Luca Candelori and I are organizing a weekly seminar on modular forms, here in the mathematics department at WSU, for the 2018-2019 academic year (and perhaps, if it maintains enough energy, it might continue into the future too). One major goal of the seminar is to cover enough background from number theory and topology for us to tackle two open, probably rather difficult, questions: 1. whether one can give K-theoretic or homotopy-theoretic descriptions of special values of L-functions of eigenvalue 1/4 Maass cusp forms, and 2. whether there exist spectral enrichments of geometric models for Maass forms (of some fixed level and eigenvalue), i.e., whether there exist “topological Maass forms,” especially in the harmonic case and the eigenvalue 1/4 case.

The seminar is 3 PM to 4 PM on Wednesdays in the Nelson library, room 1146 in the Faculty/Administration Building. All are welcome to attend.

We would like to make available fairly complete notes from the seminar which provide both the number-theoretic content that readers with a background in homotopy theory are less likely to already know, and the homotopy-theoretic content that readers with a background in number theory are less likely to already know. The notes compiled so far are available below.

Lecture 1: Candelori on the Riemann zeta-function

Lecture 2: Salch on special values of zeta-functions

Lecture 3: Candelori on theta functions and modular forms

Lecture 4: Candelori on q-expansions and Waring’s problem

Lecture 5: Candelori on geometric models for modular forms

Lecture 6: Salch on moduli and stacks

Lecture 7: Salch on formal groups, supersingularity, and L-functions