Watkins's conjecture: divisibility of the modular degree of an elliptic curve by the power of two attached to its Mordell–Weil rank

Watkins's conjecture (2002) predicts that for every elliptic curve $E$ defined over $\mathbb{Q}$, the modular degree $m_E$ — the minimal degree of a surjective morphism from the modular curve $X_0(N_E)$ to $E$ — is divisible by $2^{\mathrm{rank}\,E(\mathbb{Q})}$, where $\mathrm{rank}\,E(\mathbb{Q})$ is the Mordell–Weil rank of $E$. Equivalently, $\mathrm{rank}\,E(\mathbb{Q})\le\nu_2(m_E)$, with $\nu_2$ the 2-adic valuation. The conjecture has been verified on all computed tables, holds in many special classes (rank 1, prime conductor up to rank 2, prescribed torsion or reduction type, suitable quadratic twists), and is supported by Dummigan's symmetric-square Selmer-group heuristic, but it remains open for curves of general conductor and rank $\ge 2$. The problem asks for a proof for all elliptic curves over $\mathbb{Q}$, or an explicit counterexample.

Background

An elliptic curve $E$ over $\mathbb{Q}$ is a smooth projective curve of genus one with a specified rational point; its arithmetic is governed by two a priori unrelated invariants. The first is algebraic: by the Mordell–Weil theorem the group $E(\mathbb{Q})$ of rational points is a finitely generated abelian group, $E(\mathbb{Q})\cong\mathbb{Z}^{r}\oplus E(\mathbb{Q})_{\mathrm{tors}}$, and the nonnegative integer $r$ is the Mordell–Weil rank. The second is modular: by the modularity theorem (Wiles, Taylor–Wiles, Breuil–Conrad–Diamond–Taylor), there exists a surjective morphism $\phi_E:X_0(N_E)\to E$ defined over $\mathbb{Q}$, where $N_E$ is the conductor of $E$ (an integer encoding the primes and types of bad reduction) and $X_0(N)$ is the modular curve that classifies pairs consisting of an elliptic curve together with a cyclic subgroup of order $N$. The modular degree $m_E$ is the minimum of the degrees of all such modular parametrizations. Watkins's conjecture connects the two invariants: writing $\nu_2(n)$ for the exponent of the largest power of $2$ dividing $n$, it asserts that $2^{r}$ divides $m_E$, that is, $r\le\nu_2(m_E)$, for every elliptic curve $E$ over $\mathbb{Q}$.

The conjecture originated in computations: Watkins (2002), while developing algorithms for computing modular degrees, observed that $2^{r}$ divided $m_E$ for every curve in the then-current Cremona database and proposed the divisibility as a general conjecture. It has survived all subsequent computation, including the tables now contained in the L-functions and Modular Forms Database (LMFDB); no counterexample is known.

The structural reason the conjecture is plausible is the link between the modular degree and congruence primes. Modularity attaches to $E$ a weight-2 cusp newform $f_E$ of level $N_E$; the congruence number $r_E$ measures congruences modulo primes between $f_E$ and other cusp forms (more precisely, it is defined through the integral structure on the space of cusp forms, and it detects when $f_E$ is congruent to a form orthogonal to it). Ribet proved that $m_E$ divides $r_E$; Agashe–Ribet–Stein showed that $\nu_p(r_E)=\nu_p(m_E)$ whenever $p$ divides the conductor to order at most one, and conjectured the bound $\nu_p(r_E/m_E)\le\tfrac12\nu_p(N_E)$ on the quotient. Rank should then be visible through congruences of $f_E$ modulo powers of $2$. Dummigan (2006) turned this into a heuristic proof for a restricted class of curves: the divisibility would follow from an isomorphism $R\cong T$ between a certain Selmer-group-controlled deformation ring and a Hecke algebra for the symmetric-square representation of $E$; however, he also showed that the Taylor–Wiles patching method, the standard tool for proving such isomorphisms, is necessarily inapplicable in this setting, so genuinely new ideas appear to be required. Dummigan–Krishnamoorthy (2013) proved lower bounds of the shape $\nu_2(m_E)\ge\omega(N_E)-3$ (where $\omega(n)$ counts the distinct prime factors of $n$), showing that $\nu_2(m_E)$ is unbounded across curves, consistent with the conjecture in a world of unbounded ranks.

Many special cases are known. Calegari–Emerton (2009) proved that a curve of odd modular degree has global root number $+1$ (the sign in the functional equation of the Hasse–Weil $L$-function of $E$) and even analytic rank, with strong restrictions on the conductor; combined with the parity theorem (the statement that the root number equals $(-1)^r$, now a theorem for elliptic curves over $\mathbb{Q}$), this yields the conjecture for all curves of rank 1. Yazdani (2011) obtained the analogous statements for odd congruence numbers. Kazalicki–Kohen (2018) settled prime-conductor curves of rank 0 with odd congruence number and, conditional on a parity prediction for the coefficients of the Gross vector (a modular eigenvector on the supersingular locus, computable via Brandt matrices over a quaternion order), the rank-1 prime-conductor case; that parity prediction was proved in August 2026 by Kazalicki–Slijepčević, who thereby established the conjecture for all prime-conductor curves of rank at most 2. Further cases come from prescribed structure: curves with non-split multiplicative reduction in suitable torsion configurations (Caro–Pasten), quadratic twists $E^{(D)}$ with $\omega(D)$ large (Esparza-Lozano–Pasten), quadratic twists of curves with rational $2$-torsion and prime-power conductor and a function-field analogue over $\mathbb{F}_q(T)$ (Caro), semistable curves with full rational $2$-torsion (Bhakta–Krishnamoorthy–Pasupulati), thin families with rational $\ell$-torsion (Bhakta–Krishnamoorthy), and curves with additive reduction at $2$, good reduction outside at most two odd primes, and a rational point of order $2$ (Hatley–Kundu). Under the Birch and Swenson–Dyer conjecture (BSD) and the generalized Riemann hypothesis (GRH), the conjecture holds for almost all quadratic twists of a fixed curve with trivial rational $2$-torsion. For curves of general conductor and rank $\ge 2$ carrying no such auxiliary structure, no unconditional result is known.

Problem Statement

Prove or disprove Watkins's conjecture: for every elliptic curve $E$ defined over $\mathbb{Q}$, with Mordell–Weil rank $r=\mathrm{rank}_{\mathbb{Z}}E(\mathbb{Q})$ and modular degree $m_E$ (the minimal degree of a surjective $\mathbb{Q}$-rational morphism $X_0(N_E)\to E$ from the modular curve of level equal to the conductor of $E$), the power $2^{r}$ divides $m_E$; equivalently, $r\le\nu_2(m_E)$. The statement is quantified over all elliptic curves over $\mathbb{Q}$: no restriction on conductor, rank, torsion, reduction type, or membership in any special family is admissible. The conjecture is asserted unconditionally: an argument conditional on BSD, GRH, the parity conjecture, or an $R\cong T$ hypothesis does not by itself resolve it. Because the conjecture is a single universal statement, exhibiting one elliptic curve over $\mathbb{Q}$ with $\nu_2(m_E)<r$ would refute it.

References

1. Mark Watkins, "Computing the modular degree of an elliptic curve", Experimental Mathematics 11 (2002), no. 4, 487–502. DOI: 10.1080/10586458.2002.10504701, https://doi.org/10.1080/10586458.2002.10504701
2. Subham Bhakta, Srilakshmi Krishnamoorthy, Sunil Kumar Pasupulati, "Modular degree and a conjecture of Watkins", arXiv:2308.13708 (2023). DOI: 10.48550/arXiv.2308.13708, https://arxiv.org/abs/2308.13708
3. Neil Dummigan, "On a conjecture of Watkins", Journal de Théorie des Nombres de Bordeaux 18 (2006), no. 2, 345–355. DOI: 10.5802/jtnb.548, https://doi.org/10.5802/jtnb.548
4. Frank Calegari, Matthew Emerton, "Elliptic curves of odd modular degree", Israel Journal of Mathematics 169 (2009), no. 1, 417–444. DOI: 10.1007/s11856-009-0017-x, https://doi.org/10.1007/s11856-009-0017-x
5. Soroosh Yazdani, "Modular abelian varieties of odd modular degree", Algebra & Number Theory 5 (2011), no. 1, 37–62. DOI: 10.2140/ant.2011.5.37, https://doi.org/10.2140/ant.2011.5.37
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7. Matija Kazalicki, Daniel Kohen, "On a special case of Watkins' conjecture", Proceedings of the American Mathematical Society 146 (2018), no. 2, 541–545. DOI: 10.1090/proc/13759, https://doi.org/10.1090/proc/13759
8. Matija Kazalicki, Daniel Kohen, "Corrigendum to "On a special case of Watkins' conjecture"", Proceedings of the American Mathematical Society 147 (2019), no. 10, 4563. DOI: 10.1090/proc/14456, https://doi.org/10.1090/proc/14456
9. Jose Esparza-Lozano, Hector Pasten, "A conjecture of Watkins for quadratic twists", Proceedings of the American Mathematical Society 149 (2021), no. 6, 2381–2385. DOI: 10.1090/proc/15376, https://doi.org/10.1090/proc/15376
10. Jerson Caro, "Watkins's conjecture for quadratic twists of Elliptic Curves with Prime Power Conductor", arXiv:2206.10008 (2022). https://arxiv.org/abs/2206.10008
11. Jerson Caro, "Watkins' conjecture for elliptic curves over function fields", arXiv:2203.10932 (2022). https://arxiv.org/abs/2203.10932
12. Jerson Caro, Hector Pasten, "Watkins's conjecture for elliptic curves with non-split multiplicative reduction", Proceedings of the American Mathematical Society 150 (2022), no. 8, 3245–3251. DOI: 10.1090/proc/15942, https://doi.org/10.1090/proc/15942
13. Subham Bhakta, Srilakshmi Krishnamoorthy, "Watkins's conjecture for elliptic curves with a rational torsion", arXiv:2407.17680 (2024). https://arxiv.org/abs/2407.17680
14. Jeffrey Hatley, Debanjana Kundu, "Elliptic curves of conductor $2^m p$, quadratic twists, and Watkins' conjecture", Annales mathématiques du Québec 50 (2025), no. 1, 143–165. DOI: 10.1007/s40316-025-00249-8, https://doi.org/10.1007/s40316-025-00249-8 (preprint arXiv:2411.08321, https://arxiv.org/abs/2411.08321)
15. Matija Kazalicki, Siniša Slijepčević, "Gross vectors modulo 2 and elliptic curves of prime conductor", arXiv:2608.13371 (2026). https://arxiv.org/abs/2608.13371
16. Amod Agashe, Kenneth A. Ribet, William A. Stein, "The modular degree, congruence primes, and multiplicity one", in: Number Theory, Analysis and Geometry, Springer, 2012, pp. 19–49. DOI: 10.1007/978-1-4614-1260-1_2, https://doi.org/10.1007/978-1-4614-1260-1_2