AN INTERESTING FAMILY OF CURVES OF GENUS 1

We study the family of elliptic curvesy2 = x3−t2x+1, both overQ(t) and over Q. In the former case, all integral solutions are determined; in the latter case, computation in the range 1 ≤ t ≤ 999 shows large ranks are common, giving a particularly simple example of curves which (admittedly over a small range) apparently contradict the once held belief that the rank under specialization will tend to have minimal rank consistent with the parity predicted by the Selmer conjecture.


1.
It is the purpose of this note to draw attention to the family of elliptic curves which tend to possess many integral points for a fixed t ∈ Z; for example, at least 52 integral points when t = 8.We describe the computational investigation into this phenomenon.First, regarding (1.1) as the generic fibre of an elliptic surface defined over C, then the group G of sections of the surface is naturally identified with the group of C(t)-rational solutions of (1.1), which by abuse of notation will also be denoted by G. Indeed, we tend to refer to (1.1) as an elliptic curve over C(t) and speak about "points on the curve" rather than "sections of the surface".It is well known that G is finitely generated.For (1.1), the C(t)-rank is 4, and this allows the determination of all the "integer" points of (1.1), which we effect over Q, that is, finding all points (x, y) Second, we give numerical information about the curves (1.1) in the range 1 ≤ t ≤ 999, computing the rational Mordell-Weil rank independently of any of the standard conjectures, except in four instances where it was necessary to assume the Birch and Swinnerton-Dyer conjecture.The ranks are surprisingly large: although the Q(t)-rank of (1.1) is 3, there are 9 cases of Q-rank 7 and 82 cases of Q-rank 6 within the given range.This provides some support for the belief that a curve under specialization will not necessarily tend to have the minimal rank consistent with the parity predicted by the Selmer conjecture.

2.
The curve (1.1) occurs as Example 2.32(b) Cox and Zucker [3] where it is shown that the curve is torsion-free, and G has C(t)-rank 4, with basis where Alternatively, this is a straightforward verification using the theory of Mordell-Weil lattices as described by Shioda [7]; see Kuwata and Top [5] for a typical illustration of the method.
Let , : G × G → R be the height-pairing defined by where ĥ is the canonical height function.Using the algorithm of Silverman [8], one computes the height-pairing matrix on the basis (2.1) to be Accordingly, ĥ It is now, in theory at least, straightforward to determine all the points on (1.1) with x, y ∈ C[t].Hindry and Silverman [4,Corollary 8.5], give an explicit bound for the height of such points P , which for the curves (1.1) becomes ĥ(P ) ≤ 29.Using (2.4), the coefficients m 0 , m 1 , m 2 , m 3 of P with respect to the basis (2.1) must therefore satisfy (x, ±y) = (0, 1), (−1, t), (t, 1), (−t, 1), (t + 2, 2t + 3), (−t + 2, −2t + 3), (2.6) 3. Using Cremona's package "mrank" on a SUN workstation, we computed the rational rank of (1.1) when specialized to t in the range 1 ≤ t ≤ 999.There were just four instances where the rank was not calculated exactly, corresponding to the presence of a nontrivial 2-component of the Safarevic-Tate group.Here Connell's "Apecs" package was used to compute the appropriate derivative of the L-series, thereby establishing the rank subject to the Birch and Swinnerton-Dyer conjecture.The tabulation of ranks is as follows: what is dramatic here is that although the Q(t)-rank of (1.1) is 3, under ) with the minimally consistent rank 3. Kramarz and Zagier [9] seem to give the first recorded instance of scepticism regarding minimality of the rank under specialization: in a family with generic rank 0, 23.3% of curves with sign of functional equation equal to +1 had rank at least 2 over the range searched.It must be stressed however that the number of curves considered here is really very small, and results of Brumer [1] interpret rather differently the voluminous data of Kramarz and Zagier.See also Brumer and McGuinness [2], and Rohrlich [6] for a more contemporary discussion of this issue.A further extraordinary feature of the family (1.1) is that in all but four cases computed, the Safarevic-Tate group has trivial 2-component, the exceptions being t = 210, 285, 455, 645, where the 2-component has order 4. I have no explanation for this phenomenon.

4.
Several other similar pencils of curves may be written down with positive Q(t)rank, for example, the curves and its twist by i, E : Similar computations as in Section 2 provide the following.

Table 3 .
1.The curves (1.1) with rank 7 occur at t = 347, 443, 614, 757, 778, 784, 857, 877, 888.02% of the curves with rank 3, 41.34% with rank 4, 27.23% with rank 5, 8.21% with rank 6, and 0.90% with rank 7. It was once believed that almost all curves under specialization would have the minimal rank consistent with the generic rank and sign of the functional equation, so here, ranks 3 or 4; yet ranks 3, 4 account for only 64% of the cases.Put another way there are 502 curves in the range with sign of functional equation −1; and 272 of these, or fully 54.18%, have rank 5, compared to 220 (43.82%