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module
public import FormalConjecturesUtilSerre's uniformity conjecture over the rationals
Is there a bound $C$, independent of the non-CM elliptic curve $E/\mathbb{Q}$, such that the Galois action on $E[p]$ is surjective onto $\mathrm{GL}_2(\mathbb{F}_p)$ for every prime $p > C$? This is the $\mathbb{Q}$ case of Serre's question [Ser72], recalled in the introduction of [Lem17].
We express surjectivity without choosing a basis: every additive automorphism of $E[p]$ must be induced by an element of $G_{\mathbb{Q}}$. For prime $p$, these automorphisms are exactly the $\mathbb{F}_p$-linear automorphisms. The non-CM condition uses the classification of rational CM $j$-invariants.
References:
[Ser72] J.-P. Serre, Propriétés galoisiennes des points d'ordre fini des courbes elliptiques. Inventiones Mathematicae 15 (1972), 259–331. https://doi.org/10.1007/BF01405086
[Lem17] P. Lemos, Serre's uniformity conjecture for elliptic curves with rational cyclic isogenies, introduction. https://arxiv.org/abs/1702.01985
@[expose] public sectionnamespace SerreUniformityThe thirteen rational CM $j$-invariants.
def cmJInvariants : Finset ℚ :=
{0, 1728, -3375, 8000, -32768, 54000, 287496, -884736,
-12288000, 16581375, -884736000, -147197952000, -262537412640768000}open scoped Classical inEvery additive automorphism of $E[p]$ is induced by a Galois automorphism. For elliptic $E$ and prime $p$, this means that the mod-$p$ representation is surjective.
def HasFullTorsionAction (E : WeierstrassCurve ℚ) (p : ℕ) : Prop :=
let T := AddSubgroup.torsionBy (E.baseChange (AlgebraicClosure ℚ)).toAffine.Point (p : ℤ)
∀ f : T ≃+ T, ∃ σ : AlgebraicClosure ℚ ≃ₐ[ℚ] AlgebraicClosure ℚ,
∀ P : T, WeierstrassCurve.Affine.Point.map (W' := E) σ.toAlgHom P.val = (f P).valSerre's uniformity question over $\mathbb{Q}$ [Ser72, Lem17]: is there a bound $C$ such that every non-CM elliptic curve over $\mathbb{Q}$ has surjective mod-$p$ Galois representation for every prime $p > C$?
@[category research open, AMS 11 14]
theorem serre_uniformity :
answer(sorry) ↔
∃ C : ℕ, ∀ (E : WeierstrassCurve ℚ) [E.IsElliptic], E.j ∉ cmJInvariants →
∀ p : ℕ, p.Prime → C < p → HasFullTorsionAction E p := ⊢ True ↔
∃ C,
∀ (E : WeierstrassCurve ℚ) [inst : E.IsElliptic],
E.j ∉ cmJInvariants → ∀ (p : ℕ), Nat.Prime p → C < p → HasFullTorsionAction E p
All goals completed! 🐙Serre's uniformity conjecture over $\mathbb{Q}$, explicit form: the bound $C = 37$ works, i.e. every non-CM elliptic curve over $\mathbb{Q}$ has surjective mod-$p$ Galois representation for every prime $p > 37$. From the introduction of [Lem17]: "This conjecture remains open today, but, over the last forty years, there has been a lot of progress towards a proof for $K = \mathbb{Q}$ — it is believed that, in this case, $p_K = 37$."
@[category research open, AMS 11 14]
theorem serre_uniformity.variants.bound_37 :
answer(sorry) ↔
∀ (E : WeierstrassCurve ℚ) [E.IsElliptic], E.j ∉ cmJInvariants →
∀ p : ℕ, p.Prime → 37 < p → HasFullTorsionAction E p := ⊢ True ↔
∀ (E : WeierstrassCurve ℚ) [inst : E.IsElliptic],
E.j ∉ cmJInvariants → ∀ (p : ℕ), Nat.Prime p → 37 < p → HasFullTorsionAction E p
All goals completed! 🐙end SerreUniformity