Products of CM elliptic curves - Universität Duisburg-Essen

If H is any finite subgroup scheme of E, then its reduction H k is a subgroup schemeof the same rank, i.e. |H k | = |H|, and we have(58)r k (I(H)) ⊂ I(H k ) and r 0 k(E(H)) ⊂ E(H k ).Furthermore, both inclusions are equalities if H is an ideal subgroup or if (p, |H|) = 1.In addition, we have(59)H(I) k = H(r k (I)), for all non-zero ideals I of End(E).Proof. (a) Lang[18], Theorem 13.12 (p. 182).(b) We first recall how the reduction map r k is defined. Let O p = (O K ) p denotethe local ring (valuation ring) at p, and let Ẽ/O p denote the minimal (Néron) modelof E/K (with respect to p). Then r k is defined as the composition of the base-changemapsr k = rkE ∼: End(E) ← End(Ẽ) → End(E k).Now let H be a finite subgroup scheme of E. To define H k , we shall use thehypothesis that E/K has good reduction at p, i.e. that Ẽ/S := Spec(O p) is an abelianscheme (of relative dimension 1). Then E H ∼ E also has good reduction at p (bythe criterion of Néron-Ogg-Shafarevich or otherwise), and so π H extends by the Néronproperty to a homomorphism ˜π H : Ẽ → ẼH. By [4], 7.3/6, ˜π H is an isogeny. Moreover,since π H is proper (because ẼH/S is proper), it follows that ˜π H is finite and flat by(the argument of) [4], 7.3/2, and hence ˜H := Ker(˜π H ) is a subgroup scheme of Ẽwhich is finite and flat over S and whose generic fibre is H. Thus, its special fibreH k = ˜H × S Spec(k) is a finite subgroup scheme of E k of the same rank as H, i.e.|H| = |H k |. Note also that the above shows that (E k ) Hk is the reduction of E H .As a special case of this construction we see that(60)Ker(g) k = Ker(˜g) × S Spec(k) = Ker(r k (g)), for all g ∈ End(E).From this, equation (59) follows immediately because intersections (i.e. fibre products)commute with base-change.To prove the first part of (58), let h ∈ I(H), so h = gπ H with g ∈ Hom(E H , E)and hence ˜h = ˜g˜π H and r k (h) = ˜h k = ˜g k (˜π H ) k = g k π Hk ∈ I(H k ) because g k ∈Hom((E H ) k , E k ) (and (E H ) k = (E k ) Hk ). This proves the first inclusion of (58).The second inclusion follows immediately from the fact that(61)r 0 k(Φ H (h)) = Φ Hk (r E Hk(h)), for all h ∈ End(E H ),which in turn follows from the functoriality of the construction of r k , for we have thatn H rk 0(Φ H)(h) = r k (π H ′ hπ H) = (π H ′ ) kh k (π H ) k = π H ′ kr E H(h)π Hk = n Hk Φ Hk (r E Hk(h)).Note that it follows from (61) that(62)r 0 k(E(H)) = E(H k ) ⇔ p ∤ f E(H) = f EH21