Serre’s uniformity conjecture

Certain smooth cubic equations have a rule for adding points: a point P may satisfy P+P+P=0. Arithmetic symmetries permute these division points while respecting addition. Above one common prime cutoff, do they realize every such invertible rearrangement? The question concerns elliptic curves with rational coefficients and without complex multiplication, an extra algebraic symmetry. The proposed cutoff is 37.[1] [2]

An elliptic curve’s division points carry a finite additive structure on which arithmetic symmetries act. Can a restricted action persist at a prime above 37 for a rational curve lacking extra endomorphisms? For every non-CM $E/\mathbb Q$ and prime $p>37$, the target is $\rho_{E,p}(\operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q))=\mathrm{GL}_2(\mathbb F_p)$, where $\rho_{E,p}$ describes the action on $E[p]$. This mod-$p$ assertion with explicit cutoff 37 is stronger than the existence of some universal $C$; it excludes CM curves.[1] [2]

For each allowed curve, only finitely many primes give a restricted action. What is missing is the same upper limit 37 for every curve. Finding one prime at most 7 with a full action does not exclude a later exceptional prime.[1] [2]

The May 2026 theorem gives a smallest ℓ-adically surjective prime ≤7, leaving the largest-exception target unresolved.[1] [2]

Arithmetic symmetries reveal which division points can be exchanged while preserving rational equations and addition. A common cutoff would confine all exceptional actions to a fixed list of small primes, even as the curve varies. That gives the classification of division-point symmetries a shared range of possible prime-level exceptions, even when a new curve is considered.[1] [2]

Exceptional images translate an arithmetic symmetry restriction into a rational-point problem on a modular curve. Uniformity would bound the prime levels where such exceptions can arise independently of the original curve. This is part of the wider classification of rational elliptic curves by Galois images. The cited finite parameterization is conditional on further effective conjectures, so that motivation does not turn the cutoff 37 question into an already proved classification.[1] [2]