Elliptic curves of high rank over function fields

By the Mordell-Weil theorem the group of Q(z)-rational points of an elliptic curve is finitely generated. It is not known whether the rank of this group can get arbitrary large as the curve varies. Mestre and Nagao have constructed examples of elliptic curves E with rank at least 13. In this paper a method is explained for finding a 14th independent point on E, which is defined over k(z), with [k : Q] = 2. The method is applied to Nagao’s curve. For this curve one has k = Q( √ −3). The curves E and 13 of the 14 independent points are already defined over a smaller field k(t), with [k(z) : k(t)] = 2. Again for Nagao’s curve it is proved that the rank of E(Q̄(t)) is exactly 13, and that rankE(Q(t)) is exactly 12. 1 Mestre’s construction First a method due to Mestre [2] for constructing elliptic curves with high rank is described. Let k be any field with char k 6= 2. Choose 2n elements a1, . . . , a2n ∈ k. We are going to construct a plane curve C such that the points with x-coordinate ai are k-rational. To do so, set p(x) = ∏2n i=1(x − ai). It is easily shown that there exist polynomials q and r in k[x] with deg r ≤ n− 1 such that p = q − r. Define C by the equation y = r(x). Clearly C contains the points (ai,±q(ai)). For n = 5 almost all choices for the ai give that deg r = 4 and that C is a curve of genus 1 with 10 points of the form (ai,±q(ai)). If C is made into an elliptic curve by choosing one of these points as the zero point then the other points generate a group of rank 9 (generically). Mestre constructs an elliptic curve of rank 11 over Q(t) by taking n = 6 and ai = bi + t for i = 1, . . . , 6, and ai = bi−6 − t for i = 7, . . . , 12. Now the x-coefficient of r is of the form s · t with s ∈ Q[b1, . . . , b6]. It is not very difficult to find bi ∈ Q such that s(b1, . . . , b6) = 0. (One can for example choose b1 upto b5 at random and hope that there exists a b6 ∈ Q with s(b1, . . . , b6) = 0. Trying this often enough will almost surely give the desired bi’s.) In Mestre’s example we have b1 = −17, b2 = −16, b3 = 10, b4 = 11, b5 = 14, b6 = 17. If