Common use of Proposition 1 Clause in Contracts

Proposition 1. 4.2. (Birkenhake–▇▇▇▇▇ [6, Proposition 13.3.1] and ▇▇▇▇ [20, Theorem 1.4.1-(iii)]) With the notation above, the complex torus AΦ,m is an abelian variety and has a natural CM structure given by the action of OK on m. In this thesis, we use the complex torus Cg/Φ˜(m) instead of AΦ,m as a realization of an abelian variety over C conforming to the notation of Lang [20] and ▇▇▇▇▇▇▇–Taniyama [40]. For each α ∈ K, we let SΦ(α) be the matrix diag(φ1(α), . . . , φg (α)). Theorem 1.4.3. (Lang [20, Theorem 1.4.1-(ii)]) Let (K, Φ) be a CM pair and let (A, θ) be an abelian variety of type (K, Φ) with CM by K. Then there is a fractional ideal m I and an analytic isomorphism ι : Cg/Φ˜(m) → A(C) such that the diagram Cg/Φ˜(m) A(C) SΦ(α) θ(α) Cg/Φ˜(m) A(C) commutes for all α ∈ OK. Definition 1.4.4. We say that an abelian variety (A, θ) of type (K, Φ) is of type (K, Φ, m) if there is a fractional ideal m I and an analytic isomorphism ι : Cg/Φ˜(m) → A(C). Definition 1.4.5. Let (A, θ) be an abelian variety of type (K, Φ) and let X be an ample divisor on A. We say that (A, θ) is Φ-admissible with respect to the polarization ϕX if θ(K) is stable under the ▇▇▇▇▇▇ involution. Theorem 1.4.6. (Lang [20, Theorem 1.4.5-(iii)] If an abelian variety (A, θ) of type (K, Φ, m) is simple, then it is Φ-admissible with respect to every polarization.