Constructing
CM elliptic
curves with
minimal
Galois image
Riccardo
Pengo
3
Computing the Galois image of a non-CM elliptic curve
Fix a number field F with algebraic closure F, and an elliptic curve E/F such that EndF (E) ∼
= Z.
Set Etors := E(F)tors and E[m] := E(F)[m] for any m ∈ N. We have the Galois representation:
ρE : GF := Gal(F/F) → AutZ(Etors) ∼
= GL2(b
Z)
putting together the ℓ-adic ones ρE,ℓ∞ : GF → AutZ(E[ℓ∞
]) ∼
= GL2(Zℓ), where E[ℓ∞
] := lim
−
→n∈N
E[ℓn].
Serre (1971) The index I (E/F) := |AutZ(Etors): ρE (GF )| is finite.
How can we make this theorem effective?
Lombardo (2015) I (E/F) < exp(1.9·1010)·(dF ·max
©
1,h(E),log(dF )
ª
)
12395
, where dF := [F : Q] and h(E)
denotes the stable Faltings height of E, computed in PARI/GP by ellheight(E).
Zywina (2015) The set {I (E/Q): E/Q} ⊆ N is finite, if we assume Serre’s uniformity conjecture.
Brau-Avila (2015) There is a very slow, deterministic algorithm which computes an integer m ≥ 2 such that
ρE (GF ) = π−1
m (ρE,m(GF )), where πm : AutZ(Etors) ↠ AutZ(E[m]) is the reduction map, and ρE,m := πm ◦ρE .
Moreover, there is another algorithm which computes ρE,m(GF ), and thus can be used to compute I (E/F).