Background for the Elkies-Klagsbrun curve of rank 29

Question

Elkies and Klagsbrun have recently announced an elliptic curve over Q of rank (at least) 29, improving on the previous record from 2006!

https://web.math.pmf.unizg.hr/~duje/tors/z1.html

It has trivial torsion group.

What is the background on its discovery? Are some of its properties surprising? And what about the conjecture that ranks are bounded?

Answer 1

With regard to question 3:

In earlier work, specifically the paper "New rank records for elliptic curves having rational torsion", the same authors, Elkies and Klagsbrun, attained new records for the largest rank of an elliptic curve with torsion subgroup $H$, for various fixed (nontrivial) values of $H$. In that paper, they remarked:

At the same time, our work provides, at best, limited evidence that ranks are unbounded. We broke five different records, and found numerous new curves whose ranks tie the old records (and many more whose ranks exceed the heuristically conjectured asymptotic upper bounds). But the scale of this search was vastly larger than any previously attempted, and yet we could not break any of the previous records by more than 1, and in each case found only a handful of curves (in most cases a single curve) with the new record rank. This suggests that the growth of ranks of elliptic curves might indeed peter out at some point.

I think the new result might lead to a similar judgment. Given the 18 years between this result and the previous rank record of 28 in 2006 (due to Elkies), I would be shocked if the scale of the search to find this curve was not vastly larger than the scale of the search to find the rank 28 curve, but only an increase of 1 was obtained. So this is consistent with the idea that ranks are bounded.

More generally, until a paper is written up about the new results, looking at that old paper is probably the best source of background on this discovery.

Answer 2

The announcement on the NMBRTHRY listserv seems by Noam Elkies seems worth reproducing here.

---

The elliptic curve $$E29 :y^2 + xy = x^3 - 27006183241630922218434652145297453784768054621836357954737385 x + 55258058551342376475736699591118191821521067032535079608372404779149413277716173425636721497$$ has rank at least 29, and exactly 29 under the GRH (generalized Riemann hypothesis) for zeta functions of number fields. This is now the largest rank known for an elliptic curve over $\mathbb{Q}$ (more precisely, the largest rank known for a subgroup of $E(\mathbb{Q})$), at last incrementing the previous record rank of 28 which I found and announced here in 2006.

The new curve was found by Zev Klagsbrun last week by a sieve search on a rank-17 fibration of the same K3 surface that I used to find the rank-28 curve, using the techniques we described in our ANTS-XIV (2020) paper. For each specialization that was a candidate for high rank, Zev searched for points outside the fibration's generic $\mathbb{Z}^{17}$; for $E29$ this search found 12 more independent points. He then used the analytic methods of Klagsbrun, Sherman, and Weigandt (Math. of Computation 88 (2019), 837-846 = arXiv:1606.0717) to prove that:

• Assuming the GRH for zeta functions of number fields, the arithmetic rank of $E29$ is at most 29 and thus exactly 29 once we know 29 independent points; and
• assuming $L(E29,s)$ satisfies GRH, the analytic rank of $E29$ is at most 29, and thus exactly 29 assuming also the Birch and Swinnerton-Dyer conjecture, again because we know 29 independent points.

We intend to write up these results, including my computation of the K3 surface once we have finished searching for high-rank specializations.

For now we give some more information about the new record curve.

First to exhibit a rank-29 subgroup: there are 29 independent points with X-coordinates

2891195474228537189458255536634, 3402542165322127811451484642234, 4298760026558467240422107564794, 3728756667770947009884455714554, 5991744132052078230511185130234, 3236493534632768520540227223034, 78226686134991174232380689386234, 11492605643548859374635605140234, -5143303362384229804906088118566, 443985655575065435281568435002, -979565018904269680752629749766, 5184894285212178249566461261834, -4469171023687146502067179612166, 3606405835110925482450522970234, 16151744576785317732688993162234, 3573684355943766387962362869754, -759376049938858166436491644166, -5328058719935886182106003119366, 5380268474895377355583039694554, 17069233487425098088940203248484, 5215432542403430758248050783794, 2838942178046024039763692432122, 243146882395382015946366404808154/81, 2558229016839511149831260080762, 2361253942905600810977556672634, 2678312077644931683114439906234, 3379397084927230910084852603902, 3632407730870998917912491355514, 2428778263277521959543043930234 .

The gp code appended to this message computes that the canonical-height matrix of those points has height $1.43\ldots \times 10^{36}$; in particular they are independent in $E29(\mathbb{Q})$. This is a lexicographically minimal $\mathbb{Z}$-basis for the rank-29 subgroup generated by these points, which is saturated at least at all primes less than $2^{12}$ according to gp's "ellsaturation". The first 32 point-pairs (ordered by height) sufficed to find these 29 points, whose heights range from 46.36+ to 49.94+.

Arithmetic invariants of the curve: $E29$ has discriminant $$D = -2^{19}\cdot 3^7\cdot 5^7\cdot 7^4\cdot 11^5\cdot 13^3\cdot 17^4\cdot 31^3\cdot 41^2\cdot 43^2\cdot 61^2\cdot 233\cdot 241^2\cdot 4139\cdot D_0 $$ where the 129-digit composite $D_0$ factors as the product of three primes

678146849364709860535420504397393, 159788990966780131363155786084695062643236502969, 4402149008473369392540402625019227412319473055901

as Magma found in just over 8 hours. The conductor of $E29$ is the product of the 17 prime factors of $D$, each with multiplicity 1. The local root number is +1 at 41 and -1 at each of the other 16 prime factors, so (including the root number at infinity, which is always -1) the global root number is -1, consistent with a curve of odd rank.

Integral points: we found 1140 pairs of integral points among the rational points of height at most 100 in our rank-29 group. All but 40 have canonical height $h < 65$; all but 9 have $h < 70$; and all but the largest, with $$ x = 1000035519286187601438549887756593382867666394$$ ($h = 88.73$+), have $h < 80$. The first non-integral rational points in our rank-29 group, other than the origin, are the pair with $x = 243146882395382015946366404808154/81$ seen above.

--Noam D. Elkies with Zev Klagsbrun

{
E29 = ellinit(
 [1, 0, 0, -27006183241630922218434652145297453784768054621836357954737385,
 55258058551342376475736699591118191821521067032535079608372404779149413277716173425636721497]
);

X = [
  2891195474228537189458255536634, 3402542165322127811451484642234,
  4298760026558467240422107564794, 3728756667770947009884455714554,
  5991744132052078230511185130234, 3236493534632768520540227223034,
  78226686134991174232380689386234, 11492605643548859374635605140234,
  -5143303362384229804906088118566, 443985655575065435281568435002,
  -979565018904269680752629749766, 5184894285212178249566461261834,
  -4469171023687146502067179612166, 3606405835110925482450522970234,
  16151744576785317732688993162234, 3573684355943766387962362869754,
  -759376049938858166436491644166, -5328058719935886182106003119366,
  5380268474895377355583039694554, 17069233487425098088940203248484,
  5215432542403430758248050783794, 2838942178046024039763692432122,
  243146882395382015946366404808154/81,
  2558229016839511149831260080762, 2361253942905600810977556672634,
  2678312077644931683114439906234, 3379397084927230910084852603902,
  3632407730870998917912491355514, 2428778263277521959543043930234
];
}

PT = vector(29, n, [X[n], ellordinate(E29,X[n])[1]]);

\p 72
matdet(ellheightmatrix(E29,PT))

Answer 3

Not sure if this is surprising, but the discriminant factors as $-2^{19}\cdot 3^7\cdot 5^7\cdot 7^4\cdot 11^5\cdot 13^3\cdot 17^4\cdot 31^3\cdot 41^2\cdot 43^2\cdot 61^2\cdot 233\cdot 241^2\cdot 4139\cdot 678146849364709860535420504397393\cdot 159788990966780131363155786084695062643236502969\cdot 4402149008473369392540402625019227412319473055901.$

(It took YAFU ~1 hour to factor the 129-digit factor into 33-, 48-, and 49-digit factors on my 13700HX.)

In contrast the discriminant of the rank 28 curve factors as $2^{15}\cdot 3^6 \cdot 5^6 \cdot 7^4 \cdot 11^2 \cdot 13^4 \cdot 17^5 \cdot 19^3 \cdot 48463 \cdot 20650099 \cdot 315574902691581877528345013999136728634663121\cdot 376018840263193489397987439236873583997122096511452343225772113000611087671413.$

(The 123-digit factor wasn't factored when the curve was discovered in 2006, but was already factored into 45- and 78-digit factors when I checked factordb.com in November 2022.)