Constructing
CM elliptic
curves with
minimal
Galois image
Riccardo
Pengo
6
CM elliptic curves with minimal Galois image
Fix a number field F and an elliptic curve E/F such that EndF (E) ∼
= O ⊆ K ⊆ F.
Another good choice for a finite extension L ⊇ F such that F(Etors) = LKab, is given by L = F( u
p
α), where
u = |O×
| and α ∈ F is such that E is the twist by u
p
α of an elliptic curve E′
/F
such that F(E′
tors) = FKab.
Campagna & P. (2020) For any O such that ∆O ̸∈ {−4f 2 | f = pa1
1 ···par
r ,p1 ≡ ··· ≡ pr ≡ 1(4)}, there are
infinitely many non-isomorphic elliptic curves E′
/HO
such that HO (E′
tors) = Kab.
The previous theorem gives an algorithm to construct such an E′
. More precisely (if ∆O < −4 and K ̸= Q(i)):
• take any elliptic curve E/HO
such that EndHO
(E) ∼
= O, e.g. E = ellfromj(j(O));
• if HO (E[3]) ⊆ Kab, then I (E/HO ) = 2, and we can take E′
= E;
• if HO (E[3]) ̸⊆ Kab, continue as follows:
• take any (e.g. the smallest) prime p ∈ N which splits in K, is inert in Q(i), and such that p ∤ fO ·NHO /Q(fE );
• find α ∈ HO such that HO (E[p]) = Hp,O (
p
α), where p ·O = pp and Hp,O is the p-th ray class field of O;
• take E′
= E(α).
Challenge: Implement this algorithm completely in PARI/GP.
Theoretical challenge: Find A,B ∈ Q(u,v) such that E′
: y2 = x3 +A(j(O),∆O )·x +B(j(O),∆O ) for each O.