• LfunctionsAndModularFormsII
• CentralValues
• 32A EvenRankTwists

Updates on post-processing for even-rank twists of 32A

See also the Rank 4 page. This page is more concerned with rank 6.

\bullet Pre-post-processing: computed weight 3/2 forms to O(q^{10^{10}_{\phantom0}}) and relevant zero coefficients extracted [G.Tornaría (et al.?) in Seattle; see the first item in the top CentralValues page]. Total = 205\,200\,161

\bullet Extracted list of squarefree |D|; total = 72\,601\,560. Those are all the squarefree D < 10^{10} such that the quadratic twist Dy^2 = x^3 - x of 32A has even and positive analytic rank ("even" means D \in \{1,2,3\} \bmod 8, "positive" uses Tunnell's criterion and the weight 3/2 modular forms). NB Most of the non-squarefree D = m^2 D_0 do not come from positive-rank twists; e.g. if D_1 \equiv \pm 1 \bmod 3 then the q^{9D_1} coefficient is always zero, and this accounts for the majority of the 205\,200\,161 values of D in the first step.

\bullet For each of those curves Dy^2 = x^3 - x, computed the Selmer upper bound for the 2-isogenies between it and Dy^2 = x^3 + 4x. Happily all of these bounds (which are always even for these choices of D \bmod 8) were positive. The majority equal 2, but there are still 19\,002\,034 curves with an upper bound of 4, and 597570 with an upper bound of 6 or more (here "more" is always 8, and that in only 1140 cases). [N.D.Elkies, using an unpublished (c.1990) formula for this Selmer bound in terms of a certain matrix over {\bf Z} / 2 {\bf Z}]

\bullet For each of the candidates for rank at least 6, used mwrank -s to compute the Selmer upper bound on each of those curves using complete 2-descent for Dy^2 = x^3 - x and each of the 2-isogenous curves Dy^2 = x^3 + 4x and Dy^3 = x^3 - 11 x \pm 14 (these are the other three curves of conductor 32). Only 1033 candidates survived this test, each with an upper bound of exactly 6 on the rank (and again happily none of the others has an upper bound less than 2). This was by far the longest part of the computation: about a week for two of Sage's processors running in parallel.

\bullet For each of those 1033, ran mwrank -p80 to search for rational points using 2-descent. In only one case were more than 4 independent points found, namely

giving the rank-6 twist that Nicholas Rogers published in 2000 (see page 593 of his Experimental Mathematics paper), and for which mwrank quickly confirms rank=6. For each of the others, mwrank finds 0, 1, 2, or 4 independent points, except for the case of D = 6526531234 = 2 \cdot 17 \cdot 1361 \cdot 141041 where it finds 3 independent points on Dy^2 = x^3 - x but 4 on the isogenous curve Dy^2 = x^3 + 4x (evidently mwrank used one of the other 2-isogenies to search for points on Dy^2 = x^3 - x). This took about 40 hours on a single Sage processor.

The following table shows the distribution of rank bounds describes above, broken down by residue class mod 8 or 16 (because that's how they were computed, though the residue mod 8 seems to affect the statistics significantly). B_\phi is the upper bound using the 2-isogeny with Dy^2 = x^3 + 4x, while B_2 is the mwrank -s bound using complete 2-descent for all four curves in the isogeny class and b is the lower bound using the Mordell-Weil subgroup found by mwrank; the asterisked b=4 counts include D = 6526531234.

It seems reasonable then to conjecture that D = 6611719866 gives the only rank-6 twist up to 10^{10}. Moreover, there are probably no cases of rank 4 on the list of 1033 candidates other than the 98 for which mwrank found 4 independent points. It is true that by searching far enough we will/would eventually find a rank-6 candidate whose actual rank is 4 but on which we could find no more than 2 independent points using mwrank; but before this happens there should be quite a few cases where we can only get a lower bound of 3 on a rank-4 curve, and we have yet to run across such a twist.

\bullet For these 1033 rank 6 candidates, the 4-descent machinery of Magma was run on various 2-covers (of all curves in the isogeny class). Other than the rank 6 twist D=6611719866 already found by Rogers, this was sufficient to prove (assuming something like the Bach bound when computing of class and unit groups) an upper bound of 4 on the rank, with one exception, namely D = 9289874770 = 2 \cdot 5 \cdot 11 \cdot 47 \cdot 401 \cdot 4401. For this twist, all 2-covers of all isogenous curves lift to elements in the 4-Selmer group, but we suspect that the rank is only 4. This already proves (modulo GRH, via the Bach bound) that D=6611719866 is the first twist of rank 6, as Rogers guessed based on his searches for curves Dy^2 = x^3 - x with many small points. [M.Watkins]

\bullet Steve Donnelly reports that he eliminated the remaining rank-6 candidate D = 9289874770 using a Cassels-Tate pairing linking 2-coverings with 4-coverings. Tom Fisher has independently done the same (Oct 2) using a ersatz 8-descent (specific to curves with 2-torsion), and has listed a short certificate that proves this. This completes the rank-6 postprocessing (modulo the usual provisos of programming bugs).

Note that we did not try to prove our guess that none of the 934 curves with b \leq 2 actually has rank 4; even if we did prove this, there would be plenty of other rank-4 twists with D < 10^{10}, including some rank-4 twists with nontrivial Sha, that were discarded earlier in the search for twists of rank 6.