GENERALIZED EXPLICIT RECIPROCITY LAWS
0.1. In this paper, we give generalizations of the explicit reciprocity law to p adically complete discrete valuation fields whose residue fields are not necessarily perfect. The main theorems are 4.3.1, 4.3.4, and 6.1.9. In subsequent papers, we will apply our generalized explicit reciprocity laws to p adic completions of the function fields of modular curves, whose residue fields are function fields in one variable over finite fields and hence imperfect, and will study the Iwasawa theory of modular forms. I outline in 0.3 what is done in this paper in the case of p adic completions of the function fields of modular curves, comparing it with the case of classical explicit reciprocity law outlined in 0.2. I then explain in 0.4 how the results of this paper will be applied in the Iwasawa theory of modular forms, for these applications are the motivations of this paper, comparing it with the role of the classical explicit reciprocity laws in the classical Iwasawa theory and in related known theories.