From: victor@idaccr.org (Victor S. Miller)
Subject: Re: Points of Order 13 on Elliptic Curves
Date: 20 Mar 99 13:08:44 GMT
Newsgroups: sci.math.numberthy

Thanks for all the replies to my query.  Before I sent in my query I
was almost sure that the author was Josef Blass, at some university in
the midwest.  However, when I looked in Math. Reviews I couldn't find
anything by him that resembled what I was looking for, so I decided
that I was wrong.  However, Ken Ribet and Armand Brumer also
remembered Blass as the author -- he's now at Bowling Green State
University, so I guess that that's the answer.  Most likely the
manuscript that I saw was never published.

The problem has an interesting history.  The paper by Billing and
Mahler "On Exceptional Points on Cubic Curves" J. London
Math. Soc. 15, (1940). 32--43.  In it, they show how to find the
singular planar equations for X_1(N) -- the curve parametrizing pairs
(elliptic curves, points of order N), and give the explicit equations
for N up to 15.  They also then do a 2-descent on the elliptic curve
X_1(11) to find that its Mordell-Weil group has order 5, thus showing
that there is no rational elliptic curve with a rational point of
order 11.  Ogg later treated the case of N=17 ("Rational Points of
Finite Order on Elliptic Curves" (Invent.  Math., 12. 105-111 (1971)))
and noticed that all of the N that were known to have points had
X_1(N) genus 0.  Of course later on this conjecture (which may not
have been made in print) was proven by Mazur in his Eisenstein ideal
paper (with the hard case of N=13 being handled in the Mazur-Tate
paper cited before).

Richard Pinch did point out the paper by Connell ("Points of Order 11
on Elliptic Curve" Nieuw Archief foor Wiskunde, 13(1995), pp 257-288),
in which Connell also gives an elementary treatment of the X_1(11)
case, but treats the problem of which quadratic fields Q(sqrt(m)) have
an elliptic curve with a rational point of order 11.

Finally, Bjorn Poonen mentions:

"Back in 1995, I did a 2-descent on the Jacobian of the genus 2 curve
classifying quadratic polynomials x^2+c together with points of
period 2 and 3, and later I discovered that the curve had the same
defining equation as X_1(13)!"

Unfortunately, I still can't get in touch with Prof Blass as Bowling
Green State's mail system is loused up.  Some of you asked for copies
of the paper if I find it.  If I do, I'll send them to you.

                Victor Miller