Summary of Results

Quotes from the paper

See "The Behavior of the Mordell-Weil Group of Elliptic Curves" for the context of these quotes. Please excuse the "TeX"isms. Reference numbers are those used in the paper.

From section 1: "The important question then is to understand the behavior of the rank as $E$ varies over elliptic curves. It is still unknown whether $r$ is unbounded or not. In fact, the opinion had been expressed that, in general, an elliptic curve might tend to have the smallest possible rank, namely $0$ or $1$, compatible with the rank parity predictions of Birch and Swinnerton-Dyer. We present evidence that this may not be the case."

"Mestre and Oesterl\'{e} found the 436 modular elliptic curves of prime conductor up to 13100, using~\cite{11}. There were~80 rank~2 curves among the 233~curves of even rank. This proportion of rank~2 curves seemed too large to conform to the conventional wisdom stated above (see also~\cite{18}, pages~254--255). We decided to investigate the ranks of elliptic curves in a systematic way, over a significantly larger range. Curves of {\it prime\/} conductor only were considered for practical and theoretical reasons. This collection of curves appears to be a typical sample of the set of {\it all} curves (see section~5 for some evidence)."

"We have studied 310716 elliptic curves of prime conductor less than $10^8$. There were 155658 curves with odd rank, and 155058 curves with even rank. We found 20.06$ of {\it all\/} our curves have even rank at least~2, or about $40\%$ of all the even rank curves. Even more striking is the behavior of the average rank, as discussed in section~3. An incidental aspect of our computations is a massive corroboration of the standard conjectures on elliptic curves, recalled in section~2."

Contents of section 3: "Elliptic curves of prime conductor $N$ were conjectured to have prime discriminant, except for the Setzer-Neumann curves and for 5~other small conductor curves, see \cite{2, appendix}. This is now known for {\it modular} curves by Theorem~2 of \cite{12}. We therefore searched for curves of prime discriminant, by looking for integral solutions to the equation

c_4^3-c_6^2 = 1728\Delta, (3) where $c_4$ and $c_6$ are the usual invariants attached to equation~(1), or more precisely, by fixing $a_4$, and searching for $a_6$ for which~(3) has a solution with $\Delta$ prime and less than $10^8$. This produced $\totalfound$ curves including the $869$ expected curves with non-trivial torsion and rank~0. The set of 310716 curves that we studied is most simply described by $$ \{E :\left|\Delta\right| \le 10^8, |a_6|\le 2^{31}-1\}, $$ with $|\Delta|$ {\it prime\/}."

"We will not describe here all the details of the several thousand hours of computations, but just say that, imitating Mazur's description \cite{17} of ``infinite descent'' we searched for points by night, and calculated $L$-series derivatives or regulators by day. We may use the infinite series formulas of~\cite{3} for the derivatives of the $L$-series at $s=1$, since $E$ is assumed modular. Then an upper bound for the analytic rank is found by estimating the order of vanishing of $\Ls$ at $s=1$. Using 2000 or 4000 terms of these series provided sufficient accuracy for our purposes, since the values are either~0 to several places or else are far from~0 in most cases. The period $\Omega$ is easily calculated using the Arithmetic-Geometric mean algorithm of Gauss \cite{7}, and the height regulator $R=\det\left(\heightpair{P_i}{P_j}\right)$ is computed by using the method of Tate, as modified by Silverman~\cite{16}, once points have been found by a search."

"The rank predictions are based on a combination of three calculations: the rank parity, the analytic rank or order of vanishing of the $L$-series, and the number of independent points found which is a lower bound for the algebraic rank~$r$. When the ranks coincide, as they should, we get a prediction from~(2) of $|\Sha|$, which should be an integer square. The largest $|\Sha|$ we found was~$289$, for a curve of rank~0."

"For each curve, we keep its discriminant, parity, period, rank, the appropriate $L$-derivative value, a list of $x$-coordinates of the independent points found, and the regulator of these points."

"Of the curves analysed, 113969 had positive discriminant, and 196747 had negative discriminant.\footnote{The quotient is about 1.726, near $\sqrt{3}$. See section~5 for an explanation.} An interesting phenomenon was the systematic influence of the discriminant sign on all aspects of the arithmetic of the curve. The rank distribution is given in the following table:"

Rank	0	1	2	3	4	5
Delta > 0	31748	51871	24706	5267	377	0
Delta < 0	61589	91321	36811	6594	427	5
Totals	93337	143192	61517	11861	804	5
Percents	30.04	46.08	19.80	3.82	0.26

"Thus 20.06%$ of the curves have even rank at least~2. Note that the {\it positive\/} discriminant curves give an even higher percentage!"

"Define $N(r, X)$ to be the proportion of curves with conductor at most $X$, and with rank at least $r$. Our data shows that these functions are {\it increasing\/} functions of $X$ for $r\geq 3$ and $X\le 10^8$. In contrast \cite{6}, dealing with quadratic twists of elliptic curves, suggests a decrease for the analogous functions."

"Denote the {\it average rank\/} among the curves with discriminant sign $\epsilon$ and conductor at most $X$ by $r_\epsilon(X)$. In our data, the functions $r_\epsilon(X)$ for $X<10^8$ are quite steadily climbing to the numbers $1.04$ for $\Delta >0$, and to $0.94$ for $\Delta <0$. In particular, the average rank of our curves is {\it not\/} $0.5$, as is expected to be the case for twists~\cite{5},\cite{8}."

"In most cases, the predicted analytic rank matches with the rank of points found, and the predicted $\Sha$ is close to a non-zero integral square. More precisely, this is so for all the curves of rank at least~3, and for 95\% of the rank~2 curves. There is a very small number of rank~2 curves for which the prediction of rank~2 is based solely on the vanishing of $L$-series. Note that in \cite{8}, a numerical study of one family of cubic twists, only the analytic rank is estimated. For many rank~1 curves (i.e., curves whose rank is predicted to be odd for which $L'(E,1)$ does not vanish), we have found no points by point searches over moderate ranges, and do not expect to find any small points."

Other computations and speculations

Here we should give some information about other computations from the project, e.g., period averages versus integral points, counts of curves, shapes of lattices etc.

Recent developments

Here we should say something about the bound of Brumer for the average rank, the results and speculations of Katz-Sarnak, and Sarnak, and whatever else is appropriate.