SerreUniformity.lean`

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4 declarations, 2 contain sorry, 0 declared as axioms

Serre'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

Imports (1)

FormalConjecturesUtil

def SerreUniformity.cmJInvariants The thirteen rational CM $j$-invariants line 48 source
Source documentation
The thirteen rational CM $j$-invariants.

standard axioms: Quot.sound, propext what this means
Statement uses: Finset, Rat
Proof uses: Insert.insert, OfNat.ofNat, Neg.neg, Singleton.singleton
5 instances, coercions and auxiliary constants left out.
```
def cmJInvariants : Finset ℚ :=
…
```
def SerreUniformity.HasFullTorsionAction Every additive automorphism of $E[p]$ is induced by a Galois automorphism line 57 source
Source documentation
Every 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.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: WeierstrassCurve, Rat, Nat
Proof uses: AddSubgroup, WeierstrassCurve.Affine.Point, AlgebraicClosure, WeierstrassCurve.toAffine, WeierstrassCurve.baseChange, Classical.propDecidable, Eq, AddSubgroup.torsionBy, Int, AddEquiv, Subtype, Membership.mem, Exists, AlgEquiv, WeierstrassCurve.Affine.baseChange, AddMonoidHom, WeierstrassCurve.Affine.Point.map
31 instances, coercions and auxiliary constants left out.
```
def HasFullTorsionAction (E : WeierstrassCurve ℚ) (p : ℕ) : Prop :=
…
```
theorem SerreUniformity.serre_uniformity Serre's uniformity question over sorry line 68 source
Source documentation
Serre'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 $p > C$?

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, WeierstrassCurve, Rat, WeierstrassCurve.IsElliptic, Not, Membership.mem, Finset, SerreUniformity.cmJInvariants, WeierstrassCurve.j, Nat.Prime, LT.lt, SerreUniformity.HasFullTorsionAction
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem serre_uniformity :
    answer(sorry) ↔
      ∃ C : ℕ, ∀ (E : WeierstrassCurve ℚ) [E.IsElliptic], E.j ∉ cmJInvariants →
        ∀ p : ℕ, p.Prime → C < p → HasFullTorsionAction E p := by
…
```
Mentions: SerreUniformity.cmJInvariants, SerreUniformity.HasFullTorsionAction
theorem SerreUniformity.serre_uniformity.variants.bound_37 Serre's uniformity conjecture over sorry line 82 source
Source documentation
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, $p_K = 37$."

depends on sorry what this means
Statement uses: Iff, True, WeierstrassCurve, Rat, WeierstrassCurve.IsElliptic, Not, Membership.mem, Finset, SerreUniformity.cmJInvariants, WeierstrassCurve.j, Nat, Nat.Prime, LT.lt, OfNat.ofNat, SerreUniformity.HasFullTorsionAction
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
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 := by
…
```
Mentions: SerreUniformity.cmJInvariants, SerreUniformity.HasFullTorsionAction