Mock modular form

In mathematics, a mock modular form is the holomorphic part of a harmonic weak Maass form, and a mock theta function is essentially a mock modular form of weight 1/2. The first examples of mock theta functions were described by Srinivasa Ramanujan in his last 1920 letter to G. H. Hardy and in his lost notebook. Sander Zwegers (2001, 2002) discovered that adding certain non-holomorphic functions to them turns them into harmonic weak Maass forms.

Ramanujan's 12 January 1920 letter to Hardy, reprinted in (Ramanujan 2000, Appendix II), listed 17 examples of functions that he called mock theta functions, and his lost notebook (Ramanujan 1988) contained several more examples. (Ramanujan used the term "theta function" for what today would be called a modular form.) Ramanujan pointed out that they have an asymptotic expansion at the cusps, similar to that of modular forms of weight 1/2, possibly with poles at cusps, but cannot be expressed in terms of "ordinary" theta functions. He called functions with similar properties "mock theta functions". Zwegers later discovered the connection of the mock theta function with weak Maass forms.

Ramanujan associated an order to his mock theta functions, which was not clearly defined. Before the work of Zwegers, the orders of known mock theta functions included

Ramanujan's notion of order later turned out to correspond to the conductor of the Nebentypus character of the weight  ¹⁄₂ harmonic Maass forms which admit Ramanujan's mock theta functions as their holomorphic projections.

In the next few decades, Ramanujan's mock theta functions were studied by Watson, Andrews, Selberg, Hickerson, Choi, McIntosh, and others, who proved Ramanujan's statements about them and found several more examples and identities. (Most of the "new" identities and examples were already known to Ramanujan and reappeared in his lost notebook.) Watson (1936) found that under the action of elements of the modular group, the order 3 mock theta functions almost transform like modular forms of weight 1/2 (multiplied by suitable powers of q), except that there are "error terms" in the functional equations, usually given as explicit integrals. However for many years there was no good definition of a mock theta function. This changed in 2001 when Zwegers discovered the relation with non-holomorphic modular forms, Lerch sums, and indefinite theta series. Zwegers (2002) showed, using the previous work of Watson and Andrews, that the mock theta functions of orders 3, 5, and 7 can be written as the sum of a weak Maass form of weight  ¹⁄₂ and a function that is bounded along geodesics ending at cusps. The weak Maass form has eigenvalue 3/16 under the hyperbolic Laplacian (the same value as holomorphic modular forms of weight  ¹⁄₂); however, it increases exponentially fast near cusps, so it does not satisfy the usual growth condition for Maass wave forms. Zwegers proved this result in three different ways, by relating the mock theta functions to Hecke's theta functions of indefinite lattices of dimension 2, and to Appell–Lerch sums, and to meromorphic Jacobi forms.

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