Boundedness of Mordell–Weil ranks of elliptic curves over the rational numbers

For an elliptic curve $E$ over $\mathbb{Q}$ , Mordell's theorem asserts that the group $E(\mathbb{Q})$ of rational points is finitely generated, hence of the form (finite torsion) $\times \mathbb{Z}^{r}$ with a nonnegative integer $r$ , the Mordell–Weil rank. It is a famous open problem whether the ranks $r(E)$ are bounded as $E$ ranges over all elliptic curves over $\mathbb{Q}$ . The folklore conjecture predicts unboundedness, while a widely studied heuristic of Park–Poonen–Voight–Wood predicts boundedness (finitely many curves of rank above 21); no approach is known that decides either direction. The current record is a single curve of rank at least 31 (Alpöge–Howell, 2026), and no infinite family with provably growing rank is known. The problem asks, in its full unrestricted form over $\mathbb{Q}$ , whether $\sup_{E/\mathbb{Q}} r(E)$ is finite.

Background

An elliptic curve over $\mathbb{Q}$ is a smooth projective curve of genus one defined over $\mathbb{Q}$ with a specified rational point; its set of rational points carries a commutative group structure. Mordell's theorem (1922) states that $E(\mathbb{Q})$ is a finitely generated abelian group, so $E(\mathbb{Q}) \cong E(\mathbb{Q})_{\mathrm{tors}} \times \mathbb{Z}^{r}$ . The torsion subgroup is completely understood by Mazur's theorem (it is one of fifteen explicitly listed finite groups), but almost nothing unconditional is known about which values the free rank $r = r(E)$ , called the Mordell–Weil rank, can attain: it is not known whether the set $\{r(E) : E/\mathbb{Q}\}$ is bounded or unbounded.

The question sits at the rank side of the Birch–Swinnerton-Dyer (BSD) conjecture, which predicts that $r(E)$ equals the order of vanishing at $s=1$ of the Hasse–Weil $L$ -function $L(E,s)$ of $E$ (the analytic rank). The modularity theorem — every elliptic curve over $\mathbb{Q}$ arises from a weight-2 newform, i.e. a modular form of weight 2 that is a simultaneous eigenfunction of the Hecke operators, so that $L(E,s)$ is an $L$ -function of degree 2 with analytic continuation and a functional equation relating $s$ to $2-s$ — makes the analytic side accessible. Classical explicit-formula arguments of Mestre and Brumer bound the analytic rank logarithmically in the conductor $N_E$ of $E$ (a positive integer measuring the curve's bad reduction: a product of explicitly computable local factors, one for each prime of bad reduction); this transfers to a bound on $r(E)$ itself only through BSD, and no unconditional uniform upper bound on $r(E)$ is known. A 2025 preprint of Lakshmanan shows, within the framework of any hypothetical degree-2 $L$ -function with analytic continuation and functional equation attached to $E$ that yields a bound $r(E) \ll \log \Phi(E)$ for an arithmetic invariant $\Phi(E)$ , that necessarily $\Phi(E) \ge N_E$ ; that is, the conductor is minimal among such invariants, which rigidifies analytic routes but leaves the boundedness question itself untouched.

On the constructive side, records for the largest known rank have advanced slowly: rank at least 28 (Elkies, 2006), proven to equal 28 assuming the generalized Riemann hypothesis (GRH) (Klagsbrun–Sherman–Weigandt, 2016); rank exactly 20 unconditionally (Elkies–Klagsbrun, 2020, still the highest rank known exactly); rank at least 29 (Elkies–Klagsbrun, 2024); and rank at least 30 and 31 (Alpöge–Howell, 2026). All of these are isolated curves produced by explicit search, not members of any family with provably unbounded rank. In the function-field analogue — replacing the base field $\mathbb{Q}$ by a rational-function field $k(t)$ — the situation is different: Ulmer constructed explicit elliptic curves over $\overline{\mathbb{F}_p}(t)$ with unbounded rank, together with new high-rank curves over $\mathbb{C}(t)$ , so the obstruction is genuinely arithmetic over number fields.

Two general results frame the two competing expectations. Bhargava–Shankar proved that the mean size of the 2-Selmer group of elliptic curves over $\mathbb{Q}$ ordered by height equals 3, where the 2-Selmer group is a finite, effectively computable group containing $E(\mathbb{Q})/2E(\mathbb{Q})$ , so that $2^{r(E)} \le |\mathrm{Sel}_2(E)|$ and its size bounds the rank from above; this bounds the limsup of the average rank by 1.5, but average bounds say nothing about the supremum, since sparse high-rank curves are not excluded. Park–Poonen–Voight–Wood gave a heuristic, based on modeling ranks and Shafarevich–Tate groups (the groups measuring the failure of the local-to-global principle for rational points on $E$ ) simultaneously via a theorem on counting alternating integer matrices, that predicts ranks are bounded and that only finitely many curves have rank greater than 21 — directly contradicting the older folklore conjecture of unboundedness. Neither direction has any proof, and the field's standard reference for rank records still lists the possible values of $r$ as unknown.

Problem Statement

Determine whether the Mordell–Weil ranks of elliptic curves over $\mathbb{Q}$ are bounded. Precisely: is there an integer $B$ such that every elliptic curve $E$ over $\mathbb{Q}$ satisfies $r(E) \le B$ , where $r(E)$ is the rank of the finitely generated group $E(\mathbb{Q})$ ? The problem is posed in its full, unrestricted form: $B$ need not be effective, no restriction on conductor, discriminant, family, or method is allowed, and both answers are admissible — a proof of boundedness, or a proof of unboundedness, the latter amounting to a construction showing that for every $r \ge 0$ there exists an elliptic curve over $\mathbb{Q}$ of rank at least $r$ . In the unbounded case an explicit infinite family with provably growing rank lower bounds would resolve the question; in the bounded case even a non-effective existence proof of such a $B$ would resolve the question. This is the rank-side counterpart of the first prediction of the Birch–Swinnerton-Dyer conjecture.

References

K. Lakshmanan, 'On the Minimality of the Conductor for Elliptic Curve L-Functions', arXiv:2506.20175, https://arxiv.org/abs/2506.20175 (v1 June 2025, v2 June 2026, preprint)
Jennifer Park, Bjorn Poonen, John Voight, Melanie Matchett Wood, 'A heuristic for boundedness of ranks of elliptic curves', Journal of the European Mathematical Society 21 (2019), no. 9, 2859–2903, doi:10.4171/jems/893; preprint arXiv:1602.01431, https://arxiv.org/abs/1602.01431
Manjul Bhargava, Arul Shankar, 'Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves', Annals of Mathematics 181 (2015), no. 1, 191–242, doi:10.4007/annals.2015.181.1.3; preprint arXiv:1006.1002, https://arxiv.org/abs/1006.1002
Noam D. Elkies, Zev Klagsbrun, 'New Rank Records For Elliptic Curves Having Rational Torsion', arXiv:2003.00077, https://arxiv.org/abs/2003.00077 (2020, preprint)
Zev Klagsbrun, Travis Sherman, James Weigandt, 'The Elkies Curve has Rank 28 Subject only to GRH', arXiv:1606.07178, https://arxiv.org/abs/1606.07178 (2016, preprint)
Douglas Ulmer, 'On Mordell-Weil groups of Jacobians over function fields', arXiv:1002.3310, https://arxiv.org/abs/1002.3310 (2010)
Andrej Dujella (maintainer), 'History of elliptic curves rank records', curated web page, https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html (accessed 2026-08-24)