Constructing
CM elliptic
curves with
minimal
Galois image
Riccardo
Pengo
4
Galois images of CM elliptic curves
Fix a number field F, and an elliptic curve E/F such that EndF (E) ∼
= O ⊆ K ⊆ F, where O is an order inside an
imaginary quadratic field K. Then ρE (GF ) ⊆ AutO (Etors) ∼
= b
O×
, and I (E/F) := |AutO (Etors): b
O×
| is finite.
Campagna & P. (2021) I (E/F) = [F ∩Kab : HO ]·(|O×
|/[F(Etors): FKab]), where HO = K(j(E)) is the ring
class field of O. Moreover, for any finite L ⊇ F such that F(Etors) = LKab, one has:
I (E/F) =
|O×
|·[L∩Kab : K]
|Pic(O)|·[L: F]
and this allows us to compute I (E/F). Indeed, such an L always exists, and one can take L = F(E[I]) for any
ideal I ⊆ O such that |Z/(I ∩Z)| > max(2,|O×
|/2). This gives an algorithm to compute I (E/F).
Challenge: Implement this algorithm completely in PARI/GP.
Rouse, Sutherland & Zureick-Brown (2021): If F = Q, there is an algorithm, more efficient than
Brau-Avila’s, which computes ρE,ℓ∞ (GF ), for any prime ℓ ∈ N, using work of Lozano-Robledo (2018).
This algorithm works also without CM, is implemented in MAGMA, and was applied to the 238764310 elliptic
curves appearing in the database by Balakrishnan, Ho, Kaplan, Spicer, Stein & Weigandt (2016).