Abstract Modular Symbols

Abstract

In this chapter, we introduce the fundamental notion of modular symbols with values in a system of coefficients V , an abelian group with an action of a submonoid of \(\mathrm {GL}_2(\mathbb {Q})\). Modular symbols will be used throughout the rest of this book, with increasingly sophisticated systems of coefficients V —first in the next chapter, Chap. 5, with V being a symmetric power of the standard representation of GL2, and then in the subsequent chapters, with V being various spaces of modules of distributions. Modular symbols with this kind of coefficients have strong connections respectively with classical modular forms, and with families of p-adic modular forms and their p-adic L-functions.