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. Ramanujan's own definition of mock theta functions is notoriously vague, and it was an open problem for many years to find a better definition. This was finally solved by Sander Zwegers (2001, 2002), who discovered that adding certain nonholomorphic functions to them turns them into harmonic weak Maass forms.
Contents
History
"Suppose there is a function in the Eulerian form and suppose that all or an infinity of points are exponential singularities, and also suppose that at these points the asymptotic form closes as neatly as in the cases of (A) and (B). The question is: Is the function taken the sum of two functions one of which is an ordinary θfunction and the other a (trivial) function which is O(1) at all the points e^{2mπi/n}? ... When it is not so, I call the function a Mock θfunction."
Ramanujan's original definition of a mock theta function, from (Ramanujan 2000, Appendix II)Ramanujan's 1920 Jan 12 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. In a notoriously obscure definition, he called functions with similar properties "mock theta functions". No better definition was found for many years, until Zwegers discovered the connection 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.) 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 nonholomorphic 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.
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 numbers 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 qseries arising from the Rogers–Fine basic hypergeometric series are related to holomorphic parts of weight 3/2 harmonic weak Maass forms (Bringmann & Ono 200?), related mock theta functions to Galois representations (Bringmann & Ono 200?), and showed that the asymptotic series for coefficients of the order 3 mock theta function f(q) studied by of (Andrews 1966) and Dragonette (1952) converges to the coefficients (Bringmann & Ono 2006). In particular Mock theta functions have asymptotic expansions at cusps of the modular group, acting on the upper halfplane, 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 SL_{2}(Z) (or of the metaplectic group if k is halfintegral) and a character ρ of Γ. A modular form f for this character and this group Γ transforms under elements of Γ by
A weak Maass form of weight k is a continuous function on the upper half plane that transforms like a modular form of weight 2 − k and is an eigenfunction of the weight k Laplacian operator, and is called harmonic if its eigenvalue is (1 − k/2)k/2. 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
If F is any harmonic weak Maass form then the function g given by
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
where
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).
Zagier (2007) defines a mock theta function as a rational power of q = e^{2πiτ} times a mock modular form of weight 1/2 whose shadow is a theta series of the form
for a positive rational κ and an odd periodic function ε. (Any such theta series is a modular form of weight 3/2). The rational power of q is a historical accident.
Most mock modular forms and weak Maass forms have rapid growth at cusps. It is common to impose the condition that they grow at most exponentially fast at cusps (which for mock modular forms means they are "meromorphic" at cusps). The space of mock modular forms (of given weight and group) whose growth is bounded by some fixed exponential function at cusps is finitedimensional.
Appell–Lerch sums
Appell–Lerch sums were first studied by Paul Émile Appell (1884) and Mathias Lerch (1892). Watson studied the order 3 mock theta functions by expressing them in terms of Appell–Lerch sums, and Zwegers used them to show that mock theta functions are essentially mock modular forms.
The Appell–Lerch series is
where
and
The modified series
where
and y = Im(τ) and
satisfies the following transformation properties
In other words the modified Appell–Lerch series transforms like a modular form with respect to τ. Since mock theta functions can be expressed in terms of Appell–Lerch series this means that mock theta functions transform like modular forms if they have a certain nonanalytic series added to them.
Indefinite theta series
Andrews (1986) showed that several of Ramanujan’s fifth order mock theta functions are equal to quotients Θ(τ)/θ(τ) where θ(τ) is a modular form of weight 1/2 and Θ(τ) is a theta function of an indefinite binary quadratic form, and Hickerson (1988b) proved similar results for seventh order mock theta functions. Zwegers showed how to complete the indefinite theta functions to produce real analytic modular forms, and used this to give another proof of the relation between mock theta functions and weak Maass wave forms.
Meromorphic Jacobi forms
Andrews (1988) observed that some of Ramanujan's fifth order mock theta functions could be expressed in terms of quotients of Jacobi's theta functions. Zwegers used this idea to express mock theta functions as Fourier coefficients of meromorphic Jacobi forms.
Applications
 Lawrence & Zagier (1999) related mock theta functions to quantum invariants of 3manifolds.
 Semikhatov, Taormina & Tipunin (2005) related mock theta functions to infinite dimensional Lie superalgebras and conformal field theory.
 Troost (2010) showed that the modular completions of mock modular forms arise as elliptic genera of conformal field theories with continuous spectrum.
Examples
 Any modular form of weight k (possibly only meromorphic at cusps) is a mock modular form of weight k with shadow 0.
 The quasimodular Eisenstein series

 of weight 2 and level 1 is a mock modular form of weight 2, with shadow a constant. This means that
 transforms like a modular form of weight 2 (where τ = x + iy).
 The function studied by Zagier (1975) (Hirzebruch & Zagier 1976, 2.2) with Fourier coefficients that are Hurwitz class numbers H(N) of imaginary quadratic fields is a mock modular form of weight 3/2, level 4 and shadow ∑ q^{ n2}. The corresponding weak Maass wave form is

 where
 and y = Im(τ), q = e^{2πiτ}.
Mock theta functions are mock modular forms of weight 1/2 whose shadow is a unary theta function, multiplied by a rational power of q (for historical reasons). Before the work of Zwegers led to a general method for constructing them, most examples were given as basic hypergeometric functions, but this is largely a historical accident, and most mock theta functions have no known simple expression in terms of such functions.
The "trivial" mock theta functions are the (holomorphic) modular forms of weight 1/2, which were classified by Serre & Stark (1977), who showed that they could all be written in terms of theta functions of 1dimensional lattices.
The following examples use the qPochhammer symbols (a;q)_{n} which are defined as:
Order 2
Some order 2 mock theta functions were studied by (McIntosh 2007).
The function μ was found by Ramanujan in his lost notebook.
These are related to the functions listed in the section on order 8 functions by
 U_{0}(q) − 2U_{1}(q) = μ(q)
 V_{0}(q) − V_{0}( − q) = 4qB(q^{2})
 V_{1}(q) + V_{1}( − q) = 2A(q^{2})
Order 3
Ramanujan mentioned four order3 mock theta functions in his letter to Hardy, and listed a further three in his lost notebook, which were rediscovered by G. N. Watson. Watson (1936) proved the relations between them stated by Ramanujan and also found their transformations under elements of the modular group by expressing them as Appel–Lerch sums. Dragonette (1952) described the asymptotic expansion of their coefficients. Zwegers (2000) related them to harmonic weak Maass forms. See also (Fine 1988) The seven order3 mock theta functions given by Ramanujan are
 (sequence A053251 in OEIS).
 (sequence A053252 in OEIS).
 (sequence A053253 in OEIS).
 (sequence A053254 in OEIS).
 (sequence A053255 in OEIS).
The first 4 of these form a group with the same shadow (up to a constant), and so do the last three. More precisely, the functions satisfy the following relations (found by Ramanujan and proved by Watson):
Order 5
Ramanujan wrote down ten mock theta functions of order 5 in his 1920 letter to Hardy, and stated some relations between them that were proved by Watson (1937). In his lost notebook he stated some further identities relating these functions, equivalent to the mock theta conjectures (Andrews & Garvan 1989), that were proved by Hickerson (1988). Andrews (1986) found representations of many of these functions as the quotient of an indefinite theta series by modular forms of weight 1/2.
 (sequence A053256 in OEIS)
 (sequence A053257 in OEIS)
 (sequence A053258 in OEIS)
 (sequence A053259 in OEIS)
 (sequence A053260 in OEIS)
 (sequence A053261 in OEIS)
 (sequence A053262 in OEIS)
 (sequence A053263 in OEIS)
 (sequence A053264 in OEIS)
 (sequence A053265 in OEIS)
 (sequence A053266 in OEIS)
 (sequence A053267 in OEIS)
Order 6
Ramanujan (1988) wrote down seven mock theta functions of order 6 in his lost notebook, and stated 11 identities between them, which were proved in (Andrews & Hickerson 1991). Two of Ramanujan's identities relate φ and ψ at various arguments, four of them express φ and ψ in terms of Appell–Lerch series, and the last five identities express the remaining 5 sixth order mock theta functions in terms of φ and ψ. Berndt & Chan (2007) discovered two more sixth order functions. The order 6 mock theta functions are:
 (sequence A053268 in OEIS)
 (sequence A053269 in OEIS)
 (sequence A053270 in OEIS)
 (sequence A053271 in OEIS)
 (sequence A053272 in OEIS)
 (sequence A053273 in OEIS)
 (sequence A053274 in OEIS)
 (sequence A153251 in OEIS)
 (sequence A153252 in OEIS)
Order 7
Ramanujan gave three mock theta functions of order 7 in his 1920 letter to Hardy. They were studied by Selberg (1938), who found asymptotic expansion for their coefficients, and in (Andrews 1986). Hickerson (1988) found representations of many of these functions as the quotients of indefinite theta series by modular forms of weight 1/2. Zwegers (2001, 2002) described their modular transformation properties.
These three mock theta functions have different shadows, so unlike the case of Ramanujans order 3 and order 5 functions, there are no linear relations between them and ordinary modular forms. The corresponding weak Maass forms are
where
and
is more or less the complementary error function. Under the metaplectic group, these 3 functions transform according to a certain 3dimensional representation of the metaplectic group as follows
 M_{1}(τ + 1) = e ^{− 2πi / 168}M_{1}(τ), ,
In other words, they are the components of a level 1 vectorvalued harmonic weak Maass form of weight 1/2.
Order 8
Gordon & McIntosh (2000) found eight mock theta functions of order 8. They found 5 linear relations involving them, and expressed 4 of the functions as Appell–Lerch sums, and described their transformations under the modular group. The two functions V_{1} and U_{0} were found earlier by Ramanujan (1988, p. 8, eqn 1; p. 29 eqn 6) in his lost notebook.
 (sequence A153148 in OEIS)
 (sequence A153149 in OEIS)
 (sequence A153155 in OEIS)
 (sequence A153156 in OEIS)
 (sequence A153172 in OEIS)
 (sequence A153174 in OEIS)
 (sequence A153176 in OEIS)
 (sequence A153178 in OEIS)
Order 10
Ramanujan (1988, p. 9) listed four order10 mock theta functions in his lost notebook, and stated some relations between them, which were proved by Choi (1999, 2000, 2002, 2007).
 (sequence A053281 in OEIS)
 (sequence A053282 in OEIS)
 (sequence A053283 in OEIS)
 (sequence A053284 in OEIS)
References
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 Appell, P. (1884), "Sur les fonctions doublement périodiques de troisième espèce", Annales scientifiques de l'École Normale Supérieure Sér. 3, 1: 135–164, http://www.numdam.org/item?id=ASENS_1884_3_1__135_0
 Berndt, Bruce C.; Chan, Song Heng (2007), "Sixth order mock theta functions", Advances in Mathematics 216 (2): 771–786, doi:10.1016/j.aim.2007.06.004, ISSN 00018708, MR2351377
 Bringmann, Kathrin; Ono, Ken (2007), "Lifting cusp forms to Maass forms with an application to partitions", Proceedings of the National Academy of Sciences of the United States of America 104 (10): 3725–3731, doi:10.1073/pnas.0611414104, ISSN 00278424, MR2301875, PMC 1820651, PMID 17360420, http://www.math.wisc.edu/~ono/reprints/106.pdf
 Bringmann, Kathrin; Ono, Ken (2006), "The f(q) mock theta function conjecture and partition ranks", Inventiones Mathematicae 165 (2): 243–266, doi:10.1007/s0022200504935, ISSN 00209910, MR2231957, http://www.math.wisc.edu/~ono/reprints/098.pdf
 Bringmann, Kathrin; Ono, Ken, "Dyson's ranks and Maass forms", Annals of Mathematics, http://www.math.wisc.edu/~ono/reprints/098.pdf
 Choi, YounSeo (1999), "Tenth order mock theta functions in Ramanujan's lost notebook", Inventiones Mathematicae 136 (3): 497–569, doi:10.1007/s002220050318, ISSN 00209910, MR1695205
 Choi, YounSeo (2000), "Tenth order mock theta functions in Ramanujan's lost notebook. II", Advances in Mathematics 156 (2): 180–285, doi:10.1006/aima.2000.1948, ISSN 00018708, MR1808245
 Choi, YounSeo (2002), "Tenth order mock theta functions in Ramanujan's lost notebook. IV", Transactions of the American Mathematical Society 354 (2): 705–733, doi:10.1090/S0002994701028616, ISSN 00029947, JSTOR 2693766, MR1862564
 Choi, YounSeo (2007), "Tenth order mock theta functions in Ramanujan's lost notebook. III", Proceedings of the London Mathematical Society. Third Series 94 (1): 26–52, doi:10.1112/plms/pdl006, ISSN 00246115, MR2293464
 Dragonette, Leila A. (1952), "Some asymptotic formulae for the mock theta series of Ramanujan", Transactions of the American Mathematical Society (Transactions of the American Mathematical Society, Vol. 72, No. 3) 72 (3): 474–500, doi:10.2307/1990714, ISSN 00029947, JSTOR 1990714, MR0049927
 Fine, Nathan J. (1988), Basic hypergeometric series and applications, Mathematical Surveys and Monographs, 27, Providence, R.I.: American Mathematical Society, ISBN 9780821815243, MR956465
 Garthwaite, Sharon Anne (2008), "The coefficients of the ω(q) mock theta function", International Journal of Number Theory 4 (6): 1027–1042, doi:10.1142/S1793042108001869, ISSN 17930421, MR2483310, http://www.worldscinet.com/ijnt/04/0406/S1793042108001869.html
 Gordon, Basil; McIntosh, Richard J. (2000), "Some eighth order mock theta functions", Journal of the London Mathematical Society. Second Series 62 (2): 321–335, doi:10.1112/S0024610700008735, ISSN 00246107, MR1783627, http://journals.cambridge.org/action/displayAbstract?aid=63263
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 Lawrence, Ruth; Zagier, Don (1999), "Modular forms and quantum invariants of 3manifolds", The Asian Journal of Mathematics 3 (1): 93–107, ISSN 10936106, MR1701924, http://www.intlpress.com/AJM/AJMv03.php
 Lerch, M. (1892), "Bemerkungen zur Theorie der elliptischen Funktionen", Jahrbuch uber die Fortschritte der Mathematik 24: 442–445
 McIntosh, Richard J. (2007), "Second order mock theta functions", Canadian Mathematical Bulletin 50 (2): 284–290, doi:10.4153/CMB20070289, ISSN 00084395, MR2317449, http://journals.cms.math.ca/cgibin/vault/view/mcintosh8634
 Ramanujan, Srinivasa (1988), The lost notebook and other unpublished papers, Berlin, New York: SpringerVerlag, ISBN 9783540187264, MR947735
 Ramanujan, Srinivasa (2000), Collected papers of Srinivasa Ramanujan, Providence, R.I.: AMS Chelsea Publishing, ISBN 9780821820766, MR2280843
 Selberg, A. (1938), "Über die MockThetafunktionen siebenter Ordnung. (On the mock theta functions of seventh order)", Arch. Math. Og Naturvidenskab 41: 3–15 Reprinted in volume I of his collected works.
 Semikhatov, A. M.; Taormina, A.; Tipunin, I. Yu. (2005), "Higherlevel Appell functions, modular transformations, and characters", Communications in Mathematical Physics 255 (2): 469–512, doi:10.1007/s0022000412807, ISSN 00103616, MR2129953
 Troost, J. (2010), "The NonCompact Elliptic Genus : Mock or Modular", Journal of High Energy Physics 2010 (6), arXiv:1004.3649, doi:10.1007/JHEP06(2010)104
 Serre, JeanPierre; Stark, H. M. (1977), "Modular forms of weight 1/2", Modular functions of one variable, VI (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), Lecture Notes in Mathematics, 627, Berlin, New York: SpringerVerlag, pp. 27–67, doi:10.1007/BFb0065296, ISBN 9783540085300, MR0472707
 Watson, G. N. (1936), "The Final Problem : An Account of the Mock Theta Functions", J. London Math. Soc. 11: 55–80, doi:10.1112/jlms/s111.1.55
 Watson, G. N. (1937), "The Mock Theta Functions (2)", Proc. London Math. Soc. s242: 274–304, doi:10.1112/plms/s242.1.274
 Zagier, Don (1975), "Nombres de classes et formes modulaires de poids 3/2", Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries a et B 281 (21): Ai, A883–A886, ISSN 01510509, MR0429750
 Zagier, Don (2009) [2007], Ramanujan's mock theta functions and their applications (after Zwegers and OnoBringmann), "Séminaire Bourbaki. Vol. 2007/2008", Astérisque, Séminaire Bourbaki Exp. 986 326: 143–164, ISBN 9782856292693, ISSN 03031179, MR2605321, http://www.bourbaki.ens.fr/TEXTES/986.pdf
 Zwegers, S. P. (2002), Mock Theta Functions, Utrecht PhD thesis, ISBN 9039331553, http://igiturarchive.library.uu.nl/dissertations/20030127094324/inhoud.htm
 Zwegers, S. P. (2001), "Mock θfunctions and real analytic modular forms", qseries with applications to combinatorics, number theory, and physics (Urbana, IL, 2000), Contemp. Math., 291, Providence, R.I.: American Mathematical Society, pp. 269–277, ISBN 9780821827468, MR1874536, http://mathsci.ucd.ie/~zwegers/papers/001.pdf
 Zwegers, S. P. (2008), Appell–Lerch sums as mock modular forms, http://mathsci.ucd.ie/~zwegers/presentations/002.pdf
External links
 International Conference: Mock theta functions and applications 2009
 Papers on mock theta functions by George Andrews
 Papers on mock theta functions by Kathrin Bringmann
 Papers on mock theta functions by Ken Ono
 Papers on mock theta functions by Sander Zwegers
 Weisstein, Eric W., "Mock Theta Function" from MathWorld.
Categories: Modular forms
 Qanalogs
 Srinivasa Ramanujan
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