### A parametrization of elliptic curves y2 = x3 + ax + b with a large integral point

Hall's conjecture concerns large integral points on elliptic curves of the special form y2=x3+b. What about large integral points on the general curve y2=x3+ax+b? (This is not quite the most general elliptic curve over Z, but putting an arbitrary curve in this form introduces only bounded factors into all the estimates.) It turns out that, whereas for y2=x3+b one expects that x<<b2+o(1), there are curves y2=x3+ax+b with a constant that have an integral point with x asymptotically proportional to b4.

Standard heuristics suggest that if we consider curves with a<<H2 and b<<H3 then x can be as large as H10+o(1) but no larger. Zagier addressed this question in [3]; the addendum [3a] reports that Lang and H.Stark already conjectured in [2] that x<<HR for all R>10, but with the proviso that some families might have x growing as a power of H higher than the tenth. Soon after reading Zagier's paper I found an example of such a family with x asymptotically proportional to H12, and as it happens with a constant linear coefficient a. To my knowledge, this family and its easy variations are still the only such families known.

I wrote Zagier a letter giving this construction, which he acknowledged in the addendum [3a]. It hasn't been published otherwise, though the remarks concerning Hall's conjecture were subsequently incorporated into my paper [1] (see p.56). The letter can now be viewed in PDF or .dvi; please note the sign error on page 2: the second curve in the family has (tb)=(-15, 17424), not (15, -17424) as written.

References
[1] Elkies, N.D.: Rational points near curves and small nonzero |x3-y2| via lattice reduction, Lecture Notes in Computer Science 1838 (proceedings of ANTS-4, 2000; W.Bosma, ed.), 33-63 (math.NT/0005139 on the arXiv).
[2] Lang, S.: Arithmetic and Geometry, Vol.I, 155--171 [Progr. Math. 35, Birkhäuser, Boston, Mass., 1983].
[3] Zagier, D.: ``Large integral points on elliptic curves'', Math. Comp. 48 (1987) #177, 425-436.
[3a] Zagier, D.: Addendum to [3], Math. Comp. 51 (1988) #183, page 375.