Unboundedness and effective computation of the Mordell–Weil rank of curves over a fixed number field (the open questions of Weil's 1929 'L'arithmétique sur les courbes algébriques')

In his 1929 Acta Mathematica paper 'L'arithmétique sur les courbes algébriques', André Weil proved the finite basis theorem — today called the Mordell–Weil theorem: for a smooth projective curve $C$ of genus $p$ over a number field $k$, the group of $k$-rational virtual systems (divisor classes) of degree zero is finitely generated — and listed four open problems. Question (i): for fixed genus $p$ and number field $k$, is the rank $r$ of this group unbounded as $C$ varies? Even the case $p=1$, $k=\mathbb{Q}$ — are the ranks of elliptic curves over the rationals unbounded? — is open, with both conjectural directions live (the folklore unboundedness conjecture versus a boundedness heuristic). Question (ii): is there an effective method to compute the exact invariants of the group and an explicit finite basis? Descents give computable upper bounds and point searches give lower bounds, but no unconditional terminating procedure closing the gap is known. The other two questions of the 1929 list were settled by later literature: the effectivity question for virtual systems of degree at most $p-1$ by Faltings' proofs of the Mordell and Mordell–Lang conjectures, and the extension of the decomposition theorem to singular varieties by developments following Hironaka's resolution of singularities. The surviving open core is questions (i) and (ii), which are independent.

Background

Let $k$ be a number field, that is, a finite field extension of the rational numbers $\mathbb{Q}$, and let $C$ be a smooth projective algebraic curve of genus $p\ge 1$ defined over $k$ (the genus is a non-negative integer that classifies curves up to birational equivalence; genus-1 curves with a $k$-rational point are called elliptic curves). In the terminology of Weil's 1929 paper, a system of points on $C$ is a finite formal sum of points with non-negative integer coefficients — in modern language an effective divisor — and a virtual system is a difference of two systems, that is, an arbitrary divisor with integer coefficients; the degree is the total coefficient sum. Two virtual systems are linearly equivalent when their difference is the divisor of zeros and poles of a rational function on $C$. The linear equivalence classes of $k$-rational virtual systems of degree zero form an abelian group, namely the group $J(k)$ of $k$-rational points of the Jacobian variety $J$ of $C$ — an abelian variety (a projective algebraic variety carrying a compatible group structure) of dimension $p$ that parametrizes these classes. Mordell (1922) proved finite generation for $p=1$, $k=\mathbb{Q}$, and Weil (1929) proved the general statement, which he called the finite basis theorem and which is now the Mordell–Weil theorem: $J(k)$ is a finitely generated abelian group, hence the direct sum of a finite torsion group and $r$ copies of $\mathbb{Z}$; the integer $r$ is the Mordell–Weil rank. Weil's proof is an infinite descent on $J$, and its key tool is a decomposition theorem: the values at algebraic points of a rational function on $C$ (or, under a nonsingularity hypothesis, on a higher-dimensional variety) factor into ideal-valued 'distributions' attached to the individual zeros and poles, up to accessory ideal factors bounded independently of the point. The proof is purely existential: it shows a finite basis (a finite list of generators of $J(k)$) exists but gives no procedure for finding one. Weil attaches to $C$ and $k$ the following invariants: the rank $r$; the torsion coefficients $\tau_1,\dots,\tau_s$ of the degree-zero group, the orders of the cyclic direct summands of the torsion subgroup, each dividing the next; and $Q$, the degree of a certain $k$-rational virtual system $A_0$ — the positive generator of the module of degrees of all $k$-rational virtual systems (their degrees are all multiples of $Q$), which is a divisor of $2p-2$; adjoining $A_0$ to a basis of the degree-zero group yields a basis of the full group of $k$-rational virtual systems. For question (ii) the modern framework is the theory of descents: a descent computes a Selmer group, a finite effectively computable abelian group receiving $J(k)$ and thus giving an upper bound for $r$; the obstruction to the Selmer bound being sharp is measured by the Tate–Shafarevich group of $J$, whose finiteness is itself open, and the exact value of $r$ is predicted by the Birch and Swinnerton-Dyer conjecture (BSD), which asserts that $r$ equals the order of vanishing at the central point of the $L$-function of $C$ (an analytic object built from the counts of points of $C$ modulo prime ideals). Exact rank computations in the literature are therefore typically conditional on BSD or on the generalized Riemann hypothesis (GRH), the latter used to make class-group computations inside a descent effective. For question (i) the modern state over $\mathbb{Q}$ is: the 'folklore' conjecture asserts that ranks of elliptic curves over $\mathbb{Q}$ are unbounded, while a widely discussed heuristic of Park–Poonen–Voight–Wood (2019) suggests they are bounded; Bhargava–Shankar (2015) proved that the average rank over all elliptic curves over $\mathbb{Q}$, ordered by height (a standard measure of the size of a curve's defining coefficients), is bounded. Over function fields such as $\mathbb{F}_q(t)$ — fields of rational functions over a finite field, which behave differently from number fields — unboundedness of ranks is known (Ulmer 2002), but no technique transfers to number fields.

Problem Statement

Weil's 1929 list contains four questions. Two were settled by later literature — question (iii), on the effectivity of infinitely many virtual systems of degree at most $p-1$, by Faltings' proofs of the Mordell conjecture (1983) and of Diophantine approximation on abelian varieties (1991, the Mordell–Lang theorem), and question (iv), on extending the decomposition theorem to singular varieties, rendered moot for arithmetic applications by Hironaka's resolution of singularities in characteristic zero (1964) together with the modern theory of heights — and are excluded from the surviving core on that external evidence. The open core consists of the following two questions about the group of $k$-rational virtual systems of degree zero on curves of genus $p$ over a number field $k$ (equivalently, the Mordell–Weil group $J(k)$ of the Jacobian), posed independently in the source.

(A) Unboundedness. Fix a genus $p\ge 1$ and a number field $k$. As $C$ ranges over curves of genus $p$ defined over $k$, is the rank $r=r(C,k)$ of the group of $k$-rational virtual systems of degree zero unbounded? Unboundedness means that for every $N$ there exists such a $C$ with $r\ge N$; otherwise there is a uniform bound $r(C,k) \le B(p,k)$ for all such $C$. The base case $p=1$, $k=\mathbb{Q}$ — are the ranks of elliptic curves over the rationals unbounded? — is already open, and no fixed pair $(p,k)$ is settled in either direction.

(B) Effective computation. Give a method which, from defining equations of a curve $C$ of genus $p$ over a number field $k$, determines in every case the exact values of the invariants of the group of $k$-rational virtual systems — the rank $r$, the torsion coefficients $\tau_1,\dots,\tau_s$, and the degree $Q$ of the system $A_0$ — and effectively finds an explicit finite basis of the group of all $k$-rational virtual systems (equivalently, a finite basis of the degree-zero group $J(k)$ together with the system $A_0$). The method's termination and correctness must be established by mathematical proof — Weil's finite basis theorem is a mere existence proof, yielding at most an upper bound for $r$ — and procedures that are merely observed to work in practice, or whose correctness is conditional on conjectures such as BSD or GRH, do not determine the exact invariants in every case.

Components (A) and (B) are Weil's questions (i) and (ii), which are independent open problems. They address the same object from complementary faces — the structural (the possible range of $r$) and the computational (determination of the group and its invariants).

References

1. André Weil. L'arithmétique sur les courbes algébriques. Acta Mathematica 52 (1929) 281–315. DOI 10.1007/BF02592688. https://doi.org/10.1007/BF02592688
2. L. J. Mordell. On the rational solutions of the indeterminate equations of the third and fourth degrees. Proceedings of the Cambridge Philosophical Society 21 (1922) 179–192. zbMATH JFM 48.0140.03. https://zbmath.org/48.0140.03
3. K. Mahler. Review of A. Weil, L'arithmétique sur les courbes algébriques. Jahrbuch für die Fortschritte der Mathematik 55 (1929). zbMATH JFM 55.0713.01. https://zbmath.org/55.0713.01
4. Gerd Faltings. Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Inventiones Mathematicae 73 (1983) 349–366. DOI 10.1007/BF01388432. https://doi.org/10.1007/BF01388432
5. Gerd Faltings. Diophantine approximation on abelian varieties. Annals of Mathematics 133 (1991) 549–576. DOI 10.2307/2944319. https://doi.org/10.2307/2944319
6. Heisuke Hironaka. Resolution of singularities of an algebraic variety over a field of characteristic zero. I. Annals of Mathematics 79 (1964) 109–203. DOI 10.2307/1970486. https://doi.org/10.2307/1970486
7. Heisuke Hironaka. Resolution of singularities of an algebraic variety over a field of characteristic zero. II. Annals of Mathematics 79 (1964) 205–326. DOI 10.2307/1970547. https://doi.org/10.2307/1970547
8. Michael McQuillan. Division points on semi-abelian varieties. Inventiones Mathematicae 120 (1995) 143–159. DOI 10.1007/BF01241125. https://doi.org/10.1007/BF01241125
9. Douglas Ulmer. Elliptic curves with large rank over function fields. Annals of Mathematics 155 (2002) 295–315. DOI 10.2307/3062158. https://doi.org/10.2307/3062158
10. Marc Hindry and Joseph H. Silverman. Diophantine Geometry: An Introduction. Graduate Texts in Mathematics 201. Springer, 2000. DOI 10.1007/978-1-4612-1210-2. https://doi.org/10.1007/978-1-4612-1210-2
11. Manjul Bhargava and Arul Shankar. Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves. Annals of Mathematics 181 (2015) 191–242. DOI 10.4007/annals.2015.181.1.3. https://doi.org/10.4007/annals.2015.181.1.3
12. Jennifer Park, Bjorn Poonen, John Voight, and Melanie Matchett Wood. A heuristic for boundedness of ranks of elliptic curves. Journal of the European Mathematical Society 21 (2019) 2859–2903. DOI 10.4171/JEMS/893. arXiv:1602.01431. https://arxiv.org/abs/1602.01431
13. Zev Klagsbrun, Travis Sherman, and James Weigandt. The Elkies curve has rank 28 subject only to GRH. Mathematics of Computation 88 (2019) 837–846. DOI 10.1090/mcom/3348. https://doi.org/10.1090/mcom/3348
14. Noam D. Elkies and Zev Klagsbrun. New rank records for elliptic curves having rational torsion. Open Book Series 4 (2020) 233–250. DOI 10.2140/obs.2020.4.233. https://doi.org/10.2140/obs.2020.4.233
15. Michael Stoll. Rational points on curves. Journal de Théorie des Nombres de Bordeaux 23 (2011) 257–277. DOI 10.5802/jtnb.760. arXiv:1008.1905. https://arxiv.org/abs/1008.1905
16. Jennifer S. Balakrishnan, Alex J. Best, Francesca Bianchi, Brian Lawrence, J. Steffen Müller, Nicholas Triantafillou, and Jan Vonk. Two recent p-adic approaches towards the (effective) Mordell conjecture. arXiv:1910.12755 (2019). https://arxiv.org/abs/1910.12755
17. David Zywina. On the infinitude of elliptic curves over a number field with prescribed small rank. arXiv:2602.10865 (2026). https://arxiv.org/abs/2602.10865
18. Dragos Ghioca, Fei Hu, Thomas Scanlon, and Umberto Zannier. A variant of the Mordell-Lang conjecture. Mathematical Research Letters 26 (2019) 1383–1392. DOI 10.4310/MRL.2019.v26.n5.a7. arXiv:1804.10561. https://arxiv.org/abs/1804.10561
19. Hideaki Ikoma, Shu Kawaguchi, and Atsushi Moriwaki. The Mordell Conjecture. Cambridge University Press, 2022. DOI 10.1017/9781108991445. https://doi.org/10.1017/9781108991445
20. Vincent Cossart and Olivier Piltant. Resolution of singularities of threefolds in positive characteristic. I. Journal of Algebra 320 (2008) 1051–1082. DOI 10.1016/j.jalgebra.2008.03.032. https://doi.org/10.1016/j.jalgebra.2008.03.032
21. Vincent Cossart and Olivier Piltant. Resolution of singularities of threefolds in positive characteristic. II. Journal of Algebra 321 (2009) 1836–1976. DOI 10.1016/j.jalgebra.2008.11.030. https://doi.org/10.1016/j.jalgebra.2008.11.030
22. A. Dujella. History of elliptic curves rank records. Curated web table, University of Zagreb. https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html
23. André Weil. L'arithmétique sur les courbes algébriques. Thèses de l'entre-deux-guerres, no. 95 (1928), 40 p. Open digitization (Numdam) of the thesis reprint of Acta Mathematica 52 (1929) 281–315. https://www.numdam.org/item/?id=THESE_1928__95__1_0