By Conrad Oort Chai
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Proof. 1, without loss of generality we may replace A with an isogenous abelian variety. 2 the abelian variety A over K admits complex multiplication by a CM field L. Let Φ be the resulting CM type on L. Letting Q denote the algebraic closure of Q in K, we may view Φ as a Q-valued CM type on L. 1 (applied over the algebraically closed base field Q) there is a CM abelian variety B over Q with complex multiplication by L and CM type Φ (viewed as valued in Q). 1 over K we see that BK is L-linearly isogenous to A over K (as these two abelian varieties over K are endowed with complex multiplication by L yielding the same CM type Φ on L).
Example. Suppose that O has discriminant not divisible by p (so O(p) = OL,(p) ) and that p is totally inert in L. In such cases κ := O/(p) is a finite field of degree 2g over Fp and Aut(L/Q) injects into Gal(κ/Fp ), so the canonical complex conjugation on L induces a non-trivial involution on κ. For an algebraically closed extension F/K we can consider the eigenspace decomposition of T ⊗K F over κ ⊗Fp F = ϕ Fϕ where ϕ ranges over the 2g distinct embeddings of κ into F . 4) there may be conjugate pairs occurring among the ϕ for which T ⊗K F has a non-zero ϕ-eigenspace with respect to its F -linear κ-action.
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. 4. Example. 3 further over Q, we now prove that if p ≡ 3 (mod 4) (resp. p ≡ 2 mod 3) then E and E are not isogenous, so AK is not isotypic (in contrast with its reduction Aκ ). Suppose that there were an isogeny ψ : E → E , and choose it with minimal degree. In particular, ψ is not divisible by [p]E . 5. ABELIAN VARIETIES OVER FINITE FIELDS 35 particular, is positive). Suppose otherwise, so deg ψ = mp2n with n 0 and p m.