GaloisRepFamily.lean`

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2 declarations, 0 contain sorry, 0 declared as axioms

Compatible families of Galois representations

We define GaloisRepFamily K E d: a collection of d-dimensional E_λ-adic Galois representations indexed by the finite places λ of a number field E, together with the notion of compatibility.

Imports (3)

Imported by 2 files

def GaloisRepFamily Documented declaration: GaloisRepFamily line 38 source
Source documentation
GaloisRepFamily K E d is an (unrelated) collection of d-dimensional p-adic Galois representations of the absolute Galois group of the field K, parametrised by field maps from the number field E into the algebraic closure of ℚ_p as p runs through the primes.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, NumberField, Nat
Proof uses: Fact, Nat.Prime, RingHom, AlgebraicClosure, Padic, GaloisRep, PadicAlgCl, NNReal, Fin
20 instances, coercions and auxiliary constants left out.
```
def GaloisRepFamily (K : Type*) [Field K]
    (E : Type*) [Field E] [NumberField E] (d : ℕ) : Type _ :=
…
```
def GaloisRepFamily.isCompatible Documented declaration: isCompatible line 58 source
Source documentation
A family ρ_λ of E_λ-adic Galois representations GaloisRepFamily K E d of the absolute Galois group of a number field K is compatible if there is a finite set S of finite places of K and, for each finite place v of K not in S, a monic degree d polynomial P_v with coefficients in E, such that if v and λ do not lie above the same rational prime then ρ_λ is unramified at v and P_v is the characteristic polynomial of ρ_λ(Frob_v), where Frob_v denotes an arithmetic Frobenius element.
This is the weakest possible concept of a compatible family but it will suffice for our needs.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, NumberField, Nat, GaloisRepFamily
Proof uses: Exists, Finset, IsDedekindDomain.HeightOneSpectrum, NumberField.RingOfIntegers, Polynomial, Fact, Nat.Prime, RingHom, AlgebraicClosure, Padic, Not, Membership.mem, Ideal, IsDedekindDomain.HeightOneSpectrum.asIdeal, And, GaloisRep.IsUnramifiedAt, PadicAlgCl, NNReal, Fin, Eq and 8 more
38 instances, coercions and auxiliary constants left out.
```
def GaloisRepFamily.isCompatible {K : Type*} [Field K] [NumberField K]
    {E : Type*} [Field E] [NumberField E] {d : ℕ} (ρ : GaloisRepFamily K E d) : Prop :=
…
```
Mentions: GaloisRepFamily