Linking points on algebraic curves with class numbers

Dave Rusin · science forum Guru Joined: 25 Mar 2005 Posts: 487

A colleague asked me a question (about extensions of algebraic number fields)
which led us to do some experiments involving a parameter a . Only for
a few values of a does the experiment "succeed". Here is the list so far:
a = 1, 19, 37/17, 53/17, 89/19, 199/109, 323/197, 433/323, 629/251,
683/397, 703/487, 757/127, 901/719, 971/701, 1007/127, ...
I haven't looked as far for negative a but up to height 200 the
"successes" occur only for a = -1, -73/17, and -181/71.
My colleague is really only interested in integer solutions, I think;
I looked for them but found no more in a search up to a = 300 000.

Now the game is to decide where these numbers "come from".

I could not fail to notice that almost all these numerators and
denominators were prime.

Less obvious (but checked because of the appearance of "163") is the
fact that for every prime in this list, the class number of the
field Q(sqrt(+p)) is 1. It's also true for all the prime divisors
of the composites on this list. That's an awful lot of agreement.
That is the fact to be explained. (Note: for each composite
numerator or denominator, h(Q(sqrt(+n))) is greater than 1.)

Here is the "experiment": we have a polynomial

P := 1 + (3+9*a^3)*X^3 + (27*a^6+6+18*a^3)*X^6 + (7-27*a^6)*X^9
+ (27*a^6+6+18*a^3)*X^12 + (3+9*a^3)*X^15 + X^18;

which is irreducible in Q(a)[X]. (It's the miniminal polynomial of
some unit in some number field.) This P remains irreducible when the
parameter a is specialized to a rational value, except for some
exceptional values of a . Those are the "successes" of the experiment.

Well, the origins of the polynomial P happen to force any factorization
to be as a product of three sextics f_i, so it occurs to me that we could
take the coefficients of the sextics as 18 variables; then comparing
the coefficients of X in the equation P = f1 f2 f3 gives 18
equations in these variables together with a . So the factorizations
correspond to points on an algebraic variety. Since P is
generically irreducible, I guess this variety is a curve. The
experiments I ran amount to asking whether there is a rational point
in Q^19 lying over each tested value of a, that is, I am just
computing the projection of this curve to one of the coordinate axes.

These experiments suggest to me that the curve has an infinite number
of rational points. By Faltings' theorem, the curve must have genus
0 or 1. I guess since the values of a are not growing very rapidly
in height, I'm inclined to think the curve is of genus 0, i.e. it's
a rational curve. In particular it would be parameterizable, i.e. we
should be able to get these special values of a as values of some
rational function(s) a = a(t). (I'm just guessing that the curve is
irreducible; if not, my remarks would have to apply separately to
each component.)

OK, now compare the two ways of looking at the results of the experiment.
On the one hand, we think we're looking at the values of some rational
function, (or perhaps at the coordinates of rational points on an
elliptic curve). On the other hand, we have data that suggest all the
prime divisors of the numerators and denominators will correspond to
fields with class number 1. I can't recall such a theorem linking these
two themes. In fact, I don't even recall hearing that there are known
to be infinitely many primes for which h(sqrt(+p)) = 1 , which I
suspect could be a consequence of this observation.

So that's my question: is there any theorem known which connects
coordinates of points on an algebraic curve
with
class numbers of number fields
in this kind of fashion?

dave

PS -- it would be nice to have a curve in some low-dimensional space
which is birationally equivalent to the curve I am describing. One can
easily eliminate 7 or 8 of the variables but that gives a very messy
set of equations describing a variety which still involves nearly
a dozen variables. It's not very likely that I could compute the
genus that way, let alone a parameterization. I'm open to suggestion.

OwlHoot · science forum beginner Joined: 25 Jun 2006 Posts: 6

Dave Rusin wrote:

Gene Ward Smith · science forum Guru Joined: 08 Jul 2005 Posts: 409

OwlHoot wrote:

Gene Ward Smith · science forum Guru Joined: 08 Jul 2005 Posts: 409

Gene Ward Smith wrote:

Dave Rusin · science forum Guru Joined: 25 Mar 2005 Posts: 487

In article <e7uou2$rqd$1@news.math.niu.edu>, I wrote:

David Bernier · science forum Guru Wannabe Joined: 01 May 2005 Posts: 101

Dave Rusin wrote:

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