Modular symbols and arithmetic

In these lectures, we will explore what insight can be gained into the arithmetic of Galois representations in a given dimension through the geometry of a higher-dimensional locally symmetric space near a boundary component. The starting point for such a comparison, and the one to which we will pay the most attention, is the study of p-parts of class groups (or better, K-groups) of integer rings of cyclotomic fields through the quotient of the homology of a modular curve by an Eisenstein ideal (which annihilates the cusp at infinity). We will describe two maps, one which has an explicit description in terms of modular and Steinberg symbols, and another in the opposite direction that is constructed from Galois representations attached to newforms congruent to Eisenstein series, as in Mazur-Wiles-type proofs of the Iwasawa main conjecture. These two maps are conjectural inverses, and Fukaya and Kato in particular proved a major result towards our conjecture. After a discussion of what is known in this original setting, we intend to turn our attention to potential generalizations and give an idea of our broader program of study, including joint work with Fukaya and Kato in the function field setting.