Complex Multiplication of Abelian Varieties and Its Applications to Number Theory

a-multiplication a-transform A₁ ê abstract variety algebraic closure algebraic number field algebraic set analytic representation automorphism characteristic class-field coefficients coincides complex conjugate complex multiplication consider contained coordinate-system corresponding defined and finite denote differential form divisor dual easily verify element of G exists an element exists an isomorphism field of definition field of moduli follows functions H₁ Hence homomorphism ideal-class ideal-group integral ideal invariant differential forms isogeny isomorphism k₁ Kummer variety l-adic leaves invariant Lemma Let F matrix modulo non-degenerate Notations obtain p-complete p-finite p-simple p-variety P₁ Picard variety polarized abelian varieties polynomials positive integer prime ideal Proposition 13 rational mapping reduction modulo relation respect Riemann form separably algebraic subfield Suppose t₁ totally imaginary totally real type F U₁ V₁ variety defined variety of dimension variety of type vectors W₁ zeta-function