Early this year I wrote you asking for help analyzing the elliptic curves E : y^2 = x * (x^2 - x + (1-T^10)/5 ). You were kind enough to send a reply, so I thought I would let you know what happened in the ensuing months. In brief: it is now possible to determine painlessly much more of the structure of rational elliptic curves having 2-torsion than was previously expected. In particular, a rational section on this elliptic surface has been found. The troublesome feature of the curves was that for almost every rational specialization T=t there seemed to be rational points on the curve, yet there appeared to be no point on the curve over Q(T). (Indeed, for some time I thought I could prove no such point existed). I had no tools for finding points on curves over Q(T) or determining their rank, and the tools for finding points and ranks of curves over Q were easily swamped by the size of the coefficients. In fact, there is a point of infinite order in E(Q(T)). Its x-coordinate is 2 2 2 4 6 8 2 (1 - T ) T (1 + 2 T + 2 T + T - T ) 5 ---------------------------------------- 2 4 6 8 2 (1 + 6 T + 6 T + 11 T + T ) and more generally this (with T replacing T^2) provides a point on the related curve E4 : T y^2 = x * (x^2 - x + (1-T^5)/5 ). I found this point in a relatively pedestrian way, tying together data from a number of the specializations. Simultaneously and independently, Masato Kuwata found this point by a different and interesting method. Kuwata's method proceeds roughly as follows: The rational points are the invariants (under Galois action) of the complex points, of which there are many: for each of the possible divisors (1- zeta^i T) of (1-T^5) we may easily find a point representing this divisor in E4 / 2 E4, namely, there is a point P_i whose x coordinate is simply zeta^i T (1 - zeta^i T). None of these are rational, but Kuwata observed that the sum of the Galois conjugates of one of them, obviously then rational, is nonzero (for some reason I had thought otherwise in January). This gives the point shown above. With this notation, the point may also be written P_0 + 2 (P_1 + P_4) (not obviously rational but it is clear that the x-coordinates are the same modulo squares). Kuwata has done an interesting and more thorough analysis of this surface, which I will not attempt to summarize. What _is_ interesting about the approach I used is the fact that I was able to collect sufficient data to manage this at all! This was initially impossible as the points on the specializations of E (or E4) were of too great a height to find. However, Samir Siksek alerted me to the possibility of performing, for curves with 2-torsion, an additional descent. (Traditional descent computes homogeneous spaces whose Jacobian is the 2-isogenous curve E'; the additional descent, also of degree 2, computes homogeneous spaces whose Jacobian is again E). This makes it possible to deduce that many of the first homogeneous spaces have no rational points, that is, to decide that Sha is nontrivial. For example, when T=43/5, traditional descent found the rank to be at most 7 (i.e. 128 homogeneous spaces would be searched); with additional descent, the rank is found to be at most 3 (only 7 homogeneous spaces must be searched). Moreover, points on the remaining homogeneous spaces may more easily be found by a search over points of comparatively low height. For example, for T=29/3, the curve is [0,0,0,-473295637507012875,-2875270997630901543750] and we found the point [-10801643340547358712112188835989136980675 : 90363766441841314306728685293678456683648250 : 16519855954076266883258833345259] of height = 55.58 (as computed by Cremona's program ratpoint ). The methods turning the basic idea of an extra descent into usable algorithms are due primarily to John Cremona and myself this Spring; Cremona is implementing these ideas efficiently into his program MWRANK. In addition, Ian Connell has already updated APECS to incorporate them. (I've also encoded this procedure in a stand-alone Maple package.) With these procedures it has been possible to give a fairly complete analysis of the elliptic curves in this family for about a hundred values of T. I've placed a condensed form of some of the output at my web site (the descent package is there too). Here are some highlights: (*) Some of the curves have rather high rank, including three of rank 5. (*) The average rank is also high. I have 192 generators on 96 curves, and there are surely 20 more by parity conjectures. There are also five curves in the set whose rank I don't yet know for sure, possibly adding as many as 10 more generators. So the average rank is between 2.0 and 2.3 . (*) The curves involve generators of very large heights. For example, when T=51/5 the parameterization above gives a point on the curve whose height is 79.15, a point which I believe is member of a set of minimal generators. (*) When the curves are presented in minimal form (Laska's algorithm) there are frequently quite a few integral points, including some rather large ones. For example, for T=53/5 there is the integral point [91536875299414997 : 27694538017241769344614070 : 1] When T=1/4, the curve is [0, -1, 0, -139810, -7907108]; I found 7 pairs of integer points. I hadn't intended to work on algorithms for elliptic curves in general but in retrospect that project has been more fruitful than the simple analysis of my original family of curves will be. I appreciate your having responded when I first asked for suggestions. dave Details at: http://www.math.niu.edu/~rusin/research-math/funny.crv/