l-adic etale sheaf with l different from p, then the L-function L(H, T) 
is always rational by Grothendieck, whether H comes from algebraic geometry 
or not. The Weil conjectures extend very well to l-adic setting and are 
reasonably well understood by Deligne. 

One then wants to know if it is possible to extend the Weil conjectures to 
p-adic setting, particularly in the case when f: Y->X is an "infinite" map. 
As it is well known, the situation is very different and much less clean. 
First, if H is a constructible p-adic etale sheaf, it is NOT true that the 
L-function L(H,T) is rational, even assume that H comes from algebraic 
geometry. Second, supported by various results from Dwork-Monsky theory, a 
weaker statement was conjectured by Katz (73) which says that L(H,T) is 
p-adic meromorphic for any p-adic etale sheaf. This is again not true as 
a counter example is given in my paper (96). 

Dwork's conjecture says: ASSUME that H comes from algebraic geometry 
(as the relative p-adic etale cohomology of a family of varieties over 
a finite field), then L(H, T) is p-adic meromorphic. 


2. General case of Dwork's conjecture.


The conjecture says that if H is the unit root sub F-crystal of an ordinary 
overconvergent F-crystal defined on X, then L(H, T) is p-adic meromorphic. 
Furthermore, if k is an integer and H^k is the k-th power (k-th iterate) of 
H, then the conjecture says that L(H^k, T) is also p-adic meromorphic. Note 
that the k-th "power" H^k is just the k-th tensor power of H if H has rank 
one and k is a positive integer. If the rank of H is greater than one, then 
H^k is NOT the k-th tensor power (it is "not" even an F-crystal over Fq).   

Example. In nice geometric case f: Y->X, the relative (Berthelot) rigid 
cohomology with compact support gives an overconvergent F-crystal on X. 
By shrinking X if necessary, this overconvergent F-crystal is ordinary 
fibre by fibre at slope zero. Its unit root sub F-crystal H coincides with 
the relative p-adic etale cohomology with compact support of the family 
f: Y->X. This follows from a recent result of Etesse-Le Stum, which is the 
"constant sheaf case" of a plausible general "comparison" conjecture of 
Katz. Note that the unit root sub F-crystal H is obtained by solving the 
fixed point of a contraction map in a p-adic (complete) Banach space and 
thus H will be in the full complete convergent category, no longer 
overconvergent in general.

Dwork's conjecture grew out of his attempt to find a p-adic analytic formula 
for the roots of a given slope of a family of zeta functions of varieties 
over finite fields. He already obtained such a formula by a totally different 
method in the case of level 2 elliptic curve family in 1957. This formula was 
the starting point of his p-adic theory which led to his rationality proof in 
1959. In early seventies, he returned to the formula searching problem for a 
family of hypersurfaces and showed the existence of formula in the ordinary 
case where the Newton polygon coincides with the Hodge polygon fibre by fibre. 
In the more complicated "supersingular" case where the Newton polygon lies 
strictly above the Hodge polygon, the existence of p-adic formula (shrinking 
the parameter space if necessary) was later established by Katz (79) in his 
isogeny theorem in the context of Grothendieck's specialization theorem. There 
are still a couple of questions left open in Katz's work, but from L-function 
point of view, the formula problem was "essentially" completely solved in 
Katz's paper. Dwork attempted to understand these p-adic formulas from Weil 
conjecture point of view. He attached L-functions (unit root zeta functions) 
to these p-adic formulas and conjectured that they are p-adic meromorphic. 
Its consequences include explicit as well as asymptotic p-adic formula for 
"geometric" p-adic character sums. An easy corollary would be the existence 
of p-adic "equi-distribution" theorems of p-adic "angles" of the zeros for 
a family of zeta functions of varieties over a finite field. 


3. What is known?


It turns out that there are several possible categories of F-crystals that one 
could choose to work with, depending on various convergent conditions and 
depending on what types of problems one wants to study. The results would then 
be quite different. 

(a). Good case. 

This is the category of overconvergent F-crystals (F-isocrystals), where BOTH 
the Frobenius map AND the horizontal connection are assumed to be overconvergent. 
This category is in some sense "algebraic" in nature. Conjecturally, this 
category should be the p-adic analogue of lisse l-adic sheaves and thus the 
Weil conjectures are expected to extend to this category of overconvergent 
F-crystals. Berthelot's rigid cohomology is developed for this purpose. It 
includes, as special cases, both the Monsky-Washnitzer cohomology for smooth 
affine varieties and crystalline cohomology for proper smooth varieties. 
Much progress has been obtained, mostly in the case where the overconvergent 
F-crystal extends to a compactification of X. Much, however, remains to be 
done in general. In particular, the conjectural rationality of the L-function 
is still open in general. The rationality follows from the conjectural finite 
dimensionality of the rigid cohomology with compact support of overconvergent 
F-crystals. From analytic point of view, one DOES know the weaker result that 
the L-function is p-adic meromorphic, as the Frobenius map is overconvergent. 
The Riemann hypothesis (p-adic analogue of Deligne's l-adic theory of weights) 
also seems unknown in general, although some results were obtained by Crew and 
Faltings. It is conjectured that an irreducible overconvergent unit root 
F-crystal whose determinant is a character of finite order is the "p-part" of 
a compatible l-adic system, a p-adic "analogue" of Delinge's l-adic conjecture 
in Weil II. 

(b). 1/2 good case. 

This is the category of F-crystals whose Frobenius map is overconvergent 
but the horizontal connection is arbitrary. This category is certainly not 
"algebraic" any more but it is "analytically good" from L-function point 
of view. In fact, the L-function is p-adic meromorphic by the Dwork-Reich-
Monsky trace formula, but not a rational function in general.  Ideally, one 
would like to have a cohomological formula which would explain the L-functions. 
There are, however, serious difficulties with the spectral theory which shows 
that in general one cannot pass a chain level formula to its cohomology level.
Since the "1/2 good case" does arise from geometry (the family of elliptic 
curves, for instance), it would be interesting to find any "weak cohomological 
formula" in various cases if possible, which would give some non-trivial 
information about the cancellation of zeroes of the characteristic series of 
the Frobenius acting on chain level. This would be very useful, for instance, 
in deeper study of the Gouvea-Mazur type conjecture and the p-adic Riemann 
hypothesis to be described later in this expose. 

(c). Bad case. 

This is the full category of (convergent) F-crystals, where neither the 
Frobenius map nor the horizontal connection is assumed to be overconvergent. 
This category is not even "analytically good" but just "continuous" so that 
the L-function is well defined. For instance, my counter-example of Katz's 
meromorphy conjecture shows that the L-function is not even p-adic meromorphic 
in general. However, one DOES know that the L-function is meromorphic on the 
closed unit disk (a non-trivial result) and in some subcategories one knows a 
larger "optimal" finite disk for which the L-function is meromorphic. Ideally, 
one could hope for some sort of optimal p-adic cohomological formula which 
would "gives" the meromorphy "portion" of the L-function in the expected disk 
of meromorphy. This could be viewed as an analogue and improvement of Katz's 
conjectural p-adic etale formula which "explains" the zeros and poles on the 
closed unit disk.

We now return to Dwork's conjecture and see how it fits into the above crude 
classification. Let H be the unit root sub F-crystal of an overconvergent 
(hence good) ordinary F-crystal. If H happens to be in the "good case" or 
in the "1/2 good case", then the L-function L(H,T) is p-adic meromorphic 
and thus Dwork's conjecture is already true.  These are the ONLY cases for 
which Dwork's conjecture is known to be true and thus no single "non-trivial" 
example from the bad case has been proved. Unfortunately, in general H is 
most likely in the "bad case". It is already a rare and fortunate accident 
if H lands in the "1/2 good case", not mentioning in the "good case". This 
has something to do with the possible existence of the so-called excellent 
lifting introduced by Dwork to attack some special cases of his conjecture. 
The existence of an excellent lifting implies that H is in the 1/2 good case. 
But excellent lifting rarely exists. For instance, the ordinary family of 
curves of genus g (if g is not too small) has no excellent lifting (Dwork-Ogus, 
Oort-Sekiguchi). The family of higher dimensional Kloosterman family has no 
excellent lifting either (Sperber). My only philosophical reason for Dwork's 
conjecture in geometric case is that H is natural: it comes from geometry.  


4. An example: elliptic curves and Gouvea-Mazur conjecture. 


Let f: E->X be the ordinary family of elliptic curves of some level N over 
a finite field with N not divisible by p. Let H be the H^1 of the relative 
p-adic etale cohomology with compact support of the family f: E->X. Then, 
H is a rank one unit root F-crystal. It is rather fortunate that H is in 
the "1/2 good case" (not in the "good case"). This is because the Frobenius 
map is overconvergent with respect to the excellent lifting (Delinge-Tate 
lifting or the canonical lifting). Thus, the unit root L-functions L(H,T) 
and L(H^k, T) are p-adic meromorphic everywhere (they are "transcendental" 
non-rational functions). This is how Dwork proved his conjecture in the 
case of level 2 elliptic curve family in his short paper "On Hecke 
polynomials, Invent. Math., 12(1971), 249-256). These L-functions L(H^k,T) 
are closely related to the characteristic series of the U-operator in the 
theory of overconvergent p-adic modular forms as developed systematically 
by Katz in early seventies, motivated by Dwork's work on the so-called 
"Atkin conjecture". 

In another direction, Hida was studying families of ordinary modular forms 
and families of modular Galois representations of the absolute Galois group 
of Q. This led to Mazur's theory of deformation of Galois representations. 
This circle of ideas have been very useful in the work of Wiles. In an attempt 
to extend Hida's work from slope zero to higher slope cases, Gouvea-Mazur 
(from late eighties to early nineties) formulated a series of rather precise 
conjectures about families of classical and overconvergent modular forms. 
Qualitative versions of all of these conjectures have been established by 
Coleman using rigid overconvergent theory. 
 
There are two well defined families of p-adic entire functions f(k,T) 
and  g(k, T) parametrized by integer k with constant term one such that  
L(H^k, T) = f(k, T)/g(k, T) for each k. Furthermore, both f(k, T) and 
g(k, T) satisfy the same continuity condition as the characteristic series 
of the U-operator acting on the weight k space of overconvergent p-adic 
modular forms. Namely, if k_1 and k_2 are congruent modulo (q-1)p^{m-1}, 
then f(k_1, T) is congruent to f(k_2, T) modulo p^m. Similar congruence 
holds for g(k, T).  

For a non-negative rational number s, let m(s) be the smallest non-negative 
integer m (or infinity if such integer does not exist) such that whenever 
k_1 and k_2 are integers congruent to each other modulo (q-1)p^{m-1}, then 
the Newton polygon of f(k_1, T) (resp. g(k_1, T)) coincides with the Newton 
polygon of f(k_2, T) (resp. g(k_2, T)). The Gouvea-Mazur conjecture on 
dimension variation of modular forms is equivalent to the statement that 
m(s) is bounded by the linear polynomial s. Qualitative result of Coleman 
shows that m(s) is finite for every s. Using intuition on Dwork's unit root 
zeta functions, I further observed that m(s) is bounded by a quadratic 
polynomial in s.

There is a p-adic Riemann hypothesis which I proposed in this case. It 
says: the slopes (which are rational numbers) of the zeroes and poles of 
f(k, T) and g(k, T) have a uniformly bounded denominator. This can be 
viewed as a finiteness conjecture. I have no evidence for it, other than 
a characteristic p analogue I obtained for the "Riemann zeta function" 
studied by Drinfeld theorists. A proof of the p-adic RH might give new 
information on the full form of the Gouvea-Mazur conjecture.  

Another favorite example is to consider the family of n-dimensional 
Kloosterman sums associated to the family of Laurent polynomials 
y_1+...+y_n +x/y_1...y_n parametrized by x. The unit root F-crystal H 
in this case is also of rank one. If n=1, Dwork (74) showed the existence 
of excellent lifting. Thus, H is fortunately in the "1/2 good case". It 
follows that both the meromorphic continuation and the quadratic bound on 
m(s) for elliptic curve family carry over to the Kloosterman family if n=1. 
If n>1, Sperber (76) showed that there does NOT exist an excellent lifting. 
This means that H is indeed in the "bad case" for all n>1 and thus Dwork's 
conjecture is open for n>1. This first "non-trivial" example serves as the 
testing example and guidance for my research to be described below. 


5. New results: rank one case of Dwork conjecture. 


I have a systematic limiting approach to study Dwork's conjecture, which 
avoid excellent lifting completely. Although there may be many technical 
difficulties involved in general, I believe that the limiting approach 
should be able to prove Dwork's conjecture if H has rank one. I have worked 
out the details of my method in the technically simpler toric setting, where 
all of the most fundamental difficulties already occur. In particular, it 
settles the higher dimensional Kloosterman family for all n>1 as mentioned 
above. A preprint (which apparently needs revision) is now available in my 
www home page "http://www.math.uci.edu/~dwan/preprint.html". 

A major advantage of the limiting approach is that it can be used to study 
the arithmetic variation of the family of meromorphic functions L(H^k, T) 
parametrized by integer k, extending (weaker version of) Gouvea-Mazur's 
conjecture from the family of elliptic curves to a family of varieties with 
one p-adic unit root fibre by fibre. Our proof shows that there is a well 
defined decomposition L(H^k, T)= f(k, T)/g(k, T) of L(H^k, T) into the 
quotient of two continuous and p-adic entire families of functions. This 
decomposition came in a way which is very different from the "1/2 good 
case" where the trace formula can be applied directly. The decomposition 
formula makes it possible to define the quantity m(s). And our method can 
be used to show that m(s) is finite for every s. 

We expect that the limiting approach together with the method in my quadratic 
bound paper should also be able to give explicit bounds on m(s). Although I 
have not worked out the explicit calculation, preliminary feeling suggests 
that the methods should ultimately be able to prove something like the 
following in general.

"Conjecture": Let H be a rank one unit root F-crystal embedded in a rank r (>1) 
            overconvergent ordinary F-crystal on a variety X of dimension d>0. 
         (1). If r=2, then m(s) is bounded by a polynomial in s of degree d+2.  
         (2). If r>2, then m(s) is bounded by an exponential function of s.
                
Note that even in the case r=2 and d=1, the conjectural bound is a cubic 
polynomial which is slightly weaker than the known quadratic bound for the 
elliptic curve family. If one knows that an given H is already in the much 
better 1/2 good case, then m(s) should be bounded by a polynomial of degree 
d+1 for any r, by directly applying the trace formula and the method in my 
quadratic bound paper. 

Once we know that L(H^k, T) is p-adic meromorphic, my p-adic RH can be proposed 
for these L(H^k, T) as well. 

p-adic RH: Let H be a rank one unit root F-crystal coming from algebraic geometry. 
           Then the slopes of the zeroes and poles of the unit root zeta functions 
           L(H^k,T) have a uniformly bounded denominator. 

The elliptic curve family and the one dimensional Kloosterman family would be 
the first "easier" non-trivial "natural examples" to try. 


6. New "results":  higher rank case of Dwork's conjecture. 


In some higher rank cases, the limiting approach can be used to "imply" (it 
does not "prove" directly) that Dwork's unit root zeta function is meromorphic. 
In general, the limiting approach does not seem to work if the rank is greater 
than one. It would prove, if worked, that the weaker version of Gouvea-Mazur 
conjecture extends to higher rank case as well. But this would involve a great 
and uniform cancellation of zeros and poles of various L-functions which I 
could not explain and which I have serious doubt. 

I have, however, a different embedding approach which can be used to prove some 
higher rank cases of Dwork's conjecture without taking limit at all. I have 
just obtained more ideas which seem to show that the limiting approach and the 
embedding approach can be combined together to prove Dwork's conjecture in higher 
rank case as well! The embedding approach, however, does NOT prove the possibly 
false statement that the weaker version of Gouvea-Mazur conjecture extends to 
higher rank case. If one really wants to "extend" the GM conjecture to higher 
rank case, one could consider a fixed higher rank overconvergent F-crystal H_1 
and twist it by the k-th tensor power of another rank ONE unit root F-crystal 
H coming from geometry. If one lets the tensor power k varies, then one should 
be able to get a family of unit root zeta functions extending weak version of 
the GM conjecture. If we take $H_1$ to be the trivial unit root F-crystal 1,  
then we recover the variation result from rank one case. 
 
Remarks: All previous approaches to Dwork's conjecture, whenever successful, 
will show that the Fredholm determinant involved in the trace formula is p-adic 
ENTIRE. Conversely, assume that Dwork's conjecture is true, one can only conclude 
the weaker statement that the Fredholm determinant is meromorphic with possible 
poles (and thus NOT necessarily entire). Our approaches do NOT imply the possibly 
false statement that the Fredholm determinant is entire, it only implies that the 
Fredholm determinant is meromorphic which is all one needs for Dwork's conjecture. 
If the Fredholm determinant does have poles in various cases, what would be the 
meaning of these "new" poles?

Dwork also conjectured a functional equation for his unit root zeta functions 
in the geometric case. I do not know what this means yet and thus I did not 
mention it in the above. 

Our approaches, if they do work out as expected, raise MANY more questions than 
the "theorems" we could prove. In addition to many of the above questions, the 
notion of "F-crystals of INFINITE rank with Hodge structure" alone, which plays 
a key role in the limiting method, might already suggest quite a few reasonable 
problems to explore. The various unit root tensor categories occurring in the 
embedding approach provide new "mysterious coefficients" which are beyond the 
1/2 good case but are "arithmetically good" in nature because their L-functions 
are meromorphic. One could also ask for possible p-adic "modular" interpretation 
of various cases of Dwork's conjecture, generalizing what we know about the 
elliptic curve family. 
 
There is a notion of c\log convergent F-crystals which can be used to refine 
the above crude classification of three categories of F-crystals. This notion 
is not fully developed yet in general. For many of the above problems and 
results, it should be possible and sometimes more accurate to replace the 
overconvergent condition by the significantly weaker \infity\log convergent 
condition (c = infinity). Since these notions are still "non-standard", I 
have decided to ignore them entirely in this min-expose in order NOT to make 
the already complicated picture even more complicated and/or confusing. 


    Sincerely yours,
    Daqing Wan

    Department of Mathematics 
    University of California
    Irvine, CA 92697 


Appendix: A brief history about this investigation. 

Dwork's conjecture came to my attention in late 1989 when I first met him in 
an AMS meeting organized by Adolphson-Sperber. At that time, I already had 
some "ideas" on Katz's more general meromorphy conjecture. At the meeting, 
Dwork-Sperber gave me a copy of their manuscript which proves that Dwork's 
unit zeta function is meromorphic in a disk LARGER than the closed unit disk. 
Soon afterwards, I found a tiny idea which gave a "one-line" proof of the 
Dwork-Sperber result. I then naively tried to infinitely "amplify" the tiny 
idea in the hope to get better and better meromorphic continuation and 
eventually prove Dwork's conjecture. This led me to the beginning of the 
embedding approach. I could not continue the method any further because I 
did not even "understand" the definition of an F-crystal. 

A few years later, in spring 1993, I met Katz at the Igusa retirement conference. 
I asked him about F-crystals and an embedding property that I need on F-crystals. 
He explained the meaning of F-crystals and showed me immediately why the embedding 
property I needed cannot be true. This then "disproves" my embedding approach. 
If the embedding property I needed turned out to be true, it would prove Katz's 
meromorphic conjecture as well which implies Dwork's conjecture as a corollary. 
But as it turns out, Katz's meromorphy conjecture is not true. Thus, my initial 
embedding approach was dead!   

I then went back to my tiny "one-line" idea. After some thoughts, it occurred to 
me that a limiting approach might be reasonable to try as I feel that I could get 
better and better results on the terms of the sequence of L-functions which 
approach the unit root zeta function. With Bombieri's arrangement, I was able 
to visit IAS during 93-94 academic year. There, I met Deligne and discussed my 
limiting approach with him. He quickly realized the substance and explained to 
me what I need to prove first in order to make my limiting argument work. Namely, 
one needs to bound the number of zeros and poles in any fixed finite disk for 
the whole family L(H^k,T). But each of L(H^k,T) was quite mysterious itself, 
it was not clear if the uniform bound would be true. Deligne suggested even 
further to do some numerical computations to get a feeling if the uniform bound 
would likely be true or false. I did not have the courage to do numerical 
computations. After several weeks of theoretical thoughts, I started to have 
feeling that the uniform bound should be true if H has rank one and I developed 
explicit ideas how to do it. This is the most technical part of my investigation.  

After another discussion with Katz, I was then confident that the limiting 
approach should be able to prove the rank one case of Dwork's conjecture. This 
would then imply that Dwork's conjecture is true for the unit part part of the 
higher n-dimensional Kloosterman family which was the first "non-trivial" example 
on my mind. But this family also has a rank one unit root F-crystal coming from 
higher slope for each integer slope between 0 and n by a theorem of Sperber (76). 
Dwork's conjecture applies to all of these rank one unit root F-crystals coming 
from higher slopes. I could only do the slope zero part but could not do the 
higher slope case (still rank one). I was then "unhappy" that I could not even 
get a complete result for the n-dim Kloosterman family. The problem was that I 
needed some sort of "inductive" argument to make my proof work for higher slope. 
After several subsequent attempts, I still didn't get the induction I wanted. 

In spring 96, at Penn State Library I saw a new article by Gouvea-Mazur where 
they described their conjecture and pointed out the connection between their 
conjecture (on the characteristic series of the U-operator) and the unit root 
zeta functions for elliptic curve family. I was immediately interested in it. 
Guided by some of my feeling on general unit root zeta functions, I quickly 
realized that one should be able to get something out of it. This led to the 
quadratic bound for the GM conjecture. 

Then, in Summer 97, when I was visiting Sichuan University in Chengdu and the 
Morningside Center of Mathematics in Beijing, I returned to the inductive 
problem for higher slopes (still rank one). This time, luckily, I found ideas 
to make the inductive argument work. This would at least give me a clean result 
for the testing higher dimensional Kloosterman family. More importantly, 
it paved ways (without realizing it) for later ideas in higher rank case. 

With M. Fried's arrangement, I moved to Irvine in Fall 97. I felt that it was 
about time to return to Dwork's conjecture in full speed. I started to put 
together the ideas I had, work out all the details and try to write down what 
(if any) I could actually prove in the rank one case. After various small 
"surprises", all went as expected and as a pleasant bonus the limiting method 
also gives something more (the weak GM conjecture) in the rank one case. This 
is what contained in my preprint.  

However, I still had no ideas at all what would happen in higher rank case. 
In fact, I could not even prove a single "non-trivial" example in higher rank 
case. My mood had been shifting back and forth between the two modes of doubting 
the conjecture and believing the conjecture, depending on whether I had any idea 
to try at any particular moment. I was almost "certain" that the limiting approach 
would NOT work in higher rank case. On the other hand, it seemed to me to be 
"completely impossible" to find and prove a counter-example. Thus, either I should 
just give up or try to prove the conjecture to be correct and hope for the best. 
Then I started to write this mini-expose in the hope to put the picture in a wider 
perspective and see if I could get any idea after "refreshing" my mind a little. 
Looking at this expose again and again, I feel that philosophically my approach 
seems to be the right one. If Dwork's conjecture is going to true, the method 
should be able to prove it. Thus, I decided to give a final round of intensive 
attack on the higher rank case of the conjecture before giving up. 

I started to "refine" the limiting approach but try not to take the limit. 
After some thoughts, the method seems to be able to handle some special higher 
rank cases, not by proving them directly, but as a consequence "forced upon"  
by other easier rank one cases which we know by the limiting method. This was 
slightly encouraging. I then tried to extend this argument but it did not 
seem to get any further. It would be "simply too complicated to be realistic" 
even if it would work out. Trying this again and again, I found that I was 
actually trying some arguments resembling the embedding argument I abandoned 
and "forgotten" long time ago. Thus, I returned to my failed embedding approach, 
started all over again and tried to embed it in a  "correct" way. With more 
feeling and experiences now, I could quickly find where to embed and "see" how 
to prove all "easier" higher rank cases (already a lot of them) without even 
bothering the limiting approach. Thinking further, it occurred to me that the 
rank one result got from the limiting approach looks EXACTLY what I would 
need to make my corrected embedding approach work. It more or less creates 
the flexibility I need to "reduce" the general case to the  "easier" 
case. 

Although there are likely many "surprises" coming up when working out the 
details of the ideas, I believe that the basic ideas are now enough to prove 
Dwork's conjecture in full generality, in the context of Grothendieck's 
specialization theorem.