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Lou van den Dries vddries at
Wed Oct 22 15:59:13 EDT 1997

Let me try to asnwer Simpson's question "What does Faltings' theorem say?"
Not being a number theorist I think I know what it says, but I agree that
an honest asnwer does require notions that may be somewhat arcane from 
the FOM viewpoint. In fact, a weakness of that viewpoint is that it
admits certain questions as basic, but looses interest if the answer
emerges in terms of notions not admitted as basic, even if that is
the only reasonable way an answer can be given at all. 

So, the basic question under discussion is how many rational solutions
a given equation p(x,y) = 0 can have, where p is a polynomial with
rational coefficients, p not a constant. First, we can reduce to the
case that p is an irreducible polynomial over the field of rational
numbers. (This reduction can be done effectively). A further effective
reduction reduces this to the case that p is even absolutely irreducible
(i.e., irreducible ove rthe field of complex numbers; actually, if
p were irreducible but not absolutely irreducible, then there are
only finitely many rational solutions, and one can find them all).
 Now, for absolutely irreducible p one can compute a certain natural
number, called the genus, and if this genus is > 1, the solution has
, sorry, the equation has only finitely many rational solutions, and
I believe one can even compute in principle an upperbound for the
number of rational solutions. Now the genus of the equation p(x,y)=0
is one of those "arcane" notions that are among the great discoveries
of 19th century mathematics. But it's essential to answer the
"basic" question above.  (Not that we already have a complete answer
to the question above, as far as i know; there remains the question
of what happens when the genus is 1, but in this case there are
"arcane" conjectures of Birch & Swinnerton-Dyer that would tell us
when there are infinitely many solutions, and much more; the case
that the genus is 0 is completely understood.) 
  Perhaps it should be mentioned that "in general" a polynomial p
is already absolutely irreducible.

  In connection with the FOM discussion, what is perhaps of interest
here is that the answer to a "basic" question must necessarily be
given in terms of notions like "absolutely irreducible" and "genus"
that Steve would probably not admit as "basic". (That's why I called
them "arcane" above.)  Of course, this kind of thing happens again
and again, and things are much worse when one starts looking at the
methods used to prove such results. (I mean, much worse from the
FOM viewpoint.)  "Basic" questions often have answers in terms of
"arcane" notions. I believe the general scientific public accept
s that. 

 Best regards,
               Lou van den Dries

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