Chapter 4 - Homological interpretation

Summary

In this chapter, we will give a homological interpretation of the theory of the special values of Dirichlet L-functions over Q and will reconstruct p-adic Dirichlet L-functions by a homological method (called the “modular symbol” method). A similar theory might exist for arbitrary fields, but here we restrict ourselves to Q. The modular symbol method was introduced by Mazur [MzS] in the context of modular forms on GL(2) as we will construct later, in §6.5, p-adic L-functions of modular forms (on GL(2)) by his original method. Basic facts from cohomology theory we use in this section are summarized in Appendix at the end of this book. We use standard notations for cohomology groups introduced in Appendix without further warning and quote, for example, Theorem 1 in the appendix as Theorem A.1. If the reader is not familiar with cohomology theory, it is better to have a look at Appendix before reading this chapter.

Cohomology groups on G_m(C)

We consider the space T = C/Z, which is isomorphic to G_m(C) via z ↦ e(z) = exp(2πiz). Thus T ≅ P¹(C)-{0.∞}, where P¹(C) is the projective line. We apply the theory developed in Appendix to X = P¹(C) and Y = T. With the notation of Proposition A.5, S is first {0,∞} and later will be μ_N∪{0,∞}, and So will be {0,∞}. We have π₁(T) ≅ Z. Let A be any commutative algebra.