W H A T I S . . . a Mock Modular Form? · 2010. 11. 9. · space of mock modular forms M kof weight kby mapping h2M kwith shadow gto its completion hb! …h‡g 2cM k. Second, one

?W H A T I S . . .a Mock Modular Form?

Amanda Folsom

The tale has been told and retold over time.The year: 1913. An unlikely correspondence beginsbetween prominent number theorist G. H. Hardyand (then) poor Indian clerk S. Ramanujan. A math-ematical collaboration between the two persistsfor the remainder of Ramanujan’s lifetime: a mereseven years, until his untimely death at the ageof thirty-two. New to the tale yet rooted in theHardy-Ramanujan era is the modern notion of“mock modular form”, not defined in the literatureuntil nearly a century later, in 2007, by D. Zagier.Today, the story seems to hail from the distant past,but its mathematical harvest, its intrigue, has notlet up. For example, the Hardy-Ramanujan storyhas been described recently as “one of the mostromantic stories in the history of mathematics”by Zagier (2007), and decades prior to this, G. N.Watson describes Ramanujan’s mathematics asinspiring in him a great “thrill” (1936). Far toomany mathematicians to list, including G. Andrews,B. Berndt, K. Bringmann, K. Ono, H. Rademacher,and S. Zwegers, have perpetuated the legacy ofRamanujan’s mathematics. Ramanujan’s life storyhas even been dramatically reinterpreted in the2007 work of fiction The Indian Clerk by DavidLeavitt.

An aspect of the allure to the Hardy-Ramanujanstory is the spawn of a mathematical mysterysurrounding the content of Ramanujan’s deathbedletter to Hardy, his last. Watson, in his presiden-tial address to the London Mathematical Society,prophetically declared the subject of the last letter,Ramanujan’s “mock theta functions”, to be “the

Amanda Folsom is assistant professor of mathematics atYale University. Her email address is [email protected].

final problem”. The mathematical visions of Ra-manujan were telling, particularly the seventeenpeculiar functions of the last letter, which hedubbed “mock theta functions” and which ap-peared along with various properties and relationsbetween them, yet with little to no explanation.(A handful of other such functions appear inRamanujan’s so called “lost notebook”, unearthedby G. Andrews in 1976, and in work of G. N.Watson.) Curiously, the “mock theta functions”were reminiscent of modular forms, which, looselyspeaking, are holomorphic functions on the upperhalf complex plane equipped with certain sym-metries. In fact, Ramanujan used the term “thetafunction” to refer to what we call a modular form,so his choice of terminology “mock theta function”implies he thought of his functions as “fake”’ or“pseudo" modular forms.

For example, one of Ramanujan’s mock thetafunctions is given by f (q) :=

∑n≥0 qn

2/(−q;q)2n,where (a;q)n :=

∏n−1j=0(1− aqj). Modular forms by

nature are also equipped with q-series expansions,where q := e2πiτ , τ is the variable in the upperhalf complex plane, and Dedekind’s η-function, awell-known modular form, satisfies q1/24η−1(τ) =∑n≥0 qn

2/(q;q)2n. These series expansions for f (q)and q1/24η−1(τ) barely differ. In fact, Ramanujanobserved other analytic properties shared byhis mock theta functions and modular forms,providing a description of what he calls a “mocktheta function”—and a notoriously vague definition.

Historically, one reason why the mock thetafunctions became an object of fascination for somany lies in the theory of integer partitions. Forany natural number n, a partition of n is defined tobe any nonincreasing sequence of positive integerswhose sum is n. So, for example, there are three

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partitions of 3 : 1+ 1+ 1,2+ 1, and 3, and if p(n)denotes the number of partitions of n, p(3) = 3.A useful tool for studying p(n), as with manyfunctions on N, is its “generating function”, whichis of the form P(x) := 1+

∑n≥1 p(n)xn, where x is

a variable. It is not difficult to see by a countingargument that P(x) =

∏n≥1(1 − xn)−1. Returning

to modular forms, it is also well known that η(τ)has an infinite q-product expansion, and uponreplacing x with q := e2πiτ in P(x), one finds thefollowing relationship between the modular formη(τ) and the partition generating function P(x):P(q) = q1/24/η(τ).

Many combinatorial generating functions likeP(x) are related to modular forms with q-infiniteproduct expansions in similar ways, and havingsuch relationships often allows one to use thetheory of modular forms to further explain variousaspects of the combinatorial functions. A famousexample of this is the work of Hardy, Ramanujan,and H. Rademacher, who first used the theory ofmodular forms to describe the asymptotic behaviorof p(n) as n →∞. The mock theta functions stoodout in that they too seemed to be related tocombinatorial functions. We now know that themock theta function f (q), for example, is relatedto F. Dyson’s rank-generating function, where therank of a partition is equal to its largest part minusthe number of its parts.

By the time of the Ramanujan Centenary Con-ference in 1987, it had become clear that “themock theta functions give us tantalizing hintsof a grand synthesis still to be discovered,” asDyson said. “Somehow it should be possible tobuild them into a coherent group-theoretical struc-ture, analogous to the structure of the modularforms…This,” he said, “remains a challenge forthe future.” Despite the volumes of literatureproduced by many famous mathematicians onthe subject since the Hardy-Ramanujan era, the“what is” remained unanswered for eighty-twoyears.

Enter S. Zwegers, a 2002 doctoral student underD. Zagier, whose thesis finally provided longawaited explanations, one being: while the mocktheta functions were indeed not modular, they couldbe “completed” (after multiplying by a suitablepower of q) by adding a certain nonholomorphiccomponent, and then packaged together to producereal analytic vector valued functions that exhibitappropriate modular behavior. This one-sentencedescription does not do justice to the scope ofZwegers’s results, which are much broader andalso realize the mock theta functions within othercontexts.

Zwegers’s breakthrough didn’t simply put anend to the mystery surrounding the mock thetafunctions; it (fortunately) opened the door to manymore unanswered questions! Notably, K. Bring-mann, K. Ono, and collaborators developed an

overarching theory of “weak Maass forms" (see thework of J. Bruinier and J. Funke for a debut ap-pearance in the literature), a space of functions towhich we now understand the mock theta functionsbelong. Weak Maass forms are nonholomorphicmodular forms that are also eigenfunctions of acertain differential (Laplacian) operator. (For thereader familiar with usual Maass forms or themore modern “Langlands program”, while there isa small intersection with the theory of weak Maassforms, the theories are distinct in that the requiredgrowth condition on weak Maass forms is relaxed,for example, hence the descriptor “weak”.)

How do the mock theta functions fit into thisframework of weak Maass forms and lead to theanswer to the question “what is a mock modularform”? As alluded to above, to associate modularbehavior to a given mock theta function m(q),one first needs to 1) define a suitable multipleh(q) := qMm(q) for some M ∈ Q, and 2) addto h(q) an appropriate nonholomorphic functiong∗(τ), constructed from a modular theta seriesg(τ), dubbed the shadow of h (after Zagier).The final object ĥ(τ) := h(q) + g∗(τ) is then anonholomorphic modular form. (Implicit here isthat one must also replace q by e2πiτ , τ ∈ H, in thefunction h(q).) Thus, one gains modularity at theexpense of holomorphicity: h(q) is holomorphicbut not modular, while ĥ(τ) is modular but notholomorphic. In particular, the function ĥ(τ) isa weak Maass form, whose holomorphic part isessentially the mock theta function m(q).

Loosely speaking then, a “mock modular form”is a holomorphic part of a weak Maass form. Oneparticularly beautiful example of a mock modularform due to Zagier is the generating function forHurwitz class numbers of algebraic number theory,whose completion (associated weak Maass form) isthe so-called Zagier-Eisenstein series, and whoseshadow is given by the classical modular thetafunction

∑n∈Z qn

2.

The ability to realize the mock theta functionswithin the theory of weak Maass forms has led tomany important discoveries. One notable exampletowards Dyson’s “challenge for the future” withinpartition theory is due to Bringmann and Ono, whoshow that generalized rank-generating functions(which include the mock theta function f (q) as aspecial case) are mock modular forms. Their worknot only led to deeper results in the theory ofpartitions by making use of the theory of weakMaass forms but also exhibited a new perspectiveon the roles played by modular forms in boththeories.

To grasp a more precise formulation of mockmodular forms, note that functions exhibitingmodular behavior (said to satisfy modular “trans-formations”) come equipped with an integer or

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half-integer “weight” k. One has two clear formula-tions of the space Mk of weight k mock modularforms due to Zagier. First, consider the space Mkof real analytic functions that exhibit a modulartransformation of weight k and satisfy suitablegrowth conditions. Understanding the roles playedby the shadows g, and also the differential Lapla-cian operator, leads to the fact that the spaceM̂k := {F ∈ Mk | ∂τ(yk∂τF) = 0} not only con-sists of weight k weak Maass forms (with specialeigenvalue k2(1−

k2)) but is also isomorphic to the

space of mock modular forms Mk of weight k bymapping h ∈Mk with shadow g to its completionĥ! = h+ g∗ ∈ M̂k. Second, one may also realize Mkvia the exact sequence 0 → M !k →Mk

S→ M2−k → 0,where M2−k is the space of holomorphic modu-lar forms of weight 2 − k, M !k is the space ofweakly holomorphic modular forms of weight k(allowing additional poles at points called cusps)and S(h) := g for the mock modular form h withshadow g. While this definition as stated is arguablythe most natural, it may be of interest to considerother generalizations, some of which are currentlybeing explored.

The tale of mock theta functions, mock mod-ular forms, and weak Maass forms, while rootedin analytic number theory, has bled into manyother areas of mathematics: sometimes mockmodular forms are combinatorial generating func-tions, sometimes they answer questions about thenonvanishing of L-functions, sometimes they arerelated to class numbers, sometimes they are char-acters for affine Lie superalgebras, and sometimesthey tell us about topological invariants—to namejust a few of their many roles. Watson was right:“Ramanujan’s discovery of the mock theta functionsmakes it obvious that his skill and ingenuity didnot desert him at the oncoming of his untimelyend. As much as any of his earlier work, the mocktheta functions are an achievement sufficient tocause his name to be held in lasting remembrance.To his students such discoveries will be a sourceof delight and wonder….”

What is next?

References[1] G. E. Andrews, An introduction to Ramanujan’s lost

notebook, Amer. Math. Monthly 86 (1979), 89–108.[2] K. Ono, Unearthing the visions of a master: Harmonic

Maass forms and number theory, Proceedings of the2008 Harvard-MIT Current Developments in Math-ematics Conference, International Press, Somerville,MA, 2009, pages 347–454.

[3] D. Zagier, Ramanujan’s mock theta functions andtheir applications [d’apres Zwegers and Bringmann-Ono], Séminaire Bourbake, 60ème année, 2006–2007,no 986.

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