Period relation for the lifting of modular forms and transcendental cycles

Abstract

In this section, we discuss the period relation of Riemann (Proposition 14.5) for real Nebentype elliptic modular cusp forms of weight 2. More precisely speaking, we formulate the period relation of Riemann for the Hodge structures attached real Nebentype primitive forms of weight 2. Because the category of polarized Hodge structures of weight 1 is equivalent to the opposite category of polarized abelian varieties, the formalism attaching Hodge structures of weight 1 to real Nebentype elliptic primitive cusp forms of weight 2 follows immediately from the theory of Shimura [43]. Therefore we omit some of details and proofs of this formalism. About basic facts and definition, we refer to [43].