From: alf@MACADAM.MPCE.MQ.EDU.AU (Alf van der Poorten) Newsgroups: sci.math.numberthy Subject: Re: elliptic curves/Q of high rank Date: 6 Feb 95 23:06:48 GMT Some weeks ago I asked about elliptic curves/Q of high rank. I was quickly alerted to the appropriate names (Mestre/Nagao) and used the MR CD-ROM to get details. Since I had more responses asking {\em me\/} for answers than giving me information it seems reasonable to report on what I garnered. The story seems to be the following. No doubt this is well known to those who know it well but not so well known to those who do not know it. %AMS-TeX \def\Q{{\Bbb Q}} \def\Z{{\Bbb Z}} Mestre [\,J.-F. Mestre, {\it C. R. Acad\. Sci\. Paris\/} {\bf 313}, ser 1 (1991), 139--142; {\it ibid.\/} {\it 313\/} ser 1 (1991), 171--174; {\it ibid.\/} {\bf 314} ser 1 (1992), 453--455\,] constructed elliptic curves over $\Q[T]$ with $\Q[T]$-rank at least $11$ (later $12$). The idea is to pick a six-tuple $a=(\alpha_1,\ldots,\alpha_6)$ of integers and to set $h(X)=\prod_i(X-\alpha_i)$ and then $g(X)=h(X+T)h(X-T)\in\Q[T](X)$. The point is that there then plainly are polynomials $q(X)$, $r(X)$ in $\Q[T](X)$ with $\deg g=6$ and $\deg r\le 5$ so that $g=q^2-r$. It is now manifest that the curve $Y^2=r(X)$ contains the twelve $Q[T]$-rational points $(\pm T+\alpha_i, g(\pm T+\alpha_i))$. If moreover the six-tuple $A$ is so chosen that the leading coefficient $c_5$, say, of the polynomial $r(X)$ vanishes --- that is, if $r$ is of degree at most $4$ --- then $Y^2=r(X)$ is a model for an elliptic curve over $\Q[T]$. For appropriate choices $A\in\Z^6$ [\,for example, $(-17,-16,10,11,14,17)$ is such a choice\,] one now takes one of the cited points as origin $O$ for the group, and it is then not too painful to show that the remaining eleven points are independent $\Q[T]$-rational points. Presumably at this stage one also selects $A$ so as to facilitate actually finding more than eleven $\Q$-independent points as alluded to below. Finally one specialises $T\to t$, $t=t_1/t_2\in\Q$, so obtaining curves $E_{A,t}$ which are good candidates for having high $\Q$-rank. Loosely speaking, choosing the pairs $(t_1,t_2)$ is a matter of testing for lots of rational points $\mod p$ on $E_{A,t}$ with the $p$ running over a suitable quantity of good primes, and hoping for the best when it comes to actually finding $\Q$-independent points on $E_{A,t}$. In a series of papers, Koh-ichi Nagao, {\it Proc Japan Acad\. Ser\. A\/}, {\it 68\/} (1992), 287--289; {\it ibid.\/} {\bf 69} (1993), 291--293; and (with T. Kouya), {\it ibid.\/} {\bf 70} (1994), 104--105, gives elliptic curves over $\Q$ of rank respectively at least $17$, then $20$ and finally $21$. That these ranks are attained is shown by explicitly displaying the relevant numbers of $\Q$-independent points. Some ingenious computational tricks are necessary to make feasible the search for suitable $t_1/t_2$. One charm of the subject of elliptic curves is large numbers with some inner meaning. However, it's hard to feel much the wiser on being told that the choices $A=(95,71, 66, 58, 13, 0)$ and $t=619/195$ yield a curve $E_{A,t}$ which is $\Q$-isomorphic to the curve with minimal Weierstrass model $$\multline y^2+xy=x^3-431\;09298\;07663\;33677\;95836\;20958\;91166x \\ +5156\;28355\;53666\;43659\;03565\;27998\;71176\;90939\;15330\;88196 \,,\endmultline$$ on which one may display twenty $\Q$-independent points. Nagao does not explain how he actually finds the twenty independent points; Mestre's construction, as described, of course gives only eleven of the points. I am grateful to Nigel Smart for many helpful remarks. He is not to be blamed [other than by me] for anything foolish I have said above. Alf van der Poorten alf@mpce.mq.edu.au http://www.mpce.mq.edu.au/~alf/ fax: +61 2 850 9502 home fax: +61 2 415 6282 [will not work if I'm on line]