Complex Multiplication of Abelian Varieties and Its Applications to Number Theory

abelian variety algebraic number field analytic assertion assume automorphism basis characteristic choice CM-type coefficients coincides complex conjugate consider contained Conversely corresponding cycle defined defined and finite denote determined differential form dimension divisor dual easily verify easy equality equivalent exists an element extension fact field of definition finite follows functions Galois given gives Hence homomorphism identity imaginary implies independent integer invariant isomorphism l-adic Lemma linear matrix means modulo Moreover multiplication namely normalized Notations observe obtain p-variety polarized abelian varieties polynomials positive integer prime ideal primitive principal projection PROOF proper Proposition Proposition 13 proves rational rational mapping relation representation representatives respect result Riemann form ring satisfies shows simple space specialization subgroup Suppose Take Theorem theory totally positive type F uniquely values variety of type vectors virtue