Overconvergent Modular Symbols

The theory of overconvergent modular symbols was created by Glenn Stevens over 20 years ago, and since then the subject has had many generalizations and applications central to modern number theory (e.g. overconvergent cohomology, eigenvarieties of reductive groups, families of p-adic L-functions, just to name a few). In these notes, rather than give a systematic development of the general theory, we aim to convey the idea that spaces of overconvergent modular symbols (despite being infinite dimensional!) are fairly concrete and can be computed quite explicitly. We do not provide many formal proofs in this article, but instead only sketch arguments, perhaps only in specific cases, and challenge the reader to work out the details. We attempt to keep the tone of these notes quite informal in part to mimic the tone of the corresponding lectures series and in part to keep the barrier to entry to the theory as low as possible. The structure of these notes is as follows: in the next section, we give a detailed discussion of classical modular symbols and L-values, and work through a computation with modular symbols of level 11. In the third section, we discuss overconvergent modular symbols, their connection to classical modular symbols (i.e. Stevens’ control theorem), and their relation to p-adic L-functions. In the fourth section, we discuss how to approximate p-adic distributions which leads to a method of computing overconvergent modular symbols as well as a proof of the control theorem. In the fifth section, we return to the connection between p-adic L-functions and overconvergent modular symbols, and in the final section, we close with some numerical examples of overconvergent eigensymbols of level 11.