From: cohen@ceremab.u-bordeaux.fr (Henri Cohen) Newsgroups: sci.math.research Subject: Re: Families of elliptic curves of positive rank Date: 13 Dec 1993 13:06:45 GMT In article <2eahva$n6i@wuecl.wustl.edu>, jbuddenh@papaya.wustl.edu (Jim Buddenhagen) writes: |> Many families of ellipic curves of positive rank over the rationals |> are known. Usually these are parameterized by one, two or sometimes |> three rational parameters. Sometimes certain relations between the |> parameters must be avoided to insure positive rank. |> |> My question: Are similar families known for rank > 1? For rank > 2? |> Examples and/or references would be appreciated. |> |> --Jim Buddenhagen (jb1556@daditz.sbc.com) |> |> |> The best result is due to Mestre: J.-F. Mestre, Courbes elliptiques de rang >=12 sur Q(t), C.R. Acad. Sci. Paris (1991), pp 171-174. He builds a 4 parameter family of elliptic curves over Q(t) (which makes 5 parameters in all if you count t) all of rank >=11 if certain conditions are satisfied. By specializing, he obtains families of rank >=12 over Q(t) (this is the current record), but by doing a computer search in his initial family, several people (Tunnell, Fermigier, Nagao) have found individual curves over Q of much higher rank, the present record is of rank 20, held by Nagao. The idea of the construction is very elementary and in fact dates back to Neron, although Neron does not make it explicit (and Neron claimed only rank 9 I think, not 12). You can read Mestre's paper of course, but the basic idea is this (in TeX notation) Choose $r_1$, $r_2$, $r_3$ and $r_4$ four rational numbers, let $t$ be a parameter, and consider the polynomial $$P(X)=\prod_{1\le i,j\le 4, i\neq j} (X-(r_i+tr_j)$$ which is of degree 12. By considering the Laurent series expansion of $P^(1/3)$, you can find a monic polynomial $g$ of degree 4 such that $\deg(P(X)-g^3(X))\le 7$. Now the crucial point is that with our special choice of $P$, we have in fact $\deg(P(X)-g^3(X))\le 6$. This can be proved very elegantly by an elementary argument involving the geometry of a certain variety, but can of course be checked brutally by hand or (better) with your favorite computer algebra system. Knowing this, write $P(X)=g^3(X)+q(X)g(X)+r(X)$ where $q$ and $r$ are quotient and remainder of the Euclidean division of $P-g^3$ by $g$. Then $\deg(q)\le 2$ and $\deg(r)\le 3$. It follows that the equation $$Y^3+q(X)Y+r(X)$$ defines a CUBIC curve, on which 12 rational points are known, i.e. the points $(r_i+tr_j,g(r_i+tr_j))$ for $1\le i,j\le 4$. One of these points can then be sent to infinity, and then one must show that generically the 11 other points are independent over Q(t), using formulas of Shioda for elliptic surfaces. Henri Cohen