74 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION

1 in which p is not split. Let π = ±pζ when p is inert in Z, and let π generate the

unique prime over p in Z when p is ramified in Z, so π is a Weil

p2-integer

and

Z = Q(π). (Note that π = ±2

√

−1 = (1 ±

√

−1)2

since 2 is ramified in Q(

√

−1).)

By Corollary 1.6.2.2(3) (also see Example 1.6.2.4), a simple abelian variety

E0 over κ with

p2-Frobenius

equal to π must have endomorphism algebra Z and

dimension 1. The elliptic curve E0 is supersingular because p is not split in Z. The

isogeny class of E0 contains no member that is the scalar extension of an elliptic

curve over Fp, as otherwise π would have a square root π0 ∈ Z, which is visibly

absurd by inspection since π = ±2

√

−1.

The abelian surface A0 := Resκ/Fp (E0) satisfies (A0)κ E0 ×

E0p), (

so (A0)κ is

not simple. But A0 is simple, as otherwise there would be a non-zero homomorphism

E0 → A0 from an elliptic curve E0 over Fp and hence (by the universal property of

Weil restriction) a non-zero homomorphism (E0)κ → E0, contrary to what we just

saw concerning the isogeny class of E0. (Note that (A0)κ is isotypic, since

E0p) (

is

isogenous to E0 via the relative Frobenius morphism E0 →

E0p).)(

Taking K/Q to be a quadratic field in which p is inert, we can lift E0 over

OK,(p) to get an elliptic curve E over K having good reduction E0 at pOK. Then

A := ResK/Q(E) is an abelian surface over Q having good reduction Resκ/Fp (E0)

at p that is simple over Fp, so (via consideration of N´ eron models over Z(p)) A

is simple over Q. However, AK E × E where E is the twist σ∗(E) by the

non-trivial automorphism σ of K over Q, so AK is not simple.

1.6.4. Example. Pushing the end of Example 1.6.3 further over Q, we now prove

that if π = ±pζ and p ≡ −1 (mod 4) with

ζ2

+ 1 = 0 (resp. p ≡ −1 (mod 3) with

ζ2

+ ζ + 1 = 0) then E and E over K are not isogenous (so AK is not isotypic, in

contrast with its reduction (A0)κ). Suppose that there were an isogeny ψ : E → E ,

and choose it with minimal degree. In particular, ψ is not divisible by [p]E. We

claim that ordp(deg ψ) is odd (and in particular, is positive). Suppose otherwise,

so deg ψ = mp2n with n 0 and p m. Consider the reduction ψ0 : E0 →

E0p)(

of ψ, also an isogeny with degree

mp2n.

In particular, ker(ψ0) is a finite subgroup

scheme of E0 with order

mp2n,

so its p-part has order

p2n.

But E0 is supersingular,

so it has a unique subgroup scheme of each p-power order. Hence, the p-part of

ker(ψ0) is E0[pn], so ψ0 = ψ0 ◦ [pn]E0 with ψ0 : E0 →

E0p) (

of degree m.

Consider the composite isogeny

E0

ψ0

→

E0p)→E0p2)

( (

= E0

using the Frobenius isogeny of

E0p). (

This is an endomorphism of E0 with degree

pm. Since End(E0) is an order in Z[ζ] on which the degree is computed as the

norm to Z, we get an element of Z[ζ] whose norm in Z is divisible exactly once by

p. That is impossible since p is prime in Z[ζ], and so completes the verification that

deg ψ has p-part

pj

for some odd j.

We conclude that the finite K-subgroup N := ker(ψ) ⊂ E has non-trivial p-

part, and this p-part has cyclic geometric fiber (as otherwise it would contain E[p],

contradicting that we arranged ψ to not be divisible by [p]E). By cyclicity, N[p]

is a K-subgroup of E with order p. Consider its scheme-theoretic closure G in the

N´ eron model of E at pOK,(p). This is a finite flat group scheme over OK,(p) of

order p, and its special fiber Gκ is an order-p subgroup scheme of the supersingular