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Author: William A. Stein
Summary of Results

# 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:" Rank012345 Delta > 031748518712470652673770 Delta < 061589913213681165944275 Totals9333714319261517118618045 Percents30.0446.0819.803.820.26 "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.

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