THE BEHAVIOR OF THE MORDELL-WEIL

GROUP OF ELLIPTIC CURVES

Armand Brumer and Oisín McGuinness

Fordham University

October 30, 1989, revised March 1, 1990

§1 Introduction

Suppose that E is an elliptic curve defined over Q given by the equation

(1)  y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6},

where we assume that a_{i} ∈ Z. The set E(Q) of solutions (x, y) with x, y ∈ Q, together with the point at infinity, forms a finitely-generated abelian group, the Mordell-Weil group of E. It is isomorphic to Z^{r} ⊕ F where F is finite and where r is the rank of E. The possibilities for the finite group F are completely known [9]. 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 [8]. We present evidence that this may not be the case.

Published examples [2], [10] of curves of rank ≥ 2 might suggest that such curves are sparsely distributed.^{1} Mestre and Oesterlé found the 436 modular elliptic curves of prime conductor up to 13100, using [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 [18], pages 254–255). We decided to investigate the ranks of elliptic curves in a systematic way, over a significantly larger range.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 11G40, Secondary 11D25, 11G05, 11-04, 14K15.

Key words and phrases. Elliptic curve, Mordell-Weil group, rank, Birch-Swinnerton-Dyer conjecture, Hasse-Weil L-series, discriminant, period.

1This situation is not unrelated to the ranks of ideal class groups of quadratic fields, where similar phenomena occur [13].

Curves of prime conductor only were considered for practical and theoretical reasons. This collection of curves appears to be a typical sample of the set of 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 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.

Recent related work is described in [6] and [8]. Contrasts with our results are given in section 3.

We expect to publish a fuller account, including the behavior of other invariants of interest. This announcement reports mainly on ranks. The computations have been carried out on Macintosh II computers at Fordham University, with the partial support of an NSF SCREMS grant. We would like to thank our colleagues R. Lewis, I. Morrison, and W. Singer for the use of their machines.

§2 Definitions

We recall standard notations and definitions [15]. Associated with equation (1) is the discriminant Δ, which we will assume to be minimal among all models (1) of E. The fundamental property of the discriminant is that p | Δ if and only if the equation (1) is singular modulo p, the conductor N of E is a subtler invariant that has the same property.

The Hasse-Weil L-series of E is defined for ℜ(s) > 3/2 by

L(E, s) = ∏_{p | N} (1 − a_{p}p^{−s})^{−1} ∏_{p |̸ N} (1 − a_{p}p^{−s} + p^{1−2s})^{−1}

where for p | N, a_{p} ∈ {−1, 0, 1}, and for p |̸ N, a_{p} = p + 1 − |E(F_{p})|. We shall assume that E is a modular curve, so E is a factor^{2} of the Jacobian J_{0}(N) of the modular curve X_{0}(N) of level N. (That is, the Taniyama-Weil conjecture for E is true.) Hence L(E, s) can be continued to an entire function on C, satisfying a functional equation when s ↦ 2 − s, with a zero at s = 1 of order ρ, the analytic rank of E. For square-free conductor, the sign in the functional equation may be easily calculated from an equation of the curve, allowing a conjectural prediction of the parity of the analytic rank [1]. We also assume that the conjecture of Birch and Swinnerton-Dyer holds, so that the analytic rank equals the rank, ρ = r, and the leading term of L(E, s) at s = 1 is given by:

2Note that Mestre-Oesterlé found their curves by determining the 1-dimensional factors of J_{0}(N). Needless to say, we have the same curves in their range.

(2)  lim_{s→1} \frac{L(E, s)}{(s − 1)^{r}} = Ω\frac{|⨿⨿| det (⟨P_{i}, P_{j}⟩)}{[E(Q) : E′]^{2}} ∏_{p | Δ} c_{p}.

Here Ω is the period ∫_{E(R)} |ω|, for ω a Néron differential on E, ⨿⨿ denotes the conjecturally finite Tate-Shafarevich group of E, the P_{i} for 1 ≤ i ≤ r are an independent set of points in E(Q) generating the subgroup E′, and ⟨P_{i}, P_{j}⟩ denotes the height pairing. The fudge factors c_{p} are all 1 for the curves we consider. Recent work of Rubin and Kolyvagin [14] confirms the conjecture of Birch and Swinnerton-Dyer in many cases of rank r ≤ 1.

Examples illustrating (2) for the curves of ranks 4 and 5 of least known conductor are given in section 4.

§3 Rank Results

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 [2, appendix]. This is now known for modular curves by Theorem 2 of [12]. We therefore searched for curves of prime discriminant, by looking for integral solutions to the equation

(3)  c_{4}^{3} − c_{6}^{2} = 1728Δ,

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 Δ prime and less than 10^{8}. This produced 311243 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 : |Δ| ≤ 10^{8}, |a_{6}| ≤ 2^{31} − 1},

with |Δ| prime.

We will not describe here all the details of the several thousand hours of computations, but just say that, imitating Mazur’s description [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 [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 L(E, s) 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 Ω is easily calculated using the Arithmetic-Geometric mean algorithm of Gauss [7], and the height regulator R = det (⟨P_{i}, P_{j}⟩) is computed by using the method of Tate, as modified by Silverman [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 |⨿⨿|, which should be an integer square. The largest |⨿⨿| 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.^{3} 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
Δ > 0  31748  51871  24706  5267  377  0
Δ < 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 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 increasing functions of X for r ≥ 3 and X ≤ 10^{8}. In contrast [6], dealing with quadratic twists of elliptic curves, suggests a decrease for the analogous functions.

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

In most cases, the predicted analytic rank matches with the rank of points found, and the predicted ⨿⨿ 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 [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.

3The quotient is about 1.726, near \sqrt{3}. See section 5 for an explanation.

§4 Examples

In the literature, one finds very few verifications of the conjecture of Birch and Swinnerton-Dyer for ranks ≥ 2. The paper [3] on the curve of conductor 5077, found in [2], works out the only rank 3 case that we know of. It is perhaps not without interest to report the details for the curves^{4} of smallest known conductor of rank 4 and of rank 5.

The first rank 4 curve is:

y^{2} + y = x^{3} + x^{2} − 72x + 210, Δ = 501029.

For this curve, the predicted parity is even, the period Ω = 2.952580, the value and 2nd derivative of the L-series vanish to several places, and L^{(4)}(E, 1)/4! = 9.357978. The points with x-coordinates 5, 4, 3, 6 in order of increasing height, form a basis. The regulator is R = 3.169424, which matches the quotient of L^{(4)}(E, 1)/4! and Ω, so |⨿⨿| appears to be 1. There are 21 × 2 integral points with |x| < 10^{6}, while the second curve of rank 4 has 28 × 2.

The first rank 5 curve is:

y^{2} + y = x^{3} − 79x + 342, Δ = −19047851.

The previously found curves of rank 5 have conductors about ten times larger [2], [10].

The predicted parity is odd, the period Ω = 2.047641, and then the 1st and 3rd L-series derivatives are 0 to several places, and L^{(5)}(E, 1)/5! = 30.285711. Dividing by Ω gives 14.790539. There are 38×2 integral points with |x| < 10^{6}. Those with x = 5, 4, 3, 7, 0 form a basis for E(Q) with height regulator R = 14.790528. So ⨿⨿ is predicted to be trivial. There are four other curves of rank 5 with conductor less than 100 million. The next one has discriminant −64921931. All turn out to have trivial ⨿⨿, if we believe the conjectures.

Some of the curves found by the search give rise to rather spectacular cancellations. A particular example is the curve of rank 0

y^{2} + xy + y = x^{3} + x^{2} − 12632622x − 17287039382, Δ = 38593.

Here c_{4} = 606365857, and c_{6} = 14931454281967, and the equation 1728Δ = c_{4}^{3} − c_{6}^{2} involves the difference of two 27 digit numbers!

4These curves are new. The curves of rank 4 reported in [2] and [10] are respectively the 5th, 7th and 2nd curves of rank 4 in our list.

§5 Heuristics

While its connection with the Taniyama-Weil conjecture makes the conductor a natural invariant, the discriminant has turned out to be more useful in our heuristics. The idea is to arrange the curves in order by discriminant size, and then replace lattice-point counting problems by area computations. This works well, for instance, to count curves.

Heuristic Estimate. We have the following estimates for the number of positive and negative discriminants of absolute value at most N,

A_{+}(N) ∼ \frac{α_{+}}{ζ(10)}N^{5/6},
A_{−}(N) ∼ \frac{α_{−}}{ζ(10)}N^{5/6}.

The ζ(10) factor arises from taking into account the non-minimal discriminants. Here α_{+} = 0.4206 and α_{−} = 0.7285 are given by the elliptic integrals

α_{±} = \frac{\sqrt{3}}{10} ∫_{±1}^{∞} \frac{du}{\sqrt{u^{3} ∓ 1}},

which arise from parametrising suitably the integrals that replace the latticepoint counts. A well known identity of Legendre, related to complex multiplication, is α_{−} = \sqrt{3}α_{+}. Remark. One should compare this estimate with the results of [4]. While the number of elliptic curves grows like N^{5/6}, the number of cubic fields grows like N.

Assuming the distribution of prime discriminants among discriminants is that of prime numbers among all integers, the number of prime discriminants of sign ϵ and of size less than N is then α_{ϵ} Li(N^{5/6}), where Li(x) is the logarithmic integral. The expected number of curves with prime |Δ| < 10^{8} is then 311586, comparing rather well with the number 311243 found.

Similar heuristic arguments have been applied to other invariants. For instance, the average period of a curve with positive discriminant is \sqrt{3/2} times the average period of a curve with negative discriminant. This is also confirmed by the data. The fits with the experimental data are so good that one could hope for proofs in the near future.

We have not as yet been able to provide heuristics for the growth of the functions N(r, X). While our data may seem massive, N = 10^{8} is not sufficient to distinguish growth laws of log log N, N^{1/12} or N^{1/24} from constants. So we have to be cautious in formulating conjectures based on the numerical evidence.

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Mathematics Department, Fordham University, Bronx NY 10458, USA E-mail address: BITNET%”brumer@fordmurh”, BITNET%”mcguiness@fordmurh”