Positive Rank Examples of the Conjecture of Birch and Swinnerton-Dyer

Francis Oisín McGuinness Department of Mathematics Fordham University Bronx NY 10458

January 23 1989

Abstract

We describe some computations related to some rank 1 and rank 2 cases of the conjecture of Birch and Swinnerton-Dyer. Appendices describe height and period computations on elliptic curves.

Introduction

Suppose that E is an elliptic curve defined over Q. Then the conjecture of Birch and Swinnerton-Dyer states that the Hasse-Weil L-series L(E, s) has an analytic continuation to an entire function on C, with a zero at s = 1 of order r, the Mordell-Weil rank of E over Q (=free Z-rank of the finitely generated abelian group E(Q).). The conjecture also predicts the leading term of the Taylor series of L(E, s) at s = 1. Namely,

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}.

We recall briefly (see [18] or [15] for more details) that Ω denotes ∫_{E(R)} |ω| where ω is a Néron differential on E, so that Ω is [E(R) : E_{0}(R)] times a fundamental real period of 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. Finally Δ is the minimal discriminant of E, and the fudge factors c_{p} = [E(Q_{p}) : E_{0}(Q_{p})], where E_{0}(Q_{p}) denotes the connected component of the group of p-adic points on E.

In this paper we discuss the numerical verification of some rank 1 and rank 2 examples of the conjecture. Ligozat (see [6]) has verified many rank 0 examples when E is an elliptic modular curve, up to an assumption about the Tate-Shafarevich group. Other rank 0 examples for curves with complex multiplication have been explored by Birch and Swinnerton-Dyer[2] and by Razar[11, 12], while Stephens [17]has given some complex multiplication examples with rank 1. A rank 3 example of particular interest is described in [4]. Our main examples will be quotients of modular curves. Since this work was originally done, the work of Gross and Zagier ([5]) has shown that, in our examples, the numerical approximate equality we describe is an actual equality, up to an assumption about the Tate-Shafarevich group. Furthermore, Rubin ([13]) has proved the finiteness of the Tate-Shafarevich group for curves with complex multiplication.

The first section discusses the rank 1 examples, section 2 discusses a rank 2 example, while the remaining sections (which are in the nature of appendices) describe how the computations of the period, heights and Lseries derivative values were performed.

The work described in this paper was carried out at the University of Virginia, and made use of its computational facilities.

Some frequent notation: |S| will denote the size of a finite set S. Most of the notation is quite standard, and can be found in [15], for example.

1 Some Rank 1 Curves with Prime Conductor

The first curve (i.e., the elliptic curve over Q of least conductor) with positive rank is the curve

37A  y^{2} + y = x^{3} − x  Δ = N = 37,

for which P = (0, 0) generates E(Q) ≅ Z. In fact, this is the first of the nine elliptic curves of the form X_{0}(p)/w, corresponding to p = 37, 43, 53, 61, 79, 83, 89, 101 and 131. Recall that X_{0}(p) is the modular curve of level p that classifies pairs (E, C) where E is an elliptic curve and C is a cyclic subgroup of order p. The complex points of X_{0}(p) correspond to H^{∗}\Γ_{0}(p) where H^{∗} denotes the upper half plane together with the cusps, and

Γ_{0}(p) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) : ad − bc = 1, a, b, c, d ∈ Z, c ≡ 0 mod p\} .

The curve X_{0}(p) is defined over Q, and the canonical involution w = w_{p} (induced from the map on H given by z ↦ −1/pz or by the map on moduli (E, C) ↦ (E/C, E[p]/C)), acts Q-rationally so that the quotient X_{0}(p)/w is also a curve defined over Q. There are well-known formulas for the genus of both X_{0}(p) and its quotient X_{0}(p)/w. We first show that the quotient curve is of genus 1 for exactly the values of p given above.

Lemma 1.1 X_{0}(p)/w has genus 1 if and only if p = 37, 43, 53, 61, 79, 83, 89, 101 or 131.

To prove this we apply the technique described in section 5 of [10]. (A more abstract formulation can be found in [7], page 146, Lemma 2.6.) Denote X_{0}(p) by X and X_{0}(p)/w by X̄, and consider the double cover map

X → X̄.

For a prime l ≠ p, we can view this map over F_{l}, and estimate the number of F_{l^{2}} points on X. On the one hand, the supersingular points of X (those points classifying supersingular elliptic curves) are defined over F_{l^{2}}, and we get

|X(F_{l^{2}})| ≥ 2 + \frac{(l − 1)(p + 1)}{12},

where 2 is the number of cusps of X. On the other hand, we have

|X(F_{l^{2}})| ≤ 2 |X̄(F_{l^{2}})|
≤ 2(1 + 2ḡl + l^{2}),

using the Weil bound for the number of points on X̄ defined over F_{l^{2}}, where ḡ is the genus of X̄. Taking l = 2 and combining these two inequalities yields the bound on ḡ:

ḡ + 1 ≥ \frac{p + 1}{96},

which for ḡ = 1 yields p ≤ 191. A direct computation using the genus formula for ḡ shows that the only primes less than 191 for which the genus is 1 are those listed.

For completeness, we state the genus formula used in the computation just mentioned. If g denotes the genus of X, then we have

g = \frac{1}{12}(p − ν),

where p ≡ ν (mod 12), ν ∈ {13, 5, 7, −1}, and

ḡ = \frac{1}{4}(2g + 2 − f),

from the Riemann-Hurwitz formula, where f is the number of fixed points of w acting on X. Since w interchanges the two cusps 0 and ∞ of X = X_{0}(p), f is the number of fixed points of w on the open modular curve Y_{0}(p), and can be computed using the modular interpretation of Y_{0}(p). The result is (see [9] for the details) that f = h, 2h or 4h depending on whether p ≡ 1 (mod 4), p ≡ 7 (mod 8) or p ≡ 3 (mod 8) respectively, and where h = h(−p) is the class number of the imaginary quadratic field Q(\sqrt{−p}). (We could use the standard upper bounds on h to bound ḡ from below, but the bound on p thus obtained when ḡ = 1 is much larger than that obtained from the argument already given.)

For the rest of this section, we let E denote X_{0}(p)/w for one of the nine primes listed in Lemma (1.1) above. The two cusps of X_{0}(p) map to a rational point on E which is therefore an elliptic curve defined over Q. Our immediate task is to find the discriminant and conductor of E. Firstly, by the results of Igusa-Deligne-Rapaport on the reduction properties of X_{0}(p), E has good reduction at all primes l ≠ p. Then by the determination of the fibre J_{p} at p of the Néron model of the Jacobian of X_{0}(p) (as given in [7], appendix), it follows that E has multiplicative reduction at p. (The connected component J_{p}^{0} is a group of multiplicative type of which the fibre at p of the Néron model of E is a quotient.) Furthermore, the fibre at p of E has one irreducible component, since the quotient J_{p}/J_{p}^{0} is cylic generated by the class of the divisor (0) − (∞), and this divisor is killed by the action of w. We conclude that E has conductor p, and discriminant Δ = ±p. (See the table on page 46 of [1] for the last conclusion.)

Next we find a model for E by examining the table of curves of conductor up to 200 given in [1]. For the primes p = 43, 53, 61, 79, 83, 101 and 131, there is only one curve of conductor p in the table, and this curve must be E. For p = 37 there are three curves, and for p = 89 there are two curves to be considered. We can identify E by noting that for a Weil curve of conductor p, the eigenvalue of the action of w on the corresponding newform is −a_{p} in the usual notation. But for E, this eigenvalue must be 1, and so we have a_{p} = −1. This shows that when p = 37, we have E = 37A, and when p = 89, we have E = 89C, since the other curves have a_{p} = 1.^{1} Recall that the geometric significance of a_{p} = −1 is that the tangents to the node of Ẽ (E reduced modulo p) are not F_{p} rational.

1The table header to table 3 in [1] is misleading here.

It follows from the table that the curves E that we are considering are unique in their isogeny class. This is also a consequence of some results of Serre, see pages 306–308 and (5.5.6)–(5.5.8) of [14]. If ϕ_{l} : Gal (Q̄/Q) → Aut E[l] ≅ GL_{2}(F_{l}) denotes the Galois representation on the l-torsion points of E, then the corollary to Proposition 21 in [14] (take p = 2) shows that ϕ_{l} is surjective for all l > 5, and then computing |E(F_{l})| for l = 2, 3 (for p = 61 and 101 it is necessary to look at l = 5 also), shows that ϕ_{l} is surjective for all primes l. This also implies that E(Q) has no torsion. Since a non-trivial rational point is listed in the tables for each of our curves E, we must have that rank E(Q) ≥ 1. The 2-descent bounds in [3] then show that E has rank 1 (and that the 2-torsion in the Tate-Shafarevich group of E is zero). In the table below I give for reference the coefficients of a defining equation for E, the sign of Δ, and the coordinates of a point P of infinite order.

Curve a_{1} a_{2} a_{3} a_{4} a_{6} sign Δ  P
37A 0 0 1 −1 0  +  (0,0)
43A 0 1 1 0 0  −  (0,0)
53A 1 −1 1 0 0  −  (0,0)
61A 1 0 0 −2 1  −  (1,0)
79A 1 1 1 −2 0  +  (0,0)
83A 1 1 1 1 0  −  (0,0)
89C 1 1 1 −1 0  −  (0,0)
101A 0 1 1 −1 −1  +  (−1,0)
131A 0 −1 1 1 0  −  (0,0)

In each case P actually generates E(Q). This can be verified by analysing the standard proof of the Mordell-Weil theorem. We omit the details. We remark that the theory of Heegner points provides an intrinsic way to exhibit points on E, without making reference to a defining equation. For example,

[7] page 147 shows that E has positive rank using Heegner points. As specific examples we mention that Gross (private communication) has shown that the quadratic field Q(\sqrt{−67}) yields the Heegner point P = (0, 0) on 37A, while Q(\sqrt{−115}) yields the point 6P. We summarize the main results so far.

Theorem 1.2 For the nine values of p in {37, 43, 53, 61, 79, 83, 89, 101, 131}, the curve X_{0}(p)/w is an elliptic curve, defined over Q, of conductor p, and with E(Q) ≅ Z.

Now that we have the curves E of rank 1 to hand, let us restate the conjecture of Birch and Swinnerton-Dyer specialised to this situation. Noting that c_{p} = 1 (see [1] page 46, E has reduction type I_{1} at p), we have the statement

L(E, s) has a zero of order 1 at s = 1 and L′(E, 1) = Ω |⨿⨿| ⟨P, P ⟩.

We describe the calculations that corroborate this statement. Firstly, computing the period Ω is described in the first appendix below. It can be done very rapidly and accurately using Gauss’ Arithmetic-Geometric Mean. Next we describe how the height is computed. We have ⟨P, P ⟩ = 2h^{∗}(P) where h^{∗}(P) is the Néron-Tate canonical height of a point P on an elliptic curve. As recalled below, the canonical height function is a sum of local heights,

h^{∗}(P) = ∑_{v} λ_{v}(P),

where the sum is taken over all places v of Q. For the finite places v Tate has given explicit formulas for the local heights λ_{v}(P), and has given a rapidly converging algorithm to compute λ_{∞}(P) for points P ∈ E(R), see [16] for the details (and an improvement of Tate’s algorithm to cope with complex points also). In our cases, where E and P are as given, since P reduces to a non-singular point modulo every prime, it follows that

λ_{v}(P) = 0

for all finite places v of Q. Then h^{∗}(P) = λ_{∞}(P) is easily calculated. (We give the algorithm in the section on heights below.)

It remains to compute L′(E, 1). Since E = X_{0}(p)/w, the Hasse-Weil L-series L(E, s) of E is the Mellin transform of a newform f of weight 2 for the group Γ_{0}(p) which is invariant under w, f|w = f. The functional equation for L(E, s) takes the form that

Φ(s) = (2π)^{−s}Γ(s)L(E, s) = ∫_{0}^{∞} t^{s−1}f(it) dt

satisfies

Φ(s) = −p^{1−s}Φ(2 − s).

We recall that the functional equation just stated can be proved by splitting the integral from 0 to ∞ at p^{−1/2} (where w acting on H fixes ip^{−1/2}), and then using f|w = f to obtain the representation

Φ(s) = ∫_{p^{−1/2}}^{∞}(t^{s−1} − p^{1−s}t^{1−s})f(it) dt

which is valid for all s ∈ C and which provides the analytic continuation of L(E, s) to an entire function on C. We also see from the functional equation that L(E, 1) = 0 and that L(E, s) has an odd-order zero at s = 1. Suppose that we write

f(z) = ∑_{n=1}^{∞} a_{n}q^{n},

where as usual q = e^{2πiz}, then differentiating the last expression for Φ(s) once with respect to s, setting s = 1 and integrating the series for f term by term we get

L′(E, 1) = 2 ∑_{n=1}^{∞} \frac{a_{n}}{n} ∫_{2πn/\sqrt{p}}^{∞} e^{−x} \frac{dx}{x}.

The coefficients a_{n} have the multiplicativity property expressed by the existence of an Euler product of the expected form for L(E, s) (or alternatively by the fact that f is a newform, and therefore an eigenfunction of the Hecke operators). We have a_{1} = 1, a_{p} = −1 and a_{l} = 1 + l − |E(F_{l})| for l ≠ p a prime. Both the sum and the integrals in our formula for L′(E, 1) converge rapidly and a moderate number of terms is enough to obtain accurate values. Standard estimates show that taking the terms for n ≤ 60 and truncating the integrals at x = 64 gives L′(E, 1) correct to 11 decimal places. (See the third appendix for more details.) A convenient method to calculate the a_{l} (which can also be found in Table 3 in [1]) for l > 2 is to use

a_{l} = − ∑_{xmodl} (\frac{f(x)}{l}),

where f(x) = x^{3}+b_{2}/4x^{2}+b_{4}x/2+b_{6}/4, and where the b_{i} are calculated from the coefficients of the minimal equation for E according to Tate’s formulaire.

Finally, carrying out all the computations results in the numbers in the table below. The predicted and calculated values for L′(E, 1) agree to at least 8 places in each case, so that the predicted order for the Tate-Shafarevich group for each of our nine curves is 1.

Curve Ω  h^{∗}(P)  Ω⟨P, P ⟩  L′(E, 1)
37A 5.986917292 .025555704 .3059997736 .3059997738
43A 5.468689531 .031408754 .3435239797 .3435239746
53A 4.687641049 .046490742 .4358638242 .4358638241
61A 6.133193148 .039593865 .4856736508 .4856736514
79A 5.950800353 .048832105 .5811802165 .5811802159
83A 3.374468900 .088646147 .5982673327 .5982673327
89C 5.552626565 .056052440 .6224765415 .6224765415
101A 4.590247212 .082351726 .7560295657 .7560295656
131A 4.171609276 .108047599 .9014647353 .9014647353

There are two other curves with prime conductor and positive rank in the tables in [1], namely the curves 163A and 197A. Similar computations can be carried out in these cases to check that L′(E, 1) = Ω⟨P, P ⟩ (at least approximately). The formula for L′(E, 1) of course assumes that E is a Weil curve, which we haven’t verified in these two cases. For the record, the resulting values of L′(E, 1) are 1.04793302 and .78732047 respectively.^{2}

2 A Rank 2 Example

According to [8] the first rank 2 curve with prime conductor is

E  y^{2} + y = x^{3} + x^{2} − 2x,  N = Δ = 389.

By reducing modulo 2, 3 and 5 it is easy to check that E_{tor}(Q) is zero. (This also follows from a lemma in [8] since E(Q) contains more than 6 points.) Let A = (0, 0) and B = (−1, 1) be rational points in E(Q). It is easy to check that A and B are both distinct and non-zero in the finite quotient E(Q)/2E(Q). On the other hand, the cubic subfield F of Q(E[2]) has discriminant 4·389 and ring of integers Z[θ] where θ^{3} −θ^{2} −9θ +11 = 0. Using the Minkowski bound and the factorizations of 2, 3, 5 and 7 in Z[θ] one sees that F has class number 1. The rank bounds of [3] then imply that rank E(Q) ≤ 2. Thus we have rank E(Q) = 2, and the bounds just referred to also show that ⨿⨿^{(2)}(E/Q) = 0 also.

2All the period and height calculations were carried out on a Texas Instruments TI59 calculator, and the evaluation of L′(E, 1) was done on a CDC Cyber computer at the University of Virginia.

Since we have c_{389} = 1, the conjecture here says that firstly L(E, s) has an analytic continuation, has a zero of order 2 at s = 1, and secondly that

L′′(E, 1) = 2Ω |⨿⨿| det ( \begin{matrix}⟨A, A⟩ & ⟨A, B⟩ \\ ⟨B, A⟩ & ⟨B, B⟩ \end{matrix} ) .

(We are assuming here that A and B actually generate E(Q), which could be readily checked.) The height pairing ⟨A, B⟩ = ⟨B, A⟩ = h^{∗}(A + B) − h^{∗}(A) − h^{∗}(B) and the height and period computations give:

Ω = 4.980425122,
⟨A, A⟩ = 2h^{∗}(A) = 2λ_{∞}(A) = .3270007736,
⟨B, B⟩ = 2h^{∗}(B) = 2λ_{∞}(B) = .6866670834,
⟨A, B⟩ = ⟨B, A⟩ = λ_{∞}(A + B) − λ_{∞}(A) − λ_{∞}(B)
= −.2684780988,

and we find that the period times the height determinant is 1.518633001.

We assume for the purposes of the computation that E is indeed a Weil curve. Since it has prime conductor, and the tangents to the node in Ẽ(F_{389}) are F_{389} rational, the sign in the functional equation must be +1, and the order of vanishing of the L-series is even. Using the integral representation for Φ(s) = (2π)^{−s}Γ(s)L(E, s) given by

Φ(s) = ∫_{p^{−1/2}}^{∞}(t^{s−1} + p^{1−s}t^{1−s})f(it) dt,

where f is the newform corresponding to E, f(z) = ∑_{n} a_{n}q^{n}, we find that

L(E, 1) = 2 ∑_{n=1}^{∞} \frac{a_{n}}{n} e^{−2πn/\sqrt{389}}.

Furthermore, if we know that L(E, 1) = 0, then integrating the formula for Φ(s) by parts after differentiating twice gives

L′′(E, 1) = 4 ∑_{n=1}^{∞} \frac{a_{n}}{n} ∫_{1/\sqrt{389}}^{∞} log(\sqrt{389}t) e^{−2πnt} \frac{dt}{t}.

It is then routine to find the a_{n}, and then to evaluate these series. Using 130 terms gives |L(E, 1)| < 10^{−8} and that L′′(E, 1) = 1.518633009, agreeing with the period times height term. And the predicted ⨿⨿ is trivial here.

A Computing Periods of Elliptic Curves

We describe how to compute the fundamental period of an elliptic curve using Gauss’ arithmetic geometric mean.

B Computing Heights

We recall the definition and main properties of heights of points on an elliptic curve, and give a method of Tate’s to calculate the local height function at the infinite real place of Q.

C Computing L-series

This summarizes some estimates needed to evaluate the L-series derivative values accurately.

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