Modulo Errors

The Szpiro conjecture asserts that only finitely many elliptic curves have a Szpiro ratio greater than six, so I thought it’d be interesting to see if I could find any. Moreover, the result of Hindry and Silverman ensures that the greater the Szpiro ratio, the smaller the lower bound for the height of points on a curve; so there was a natural place to start looking- Elkies’ list of the nontorsion points of low height. Armed with PARI, and learning how to use it as I went, I set about computing the ratios for these curves of interest, alongside a couple of thousand test cases for comparison.

The largest ratio uncovered was around 5.6 (curve 3822bg1), but more importantly the lowest was around 2.96 (curve 1110m1), a not particularly inspiring value compared to the other data set. Further, there wasn’t a particularly strong correlation between the rankings according to ratio and height- the four largest Szpiro ratios arising from entries 6, 33, 52 and 38 on Elkies list by height. It seemed, therefore, that points could have impressively small height despite coming from curves of moderate ratio; whilst a high ratio was no guarantee of small height values.

To be thorough, I then set PARI and its elldata package loose on the curves of conductor 2000 or less, of which there are about eleven thousand, to see what kind of ratios could be reached. However, I quickly noticed that rank zero curves could generate high values, but are only relevant to the Szpiro conjecture itself; they have no nontorsion rational points to bound the height of! Below the cut I’ve therefore given the top 10 ratios observed (the best being 8.9), and then all the curves (23) of non-zero rank that attain a ratio above 6.

These I had hoped would therefore be good candidates for small heights, but as the tabulated values show this is very much not the case. Indeed, it seems that the ratio-dependent bounds given by Hindry & Silverman’s result are many orders of magnitude smaller than any heights known. Since no known points are anywhere near close enough to these bounds, ratio-based variations won’t create meaningful extra breathing space; and thus deliberately crafting curves with high Szpiro ratio is unlikely to be a sensible way to seek small heights. Which is a disappointing result, but one probably worth knowing; and there may still be interesting things to say about the distribution of values of the ratio. From my computations on 11308 curves: The lower bound is 1; the average is about 2.82; and the proportion of curves clearing the magical 6 mark is 1.19%, which must of course tend to zero as the sample size increases if the conjecture is to hold.

The top ten Szpiro ratios, elliptic curves with conductor less than 2000

Elliptic Curves with conductor less than 2000, Rank non-zero, Szpiro ratio greater than 6

This entry was posted on Monday, December 4th, 2006 at 3:13 pm and is filed under Algebraic Geometry, Number Theory, PhD.