Serre's uniformity question: is the mod-$\ell$ Galois representation of every non-CM elliptic curve surjective for all primes $\ell$ beyond a constant depending only on the base field?

Serre's open image theorem (1972) shows that for an elliptic curve $E$ without complex multiplication over a number field $K$, the Galois representation on the torsion points of $E$ has open image in $\mathrm{GL}_2(\widehat{\mathbb{Z}})$; equivalently, the mod-$\ell$ representation $\bar\rho_{E,\ell}: \mathrm{Gal}(\bar K/K) \to \mathrm{GL}_2(\mathbb{F}_\ell)$ is surjective for all primes $\ell$ above a constant $c(E)$ depending on the curve. Serre asked whether $c(E)$ can be bounded independently of $E$: does there exist $N$, depending only on $K$, such that $\bar\rho_{E,\ell}$ is surjective for every non-CM $E/K$ and every prime $\ell > N$? Over $\mathbb{Q}$ the expected answer is $N = 37$; this constant is known to be optimal (non-CM curves over $\mathbb{Q}$ with a rational $37$-isogeny have non-surjective mod-$37$ image), so what remains open is whether $37$ suffices. This is Serre's uniformity problem, open since 1972. Over $\mathbb{Q}$ it has been reduced, through work of Mazur, Bilu–Parent, Zywina, Le Fourn–Lemos, and Furio–Lombardo, to a single case: excluding images contained in the normalizer of a non-split Cartan subgroup, i.e. proving that the modular curves $X_{\mathrm{ns}}^+(p)$ have no non-cuspidal, non-CM rational points for $p > 37$. Over general number fields even comparable reductions are unavailable.

Background

An elliptic curve $E$ over a number field $K$ (a finite extension of $\mathbb{Q}$) is a smooth projective curve of genus 1 with a distinguished $K$-rational point; over an algebraic closure it is a plane cubic, and its points form an abelian group. Fix an algebraic closure $\bar K$ of $K$. For a prime $\ell$, the $\ell$-torsion subgroup $E[\ell]$ of points annihilated by $\ell$ is a two-dimensional vector space over the finite field $\mathbb{F}_\ell$. The absolute Galois group $\mathrm{Gal}(\bar K/K)$ acts on $E[\ell]$, yielding the mod-$\ell$ Galois representation $\bar\rho_{E,\ell}: \mathrm{Gal}(\bar K/K) \to \mathrm{Aut}(E[\ell]) \cong \mathrm{GL}_2(\mathbb{F}_\ell)$. The curve has complex multiplication (CM) if its endomorphism ring over $\bar K$ is strictly larger than $\mathbb{Z}$; for CM curves the image of $\bar\rho_{E,\ell}$ is never all of $\mathrm{GL}_2(\mathbb{F}_\ell)$, so CM curves are excluded from the question.

Serre's open image theorem (1972) asserts that for non-CM $E$ the combined action on all torsion — the adelic representation $\rho_E: \mathrm{Gal}(\bar K/K) \to \mathrm{GL}_2(\widehat{\mathbb{Z}})$, where $\widehat{\mathbb{Z}}$ is the profinite completion of $\mathbb{Z}$ — has open image; equivalently there is a constant $c(E)$ such that $\bar\rho_{E,\ell}$ is surjective for every prime $\ell > c(E)$. Serre asked whether $c(E)$ can be bounded independently of the curve (1972, §4.3), and in the case $K = \mathbb{Q}$ suggested specifically that $c(E) \le 37$ should always hold (this is Question 2 closing §8.4 of his 1981 paper, at p. 399 of the Œuvres pagination). This is Serre's uniformity problem.

The classification of maximal subgroups of $\mathrm{GL}_2(\mathbb{F}_\ell)$ organizes the problem. If $\bar\rho_{E,\ell}$ is not surjective, its image lies in a proper maximal subgroup, of one of the following types: (a) a Borel subgroup (upper-triangular matrices), which is equivalent to $E$ admitting a $K$-rational $\ell$-isogeny — a non-constant homomorphism of elliptic curves with cyclic kernel of order $\ell$, defined over $K$; (b) the normalizer of a split Cartan subgroup (conjugate to the diagonal matrices, plus their permutations); (c) the normalizer of a non-split Cartan subgroup (the $\mathbb{F}_{\ell^2}$-multiplicative group embedded in $\mathrm{GL}_2(\mathbb{F}_\ell)$, together with the Frobenius involution); or (d) an exceptional subgroup whose projective image is $A_4$, $S_4$, or $A_5$, excluded by Serre for $\ell > 13$. Each case is governed by rational points on a modular curve: the curve $X_H$ parameterizing elliptic curves with level structure $H$ has a $K$-rational point of non-cuspidal, non-CM type exactly when some non-CM curve realizes the corresponding small image; the finitely many points of $X_H$ not corresponding to smooth elliptic curves are the cusps. Ruling out case (c) for a prime $p$ over $\mathbb{Q}$ is equivalent to showing that $X_{\mathrm{ns}}^+(p)(\mathbb{Q})$ consists only of cusps and CM points.

Over $\mathbb{Q}$ the state of the art is: Mazur (1978) proved that a non-CM elliptic curve has no rational $\ell$-isogeny for $\ell > 37$, closing case (a). Bilu–Parent (2011) closed case (b) for all sufficiently large $p$, using a variant of Runge's method — a technique that bounds rational points on a curve by comparing the growth of auxiliary functions at the cusps, and which requires at least two of them; Bilu–Parent–Rebolledo (2013) made the range effective, $p \ge 11$ with $p \ne 13$, via rational points on $X_0^+(p^r)$, and the remaining prime $p = 13$ was settled by Balakrishnan–Dogra–Müller–Tuitman–Vonk (2019) using explicit quadratic Chabauty–Kim methods. Zywina's dichotomy, sharpened by Le Fourn–Lemos (2021) and Furio–Lombardo (2023), shows that for $p > 37$ the image is either $\mathrm{GL}(E[p])$ or contained in the normalizer of a non-split Cartan. Case (c) is the sole remaining obstacle. The genus of $X_{\mathrm{ns}}^+(p)$ grows quadratically in $p$, and the curve has a single cusp, so Runge's method — decisive in the split case — provably does not apply; Chabauty-type methods require control of the Mordell–Weil rank (the rank of the finitely generated group of rational points) of the Jacobian $J_{\mathrm{ns}}^+(p)$ (the abelian variety attached to the curve), and no such control is available for general $p$.

Per-curve effectivity is well understood: Masser–Wüstholz (1993) gave the first unconditional effective bound on $c(E)$; Serre's 1981 paper proves under the Generalized Riemann Hypothesis (GRH) a bound in terms of the conductor $N_E$ of $E$ (the integer measuring its bad reduction); Zywina (2011) and Lombardo (2015, over number fields, via the Faltings height — a measure of the arithmetic complexity of the curve) bound the adelic index $[\mathrm{GL}_2(\widehat{\mathbb{Z}}):\mathrm{Im}\,\rho_E]$; Mayle–Wang (2024) prove under GRH an explicit bound linear in $\log N_E$; Furio (2024) gives unconditional explicit bounds and Chen–Swidinsky (2026) improve the GRH bounds. Every one of these depends on invariants of the individual curve; none is uniform in $E$. For comparison, the torsion analogue of uniformity — Merel's theorem that torsion is uniformly bounded over number fields of bounded degree — is a theorem, but it concerns rational torsion points and does not control images. For a single non-CM curve over $\mathbb{Q}$, the full adelic image is in fact computable (Zywina, arXiv:2206.14959), but the algorithm's termination on all inputs is itself conditional on a positive answer to this problem.

Problem Statement

Let $K$ be a number field. Does there exist a constant $N$, depending only on $K$, such that for every elliptic curve $E$ over $K$ without complex multiplication and every prime $\ell > N$, the mod-$\ell$ Galois representation $\bar\rho_{E,\ell}: \mathrm{Gal}(\bar K/K) \to \mathrm{Aut}(E[\ell]) \cong \mathrm{GL}_2(\mathbb{F}_\ell)$, describing the action of the absolute Galois group of $K$ on the $\ell$-torsion points of $E$, is surjective? In the strongest expected form over $K = \mathbb{Q}$: is $\bar\rho_{E,\ell}$ surjective for every non-CM $E/\mathbb{Q}$ and every prime $\ell > 37$? (Here $37$ is already known to be the optimal constant: non-CM curves over $\mathbb{Q}$ admitting a rational $37$-isogeny have non-surjective mod-$37$ image, so only the sufficiency of $37$ is open.) By the known reductions, over $\mathbb{Q}$ this is exactly equivalent to: for every prime $p > 37$, the modular curve $X_{\mathrm{ns}}^+(p)$ — the coarse moduli space of elliptic curves whose mod-$p$ Galois image is contained in the normalizer of a non-split Cartan subgroup — has no $\mathbb{Q}$-rational points other than cusps and points corresponding to curves with complex multiplication. The task is resolved by a complete, correct answer — a proof or a disproof — to either of the two questions posed above, each taken at its own scope. For the general question over an arbitrary number field $K$: a proof exhibits a constant $N = N(K)$, depending only on $K$, such that $\bar\rho_{E,\ell}$ is surjective for every non-CM $E/K$ and every prime $\ell > N$; a disproof exhibits one fixed number field $K$ and, for every $N$, a non-CM curve $E/K$ and a prime $\ell > N$ with non-surjective $\bar\rho_{E,\ell}$. For the sharp question over $K = \mathbb{Q}$: a proof establishes surjectivity for every non-CM $E/\mathbb{Q}$ and every prime $\ell > 37$; a disproof is a single non-CM curve $E/\mathbb{Q}$ and a prime $\ell > 37$ with non-surjective mod-$\ell$ image. A bound that is neither the general constant $N(K)$ nor the sharp constant $37$ over $\mathbb{Q}$ — for instance, an explicit $N$ for one number field other than $37$ — is a partial result and does not by itself resolve the task. Any unproven hypothesis invoked (such as GRH) must be stated explicitly, and the answer is then the correspondingly conditional resolution of the question.

References

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