The mock modular forms introduced by Ramanujan (1920) in his last letter to Hardy remained an un-explained oddity for many decades. While they share some formal properties with usual modular forms, in particular, they are given by a power series in a parameter q, they lack the standard transformation law under the modular group. In his thesis, Zwegers (Utrecht, 2002) showed that Ramanujan's mock theta functions can be completed into true (but non-holomorphic) modular forms by adding a suitable non-holomorphic series. Thus, it turns out that the mock modular forms are, in fact, the holomorphic parts of certain harmonic weak Maass forms. The theory of mock modular forms and, more generally, the related (or unrelated) harmonic weak Maass forms has been a very active research area in the ensuing period. Significant contributions by Zwegers, Ono, Bringmann, Fosum, Bruinier, and many others have clarified the previously mysterious nature of Ramanujan's mock modular forms and revealed many new phenomena, in particular, deep connections with arithmetic.
One aspect of the theory is the notion of the `shadow' of a harmonic weak Maass form f, the (often) holomorphic modular form of complementary weight that arises as the complex conjugate of the image of f under the ‾∂, or lowering, operator. Such an operation, known as the ξ-operator, was introduced by Bruinier and Funke, who used it in an essential way to connect the Borcherds and classical theta lifts. From the point of view of representation theory, such a pair consisting of a harmonic weak Maass form and its shadow correspond to an extension of Harish-Chandra modules occurring in the space of modular forms. Such non-trivial extensions have been little studied, especially for higher rank groups, even though they are, in fact, ubiquitous! Indeed the traditional focus of the Langlands program has been on the irreducible constituents that arise in the space of square integrable functions on arithmetic quotients and their relations with Galois representations and pure motives. Yet non-semisimple Galois representations (arising, say, as extensions of one irreducible Galois representation by another) are also of great number-theoretic significance. For example the classes in Ext1(Vp(E); Qp) arising from rational points on an elliptic curve E, where Vp(E) is the p-adic Galois representation arising from the Tate module of E and Qp is the trivial p-adic representation, are at the center of the Birch and Swinnerton-Dyer conjecture. One might naturally wish to extend the Langlands philosophy to better understand non-semisimple Galois representations. The rich structures exhibited by weak harmonic Maass forms (notably, the fact that they can generate non-semisimple Harish-Chandra modules over the relevant Hecke algebras) might be viewed as the first faint intimations that such a theory is possible.
Another strong motivation for the further study of weak harmonic Maass forms and their relatives is that, just as in the case of classical modular forms, their Fourier expansions appear to contain beautiful number theoretic information. For example, the Fourier coeficients of the holomorphic part of Zagier's weight 3/2 Eisenstein series are the Gauss class numbers of binary quadratic forms. A tantalising result of Bruinier and Ono shows that the Fourier coeficients of a mock modular form of weight 1/2 having as shadow the classical weight 3/2 modular form mapping to a classical form of weight 2 (attached to an elliptic curve E, say) under the Shimura lift encode the properties of a collection of Heegner points on a varying collection of quadratic twists of E. The Fourier coe cients of the holomorphic part of the central derivative of an incoherent weight 1 Eisenstein series introduced by Kudla, Rapoport and Yang are the arithmetic degrees of certain 0-cycles on the moduli stack of CM elliptic curves. Similar properties have been observed, in independent work by Duke-Li, Ehlen, and Viazovska, for weak harmonic Maass forms of weight 1 having a cuspidal weight 1 form g as their shadow: the associated Fourier coeficients in this case being expressed as logarithms of algebraic numbers lying in the eld cut out by the adjoint of the two-dimensional Artin representation attached to g.
There are - for the moment, extremely fragmentary - indications that analogous phenomena might arise in the p-adic setting. Results of Darmon-Tornaria, Longo-Nicole, Venkat, and others show that the Fourier coeficients of certain non-classical p-adic modular forms of weight 3/2 arising from derivatives of p-adic families encode the p-adic logarithms of essentially the same collection of Heegner points arising in the theorem of Bruinier-Ono. A more recent work of Darmon, Lauder and Rotger has produced a p-adic analogue of the work of Duke-Li, Ehlen, Viazovska et al., where the associated Fourier coeficients are shown to be the p-adic logarithms of algebraic numbers in ring class fields of real quadratic fields. Such results seem to cry out for a larger conceptual framework. Laying the foundations for such a theory could provide a fertile meeting ground for the number theory groups in Toronto and Montreal. The general desire for a p-adic theory of mock modular forms was part of the motivation behind Luca Candelori's McGill PhD thesis, and his recent collaboration with Francesc Castella (another recent McGill graduate) develops similar themes. They currently hold post-doctoral appointments in Baton Rouge and Princeton respectively, but both are being invited to take part in this year's Montreal-Toronto workshop.