Final Lecture: the Proof of the Galois Twisting Form and the Algebraic Form of the Main Theorem of Cm

Recall the premise of the previous lecture. We considered (A, i,Φ) an abelian variety with CM structure, defined over Q, which (with all the relevant data) descended to A0/K where K is a large enough finite Galois extension of E. For an element σ ∈ Gal(Q/E) the Galois twist (A0 , i,Φ) is another CM abelian variety of the same type, and Jeremy and Brandon constructed an explicit isogeny ξσ,P : A0 → A0 , together with an L-isomorphism θσ,P : NΦ(p) ⊗OL A0 ' A0 that made the following diagram commute: