Investigations in the arithmetic of cyclotomic fields and modular curves

Class groups of cyclotomic fields have interested number theorists since the 19th century. In the final quarter of the 20th century, Ribet and Mazur-Wiles demonstrated that aspects of their structure can be ascertained from the study of modular representations. The goal of this talk is to give a modular interpretation of particular elements in these class groups that arise as values of a cup product pairing on cyclotomic units. These pairing values yield information on a wealth of algebraic objects, but any analytic interpretation of them had been heretofore quite mysterious. We will describe how, conjecturally, modular representations can be used to relate the pairing values to p-adic L-values of cusp forms.