Mock modular form

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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 discovered that adding certain non-holomorphic functions to them turns them into harmonic weak Maass forms.^[1][2]

History[]

Ramanujan's 12 January 1920 letter to Hardy^[3] listed 17 examples of functions that he called mock theta functions, and his lost notebook^[4] 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

3, 5, 6, 7, 8, 10.

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.) In 1936, Watson 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.^[5] 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 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.^[2] 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.

Zwegers's fundamental result shows that mock theta functions are the "holomorphic parts" of real analytic modular forms of weight 1/2. This allows one to extend many results about modular forms to mock theta functions. In particular, like modular forms, mock theta functions all lie in certain explicit finite-dimensional spaces, which reduces the long and hard proofs of many identities between them to routine linear algebra. For the first time it became possible to produce infinite number of examples of mock theta functions; before this work there were only about 50 examples known (most of which were first found by Ramanujan). As further applications of Zwegers's ideas, Kathrin Bringmann and Ken Ono showed that certain q-series arising from the Rogers–Fine basic hypergeometric series are related to holomorphic parts of weight 3/2 harmonic weak Maass forms^[6] and showed that the asymptotic series for coefficients of the order 3 mock theta function f(q) studied by George Andrews^[7] and Leila Dragonette^[8] converges to the coefficients.^[9] In particular Mock theta functions have asymptotic expansions at cusps of the modular group, acting on the upper half-plane, that resemble those of modular forms of weight 1/2 with poles at the cusps.

Definition[]

A mock modular form will be defined as the "holomorphic part" of a harmonic weak Maass form.

Fix a weight k, usually with 2k integral. Fix a subgroup Γ of SL₂(Z) (or of the metaplectic group if k is half-integral) and a character ρ of Γ. A modular form f for this character and this group Γ transforms under elements of Γ by

${\displaystyle f\left({\frac {a\tau +b}{c\tau +d}}\right)=\rho {\begin{pmatrix}a&b\\c&d\end{pmatrix}}(c\tau +d)^{k}f(\tau )}$

A weak Maass form of weight k is a continuous function on the upper half plane that transforms like a modular form of weight k and is an eigenfunction of the weight k Laplacian operator, and is called harmonic if its eigenvalue is (1 − k/2)k/2.^[10] This is the eigenvalue of holomorphic weight k modular forms, so these are all examples of harmonic weak Maass forms. (A Maass form is a weak Maass form that decreases rapidly at cusps.) So a harmonic weak Maass form is annihilated by the differential operator

${\displaystyle {\frac {\partial }{\partial \tau }}y^{k}{\frac {\partial }{\partial {\overline {\tau }}}}}$

If F is any harmonic weak Maass form then the function g given by

${\displaystyle g=y^{k}{\frac {\partial {\overline {F}}}{\partial \tau }}=\sum _{n}b_{n}q^{n}}$

is holomorphic and transforms like a modular form of weight k, though it may not be holomorphic at cusps. If we can find any other function g^* with the same image g, then F − g^* will be holomorphic. Such a function is given by inverting the differential operator by integration; for example we can define

${\displaystyle g^{*}(\tau )=\left({\frac {i}{2}}\right)^{k-1}\int _{-{\overline {\tau }}}^{i\infty }(z+\tau )^{-k}{\overline {g(-{\overline {z}})}}\,dz=\sum _{n}n^{k-1}{\overline {b_{n}}}\beta _{k}(4ny)q^{-n+1}}$

where

${\displaystyle \displaystyle \beta _{k}(t)=\int _{t}^{\infty }u^{-k}e^{-\pi u}\,du}$

is essentially the incomplete gamma function. The integral converges whenever g has a zero at the cusp i∞, and the incomplete gamma function can be extended by analytic continuation, so this formula can be used to define the holomorphic part g^* of F even in the case when g is meromorphic at i∞, though this requires some care if k is 1 or not integral or if n = 0. The inverse of the differential operator is far from unique as we can add any homomorphic function to g^* without affecting its image, and as a result the function g^* need not be invariant under the group Γ. The function h = F − g^* is called the holomorphic part of F.

A mock modular form is defined to be the holomorphic part h of some harmonic weak Maass form F. So there is an isomorphism from the space of mock modular forms h to a subspace of the harmonic weak Maass forms.

The mock modular form h is holomorphic but not quite modular, while h + g^* is modular but not quite holomorphic. The space of mock modular forms of weight k contains the space of nearly modular forms ("modular forms that may be meromorphic at cusps") of weight k as a subspace. The quotient is (antilinearly) isomorphic to the space of holomorphic modular forms of weight 2 − k. The weight-(2 − k) modular form g corresponding to a mock modular form h is called its shadow. It is quite common for different mock theta functions to have the same shadow. For example, the 10 mock theta functions of order 5 found by Ramanujan fall into two groups of 5, where all the functions in each group have the same shadow (up to multiplication by a constant).

Don Zagier^[11] defines a mock theta function as a rational power of q = e^2πi