From: elkies@math.harvard.edu (Noam Elkies)
Subject: Rational points on transcendental curves
Date: 27 Mar 00 12:38:37 GMT
Newsgroups: sci.math.numberthy
Summary: [missing]

I believe I have a proof of the following theorem.  By a "transcendental
curve" I mean a subset of P^2(C) of the form f(K), where f is an analytic
function from an open set E in C to P^2(C) whose image is not contained
in an algebraic curve, and K is a compact subset of E.  Fix such a curve,
and a number field F and an embedding of F into C.  It then makes sense
to speak of F-rational points of P^2(C), and thus of F-rational points
on our transcendetal curves.  In particular, how many such points are
there of height at most H?  Here we are using exponential height, and
as will be seen I don't have to specify how it is normalized.

The theorem I am claiming is: for each epsilon>0, there exists an
effectively computable constant K such that the number of F-rational
points on a given transcendental curve of height at most H is less
than  K H^epsilon.

Is this known already, and if so by whom and how?  Has this question
even been studied?

--Noam D. Elkies (elkies@math.harvard.edu)
==============================================================================

From: elkies@math.harvard.edu (Noam Elkies)
Subject: Rat'l pt. query, simplified
Date: 30 Mar 00 14:16:45 GMT
Newsgroups: sci.math.numberthy

I see that the phrasing of my recent query was needlessly complicated,
and also confused matters further by using K for both a compact subset
of C and a real constant.  Here is a simplified statement -- and a
further question along the same lines:

Theorem: fix a number field F imbedded in C, a compact subset K of C,
and an analytic function f from a neighborhood of K to C.  Assume that
f is not algebraic.  Then for each epsilon>0 there exists a constant A
such that: for each H, the number of z in K such that both z and f(z)
are elements of F of (exponential) height at most H is less than
A H^epsilon.

Again, my question was: is this already known, and if so where and how?

Further question: I have not been even able to find F,K,f for which I
can show the existence of infinitely many z such that z,f(z) are both
elements of F -- of whatever height.  Can such a function exist?  Is it
perhaps already Someone's Conjecture (or even Someone's Theorem) that
there are at most finitely many such z for any choice of F,K,f?

Thanks,
--Noam D. Elkies