TheLastWordsofaGenius Ken Ono

his story begins with a cryptic letter and he was elected a Fellow of the Royal Society written by a dying genius, the clues (F.R.S.), an honor shared by Sir Isaac Newton. of which inspired scores of mathemati- Sadly, the occasion of Ramanujan’s homecoming cians to embark on an adventure which was not one of celebration. He was a very sick man; resembles an Indiana Jones movie. It he was much thinner than the rotund Ramanujan T his Indian friends remembered. One of the main is reminiscent of the quest for the Holy Grail, in which skillful knights confront great obstacles. But reasons for his declining health was malnutrition. these knights are mathematicians, and the Grail is He had been adhering to a strict vegetarian diet replaced by a mathematical “Rosetta Stone” that in a time and place with no adequate resources to promises to reveal hidden truths in new worlds. support it. He also struggled with the severe change in climate. Accustomed to the temperate weather The Saga of south India, he did not have or did not wear appropriate clothing to protect him from the cool Our drama begins on March 27, 1919, the date and damp Cambridge weather. These conditions of Srinivasa Ramanujan’s triumphant, but bit- took their toll, and he became gravely ill. He was tersweet, Indian homecoming. Five years earlier, diagnosed2 with tuberculosis, and he returned to accepting an invitation from the eminent British India seeking familiar surroundings, a forgiving mathematician G. H. Hardy, the amateur Ramanu- climate, and a return to good health. Tragically, jan had left for Cambridge University with the Ramanujan’s health declined over the course of dream of making a name for himself in the the following year, and he passed away on April world of mathematics. Now, stepping off the ship 26, 1920, in Madras (now Chennai), with his wife Nagoya in Bombay (now Mumbai), the two-time Janaki by his side. college dropout, who had intuited unimaginable formulas, returned as a world-renowned number Ramanujan’s Last Letter theorist. He had achieved his goal. At the young Amazingly, in spite of his condition, Ramanujan age of thirty-one, Ramanujan had already made spent his last year working on mathematics in important contributions to a mindboggling array 1 isolation. of subjects: the distribution of prime numbers, …through all the pain and fever,…Ramanujan, hypergeometric series, elliptic functions, modular lying in bed, his head propped up on pillows, kept forms, probabilistic number theory, the theory of working. When he required it, Janaki would give partitions, and q-series, among others. He had him his slate; later she’d gather up the accumu- published over thirty papers, including seven with lated sheets of mathematics-covered paper …and Hardy. In recognition of these accomplishments, place them in the big leather box which he brought Ramanujan was named a Fellow of Trinity College, from England (see p. 329 of [29]). Janaki would later remember these last days Ken Ono is the Asa Griggs Candler Professor of Math- (see p. 91 of [38]): ematics at Emory University, and he is the Solle P. and He was only skin and bones. He often complained Margaret Manasse Professor of Letters and Science and of severe pain. In spite of it he was always busy the Hilldale Professor of Mathematics at the University of Wisconsin at Madison. His email addresses are ono@ 2The diagnosis of tuberculosis is now believed to be incor- mathcs.emory.edu and [email protected]. rect. D. A. B. Young examined the evidence pertaining to 1See [4, 6, 7, 8, 25, 29, 35] for more on Ramanujan and Ramanujan’s illness, and he concluded that Ramanujan his achievements. died of hepatic amoebiasis [42].

1410 Notices of the AMS Volume 57, Number 11 doing his mathematics. …Four days before he died Watson chose the title The Final Problem: An he was scribbling. Account of the Mock Theta Functions. In a fateful last letter to Hardy, dated January He proceeded to describe his findings, a 12, 1920, Ramanujan shared hints (see p. 220 of medley of identities and formulas. Using [7]) of his last theory. “q-hypergeometric series”, he reformulated I am extremely sorry for not writing you…I dis- Ramanujan’s examples. For f (q) he proved: covered very interesting functions recently which I n (3n2 n)/2 2 ( 1) q + call “Mock” theta functions.…they enter into math- (1.3) f (q) − . ∞ n n = n 1(1 q ) n Z 1 q ematics as beautifully as the ordinary theta func- = − ∈ + tions. I am sending you with this letter some exam- He also proved identities relating the mock theta ples. functions to Mordell integrals, such as This mysterious letter set off a great adventure: ∞ 3πx2 sinh πx 1 the quest to realize the meaning behind these last (1.4) e− dx 2π/3√ words and to then unearth the implications of 0 sinh 3πx = e 3 2n(n 1)π this understanding. These words exposed, in an ∞ e− + . unexplored territory of the world of mathematics, × (1 e π )2(1 e 3π )2 (1 e (2n 1)π )2 n 0 + − + − + − + a padlocked wooden gate, beyond which was the = He concluded by entrusting the quest to the next promise of unknown mathematical treasures. generation of mathematicians. Ramanujan’s discovery of the mock theta func- The Early Years tions makes it obvious that his skill and ingenuity The letter, roughly four typewritten pages, consists did not desert him at the oncoming of his untimely of formulas for seventeen strange power series and end. …To his students such discoveries will be a a discussion of their asymptotics and behavior near source of delight and wonder until the time shall the boundary of the unit disk. There are no proofs come when we too shall make our journey to that of any kind. Ramanujan also grouped these series Garden of Proserpine [Persephone] … based on their “order”, a term he did not define. Mathematicians continued the pursuit. In the As a typical example, he offered late 1930s A. Selberg, as a high school student, published his first two mathematical papers on ∞ n (1.1) f (q) af (n)q the subject. In the 1950s and 1960s, G. E. Andrews = n 0 and L. Dragonette, employing Watson’s results, fi- = 2 ∞ qn nally confirmed Ramanujan’s claimed asymptotic : 3 (1 q)2(1 q2)2 (1 qn)2 (1.2). Many mathematicians, among them B. C. = n 0 = + + + Berndt, Y.-S. Choi, B. Gordon, and R. McIntosh, pro- 99 1 q 17503q ..., gressed further along the lines set by Watson. After = + − + + many technical calculations that run on for pages, which he called a third-order mock theta function. these mathematicians mastered the asymptotics He then miraculously claimed that of Ramanujan’s examples, amassed identities such n 1 ( 1) − π n 1 as (1.3), and obtained analytic transformations (1.2) af (n) − e 6 − 144 . ∼ 2 n 1 relating these examples to integrals such as (1.4). − 24 Mathematicians now had a grasp of the padlock Obviously Ramanujan knew much more than he which secured the wooden gate. But they still did revealed. not know the meaning behind Ramanujan’s last G. N. Watson was the first mathematician to words. Mathematicians had gathered a box full of take on the challenge. He worked for years, and on formulas, but there would be little progress for the November 14, 1935, at a meeting of the London next ten years. Mathematical Society, he celebrated his retirement as president of the Society with his now famous The “Lost Notebook" address [40]: In the spring of 1976, while searching through It is not unnatural that [one’s] mode of ap- archived papers from Watson’s estate in the Trinity proach to the preparation of his valedictory College Library at Cambridge University, Andrews address should have taken the form of an in- discovered the “Lost Notebook” [2]. The notebook, vestigation into the procedure of his similarly consisting of over 100 pages of Ramanujan’s last situated predecessors….I was, however, deterred works, was archived in a box among assorted from this course…[Ramanujan’s last] letter is the papers collected from Watson’s estate. subject which I have chosen…; I doubt whether …the notebook and other material was discov- a more suitable title could be found for it than ered among Watson’s papers by Dr. J. M. Whittaker, the title used by John H. Watson, M.D., for what he imagined to be his final memoir on Sherlock 3K. Bringmann and the author have obtained an exact Holmes. formula for af (n) [12].

December 2010 Notices of the AMS 1411 The five pages of Ramanujan’s last letter to Hardy... (above and following pages)

who wrote the obituary of Professor Watson for the only make lucky dips [into the rubble] and, as Wat- Royal Society. He passed the papers to Profes- son never threw away anything, the result might sor R. A. Rankin of Glasgow University, who, in be a sheet of mathematics, but more probably a re- December 1968, offered them to Trinity College ceipted bill or a draft of his income tax return for so that they might join the other Ramanujan 1923. By an extraordinary stroke of luck one of my manuscripts…[2]. dips brought up the Ramanujan material. Janaki presumably sent Hardy the large leather The Lost Notebook allowed mathematicians to box, the one filled with Ramanujan’s last papers. escape the seemingly eternal morass. In addition Hardy passed it on to Watson in turn. to listing some new mock theta functions, the Although never truly lost, the sheaf of papers scrawl contained many valuable clues: striking had survived the long journey from India only to identities and relations, recorded without proofs then lie forgotten in the Trinity College Library. The of any kind. Thanks to these clues, mathemati- journey was indeed extraordinary, for the manu- cians found many applications for the mock theta script almost met a catastrophic end. Whittaker, functions: L-functions in number theory, hyperge- the son of Watson’s famous coauthor E. T. Whit- ometric functions, partitions, Lie theory, modular taker, recalled, in a letter to G. E. Andrews dated forms, physics, and polymer chemistry, to name a August 15, 1979, the scene of Watson’s study at few. the time of his death in 1965 (see p. 304 of [7]): The Lost Notebook notably surrendered new …papers covered the floor of a fair sized room sorts of identities that, as we shall see, go on to to the depth of about a foot, all jumbled together, play a crucial role in the quest. Andrews proved and were to be incinerated in a few days. One could identities [3] relating mock theta functions to

1412 Notices of the AMS Volume 57, Number 11 indefinite binary quadratic forms. For example, he A modular form is a holomorphic function proved: on the upper half complex plane H which is aτ b tamed by Möbius transformations γτ : + for (1.5) = cτ d a b 4 + 2 γ SL2(Z). Loosely speaking, a weight ∞ qn 1 = c d ∈ f0(q) : k modular form on a subgroup SL2(Z) is a n ∞ n = n 0 (1 q) (1 q ) = n 1(1 q ) ⊂ = + + = − holomorphic function f : H C that satisfies → Γ (1.6) f (γτ) (cτ d)kf (τ) j 5 n2 1 n j2   ( 1) q 2 + 2 − . = + n j 0 − n j<0 − for all γ and is meromorphic “at the cusps”.  +≥ +  n j 0 n j<0 ∈  − ≥ −  Despite this evidence, the essence of Ramanu- D. Hickerson confirmed [26] identities in which jan’s theoryΓ continued to elude mathematicians. sums of mock theta functions are infinite products. The problem was that the Lost Notebook is merely For example, for f0(q) and the mock theta function a bundle of pages that “…contains over six hun- (q), he showed that dred mathematical formulae listed one after the 5 ∞ (1 q5n)(1 q10n 5) other without proof …there are only a few words Φ f (q) 2 (q2) − . 0 − 5n 4 − 5n 1 scattered here and there…” (p. 89 and p. 96 of [2]). + = n 1 (1 q − )(1 q − ) = − − Instead of furnishing the missing key, the Notebook Φ As indefinite binary quadratic forms and infinite products appear in modular form theory, these 4If k is not an integer, then (1.6) must be suitably identities finally provided evidence linking mock modified. theta functions to modular forms, the “ordinary 5Almost all of the results on q-series in the Lost Notebook theta functions” of Ramanujan’s last letter. have now been proved [4].

December 2010 Notices of the AMS 1413 Zwegers’s Thesis By the late 1990s little progress had been made. Then in 2002, in a brilliant Ph.D. thesis [45] written under D. Zagier, S. Zwegers made sense of the mock theta functions. By understanding the meaning behind identities such as (1.3–1.5) and by notably making use of earlier work of Lerch and Mordell, he found the answer: real analytic modular forms. In the solution, one must first “complete” the mock theta functions by adding a nonholomorphic function, a so-called “period integral”. For the mock theta functions f (q) (see (1.1)) and (1.7)

∞ n ω(q) aω(n)q = n 0 = ∞ q2n(n 1) : + , 2 3 2 2n 1 2 = n 0 (1 q) (1 q ) (1 q + ) = − − − Zwegers [44] defined the vector-valued mock theta function (here q : e2πiτ ) = T F(τ) (F0(τ), F1(τ), F2(τ)) = 1 1 1 1 1 T : (q− 24 f (q), 2q 3 ω(q 2 ), 2q 3 ω( q 2 )) . = − Then using theta functions g0(z), g1(z), and g2(z), where

2 ∞ n 1 3πi(n 1 ) z g0(z) : ( 1) n e + 3 = n − + 3 =−∞ (g1(z) and g2(z) are similar), he defined the vector- valued nonholomorphic function (1.8)

T G(τ) (G0(τ), G1(τ), G2(τ)) = i T ∞ (g1(z), g0(z), g2(z)) : 2i√3 − dz. = τ i(τ z) − − + provided a hammer in the form of countless identi- He completed F(τ) to obtain H(τ) : F(τ) G(τ), ties. So armed, mathematicians burst through the and he proved that [44] = −

wooden gate, only to find a long dusty hallway 1 ζ24− 0 0 lined with locked iron doors. When the dust set- H(τ 1) 0 0 ζ3 H(τ) + = 0 ζ3 0 tled, they could read the signs on the doors, and with this knowledge they finally understood the and widespread scope of the mock theta functions. At 0 1 0 H( 1/τ) √ iτ 1 0 0 H(τ), the Ramanujan Centenary Conference in 1987, F. 0 0 1 − = − − Dyson eloquently summed up the dilemma [21]: 2πi/n 1 1 0 1 where ζn : e . As SL2(Z) 0 1 , 1− 0 , this The mock theta-functions give us tantalizing = = gives a vector version of (1.6), and so the vector- hints of a grand synthesis still to be discovered. valued mock theta function F(τ) is the holomorphic Somehow it should be possible to build them into part of the vector-valued real analytic modular a coherent group-theoretical structure, analogous form H(τ). to the structure of modular forms which Hecke Generalizing identities such as (1.3), in which built around the old theta-functions of Jacobi. This mock theta functions are related to “Appell-Lerch” remains a challenge for the future. My dream series, Zwegers also produced infinite families of is that I will live to see the day when our young mock theta functions that eclipse Ramanujan’s list. For τ H and u, v C (Zτ Z), he defined the physicists, struggling to bring the predictions of ∈ ∈ \ + superstring theory into correspondence with the function 1/2 n n(n 1)/2 facts of nature, will be led to enlarge their analytic z ( w) q + (1.9) (u, v; τ) : , − n machinery to include mock theta-functions… = ϑ(v; τ) n Z 1 zq ∈ −

1414 Notices of the AMS Volume 57, Number 11 2πiu 2πiv 2πiτ where z : e , w : e , q : e and and its complement M−(τ), the nonholomorphic = πiν =ν ν2/2 = ϑ(v; τ) : ν Z 1 e w q . Using a function part. The mock theta functions and Zwegers’s = ∈ + 2 -function give holomorphic parts of weight 1/2 R(u v; τ)which resembles the components of (1.8),− he then defined harmonic Maass forms. Harmonic Maass forms are generalizations of (1.10) (u, v; τ) : (u, v; τ) = modular forms; a modular form is a harmonic i Maass form M(τ) where M−(τ) 0. Because mod- R(u v; τ). = + 2 − ular forms appear prominently in mathematics, He proved that (u, v; τ) is a nonholomorphic one then expects the mock theta functions and Jacobi form, a function whose specializations at harmonic Maass forms to have far-reaching impli- “torsion points” give weight 1/2 real analytic mod- cations. Our first forays in the long dusty hallway ular forms. This function satisfies transformations have been profitable, and we have obtained results that imply (1.6) for these specializations; for exam- [12, 13, 14, 15, 17, 32, 36, 37] on a wide array of a b subjects: partitions and q-series, Moonshine, Don- ple, if γ SL2(Z), then there is an explicit = c d ∈ aldson invariants, Borcherds products, and elliptic root of unity χ(γ) for which curves, among others. We now describe some of u v aτ b these results. , ; + cτ d cτ d cτ d + + 1+ 3 πic(u v)2/(cτ d) Partitions χ(γ)− (cτ d) 2 e− − + (u, v; τ). = + A partition of a nonnegative integer n is any Exploring New Worlds nonincreasing sequence of positive integers that Armed with Zwegers’s landmark thesis, mathe- sum to n. If p(n) denotes the number of partitions maticians have begun to explore [36, 43] the worlds of n, then Ramanujan famously proved that behind the iron doors. Here we sample some of p(5n 4) 0 (mod 5), + ≡ the discoveries that this author has obtained with p(7n 5) 0 (mod 7), his collaborators. + ≡ p(11n 6) 0 (mod 11). + ≡ Harmonic Maass Forms In an effort to provide a combinatorial expla- The mock theta functions turn out to be holomor- nation of these congruences, Dyson defined the phic parts of distinguished real analytic modular rank of a partition to be its largest part minus the forms, the harmonic Maass forms, which were re- number of its parts. For example, the table below cently introduced by J. H. Bruinier and J. Funke includes the ranks of the partitions of 4. [16]. Partition Rank Rank mod 5 Loosely speaking, a weight k harmonic Maass 4 4 1 3 3 form is a smooth function M : H C satisfying 3 1 3 − 2 = 1 1 → 6 + − = (1.6) and k(M) 0, which also has (at most) linear 2 2 2 2 0 0 exponential growth= at cusps. Here the hyperbolic 2 +1 1 2 −3 = 1 4 + + − = − Laplacian k, where τ x iy H with x, y R, 1 1 1 1 1 4 3 2 is given by = + ∈ ∈ + + + − = − 2 2 Based on numerics, Dyson [20] made the following 2 ∂ ∂ ∂ ∂ k : y iky i . conjecture whose truth provides a combinatorial = − ∂x2 + ∂y 2 + ∂x + ∂y explanation of Ramanujan’s congruences modulo The Fourier expansions of these forms have been 5 and 7. the object of our explorations. In terms of the in- t α 1 Conjecture ( Dyson). The partitions of 5n 4 (resp. complete gamma function (α; x) : x∞ e− t − dt, + every weight 2 k harmonic Maass= form M(τ), 7n 5) form 5 (resp. 7) groups of equal size when − + where k > 1, has an expansionΓ of the form sorted by their ranks modulo 5 (resp. 7). n In 1954 A. O. L. Atkin and H. P. F. Swinnerton-Dyer (2.11) M(τ) cM+ (n)q 7 = n proved [5] Dyson’s conjecture. ≫−∞ There is now a robust theory of partition con- c (n) (k 1, 4π n y)qn. M− gruences modulo every integer M coprime to 6 + n<0 − | | [1, 34], and typical congruences look more like Obviously, M(τ) naturallyΓ decomposes into two p(48037937n 1122838) 0 (mod 17). pieces, a holomorphic part + ≡ n M+(τ) : cM+ (n)q One naturally asks: what role, if any, do Dyson’s = n original guesses play within this theory? ≫−∞

6In this paper we use a slightly stronger condition; we 7A short calculation reveals that the obvious generaliza- assume the existence of “principal parts” at cusps. tion of the conjecture cannot hold for 11.

December 2010 Notices of the AMS 1415 K. Bringmann and the author [13] investigated whether they might this question, and in their work they related be related to N(r, t; n), the number of partitions of n with harmonic Maass rank congruent to r (mod t), to harmonic Maass forms. Bringmann forms. They essentially proved that and the author have confirmed ∞ p(n) N(r, t; n) qn [14] this spec- t n 0 − = ulation; these is the holomorphic part of a weight 1/2 harmonic characters are Maass form. This result, combined with Shimura’s pieces of nonholo- theory of half-integral weight modular forms and morphic modular the Deligne-Serre theory of Galois representations, functions. We con- implies that ranks “explain” infinite classes of sider the character congruences. for the sℓ(m, 1)∧ Theorem 2.1 (Th. 1.5 of [13]). If Q 5 is prime modules L( (s)), and j 1, then there are positive integers≥ t and where L( (s)) is arithmetic≥ progressions An B such that the irreducibleΛ + sℓ(m, 1)∧ Λ mod- N(r, t; An B) 0 (mod Qj ) + ≡ ule with highest for every 0 r < t. In particular, we have that weight (s). If p(An B) ≤0 (mod Qj ). m 2 and s Z, ≥ ∈ Ramanujan passport photo. + ≡ then theΛ work of Moonshine Kac and Wakimoto implies that In the late 1970s J. McKay and J. Thompson [39] (2.12) 2 observed that the first few coefficients of the m 2 12s L − − η(2τ) tr q 0 2q 24 classical elliptic modular function L( (s)) m 2 = η(τ) + Λ 1 m 1 j(z) 744 2 i −1 ki (ki 1) − q = + 1 2 , q− 196884q 21493760q 1 q k s Zm 1 | |− = + + k (k1,k2,...,km 1 ) − + 3 = − ∈ 864299970q m 1 1/24 + + where k : i −1 ki and η(τ) : q n∞ 1(1 are certain linear combinations of the dimen- n | | = = = = − q ) is Dedekind’s eta function. Using the function sions of the irreducible representations of the R in (1.10), Bringmann and the author defined the Monster group. For example, the degrees of the function four “smallest” irreducible representations are: 1, (2.13) 196883, 21296876, and 842609326, and the first L0 few coefficients are: m,s(τ) : trL( )q T = (s) 2m m 2 mΛ 1 − η(2τ) 1 1 2 q 24 R( sτ; (m 1)τ), = − 2m 1 196884 196883 1 − η(τ) + − − = + 21493760 21296876 196883 1 = + + and they showed that 864299970 842609326 21296876 2 196883 2 1. = + + + η(τ)2m 1 + L0 J. Conway and S. Norton [18] expanded on these trL( (s))q η(2τ)2m observations, and they formulated a series of deep Λ conjectures, the so-called Monstrous Moonshine is (up to a power of q) a mock theta function. As a Conjectures. These conjectures have now been set- consequence, they proved: tled, and thanks to the work of many authors, Theorem 2.2 (Th. 1.1 of [14]). If m 2 and s Z, most notably R. E. Borcherds [9], there is a beauti- ≥ ∈ then m,s (τ) is (up to a power of q) a nonholomor- ful theory, involving string theory, vertex operator phic modularT function. algebras, and generalized Kac-Moody superalge- bras, in which connections between objects like Donaldson Invariants the j-function and the Monster are revealed. In recent work with A. Malmendier [32], it is shown In the 1980s, in the spirit of Moonshine, V. G. Kac and D. H. Peterson [27] established the modu- that the mock theta function 1 larity of similar characters that arise in the study M(q) : q− 8 = of infinite-dimensional affine Lie algebras. As a n 1 (n 1)2 3 2n 1 ∞ ( 1) + q + (1 q)(1 q ) (1 q − ) generalization of this work, ten years ago Kac − 2 − 3 2 − 2n 1−2 × n 0 (1 q) (1 q ) (1 q + ) and M. Wakimoto [28] computed characters of the = + + + 2 affine Lie superalgebras gℓ(m, 1)∧ and sℓ(m, 1)∧. is a “topological invariant” for CP . This claim per- These characters are not modular, and Kac asked tains to the differentiable topology of 4-manifolds.

1416 Notices of the AMS Volume 57, Number 11 In the early 1980s S. Donaldson proved (for ex- Borcherds Products ample, see [19]) that the diffeomorphism class Recently R. E. Borcherds provided [10, 11] a strik- of a compact, simply connected, differentiable ing description for the exponents in the infinite 4-manifold X is not necessarily determined by product expansion of many modular forms with its intersection form. In his work, he famously a Heegner divisor. He proved that the exponents defined the Donaldson invariants, diffeomorphism in these expansions are coefficients of weight invariants of X obtained as graded homogeneous 1/2 modular forms. As an example, the classical polynomials on the homology ring with integer Eisenstein series E4(τ) factorizes as coefficients.

There are two families of invariants correspond- ∞ 3 n E4(τ) 1 240 d q ing to the SU(2) and the SO(3) gauge theories. = + n 1 d n The author and Malmendier considered the SO(3) = | 240 2 26760 ∞ n c(n) case for the simplest manifold, the complex pro- (1 q)− (1 q ) (1 q ) , 2 = − − = − jective plane CP with the Fubini-Study metric. n 1 = The invariants are difficult to work out even in where the c(n) are the coefficients b(n2) of a this case; indeed, they were not computed until weight 1/2 meromorphic modular form the work of L. Göttsche [22] in 1996, assuming the n Kotschick-Morgan Conjecture. Göttsche, H. Naka- G(τ) b(n)q = n 3 jima, and K. Yoshioka have recently confirmed ≥− 3 [23] this provisional description of the Donaldson q− 4 240q = + − “zeta-function” 26760q4 4096240q9 . pmSn + + − + (2.15) Z(p, S) φmn . Bruinier and the author [17] have generalized = m!n! m,n this phenomenon to allow for exponents that In the mid-1990s G. Moore and E. Witten con- are coefficients of weight 1/2 harmonic Maass jectured [33] “u-plane integral” formulas for this forms. For brevity, we give examples of generalized zeta function. Their work relies on the following Borcherds products that arise from the mock theta identifications: functions f (q) and ω(q). To this end, let 1 < D 23 (mod 24) be square-free, and for 0 j 11 let≡ u-plane integrals Donaldson invariants for CP 2 ←→ ≤ ≤ (2.17) elliptic surface Hj (τ) Here the rational elliptic surface is the universal n curve for the modular group (4), which can be C(j; n)q 0 = n identified with CP 1 minus 3 points with singular ≫−∞ 0 if j 0, 3, 6, 9, fibers. In addition, the rationalΓ elliptic surface is = j 1 24  q− f (q ) if j 1, 5, 7, 11, endowed with an analytical marking such that the :  3 = j 8 12 12 generic fibers correspond to elliptic curves that are =  3 2q ω(q ) ω( q ) if j 2, 10,  j − − = parameterized by C-lattices ω, τω in the usual ( 1) 4 2q8 ω(q12) ω( q12) if j 4, 8.  − + − = way. Then the u-plane zeta function is given as a   2πiα D “regularized” integral If e(α) : e and − is the Jacobi-Kronecker = b (2.16) quadratic residue symbol, then define D reg (− ) 8 du du (2.18) PD(X) : (1 e( b/D)X) b . ZUP (p, S) : ∧ = − − = − √2π UP √Imτ b mod D 1 8 Using this rational function, we then define the dτ 2up S2T 3 1 e + η(τ) . generalized Borcherds product D(τ) by du 2 ω Here is the discriminant of the correspond- ∞ m C(m;Dm2) (2.19) D(τ) : PD(qΨ ) . ing elliptic curves, and T is defined by the = m1 renormalization flow on the elliptic surface. = The exponentsΨ come from (2.17), and m is the The Moore-Witten Conjecture in this case is that residue class of m modulo 12. Z(p, S) ZUP (p, S). Using (2.16), the author and = Malmendier reformulated this conjecture in terms Theorem 2.4 (§8.2 of [17]). The function D(τ) is a of harmonic Maass forms arising from M(q), and weight 0 meromorphic modular form on 0(6) with they then used Zwegers’s -function to prove the a discriminant D Heegner divisor (seeΨ §5 of [17] − following theorem. for the explicit divisor). Γ Theorem 2.3 (Th. 1.1 of [32]). The Moore-Witten This theorem has an interesting consequence Conjecture for the SO(3)-gauge theory on CP 2 for the parity of the partition function. Very little is true. In particular, we have that Z(p, S) is known about this parity; indeed, it was not even = ZUP (p, S). known that p(n) takes infinitely many even and

December 2010 Notices of the AMS 1417 odd values until 1959 [30]. Using Theorem 2.4 and (1) If < 0 is a fundamental discriminant for the fact that which p 1, then n2 = ∞ q f (q) : 2 2 2 2 n 2 L(E( ), 1) α(E) cM− ( ) . = n 0 (1 q) (1 q ) (1 q ) = | | = + + + (2) If > 0 is a fundamental discriminant for ∞ n p(n)q (mod 4), which 1, then L′(E( ), 1) 0 if and ≡ p = = n 0 = only if cM+( ) is algebraic. the author proved the following result for the partition numbers evaluated at the values of certain The Path Ahead quadratic polynomials. As we have seen, Ramanujan’s deathbed letter Theorem 2.5 (Corollary 1.4 of [37]). If ℓ 23 set into motion an implausible adventure, one (mod 24) is prime, then there are infinitely many≡ whose first act is now over. It was about the ℓm2 1 m coprime to 6 for which p + is even. More- theory of harmonic Maass forms and its implica- 24 tions for many subjects: partitions and q-series, over, the first such m is bounded by 12h( ℓ) 2, − + Moonshine, Donaldson invariants, mathematical where h( ℓ) is the class number of Q( ℓ). − − physics, Borcherds products, and L-functions of elliptic curves, to name a few. This theory has Elliptic Curve L-Functions provided satisfying answers to the first challenges: If E/Q is an elliptic curve to understand the meaning behind Ramanujan’s E/Q : y 2 x3 Ax B, last words and to realize the expectation that this = + + understanding would reveal and open new doors Q Q then E( ), its -rational points, forms a in the interconnected world of mathematics. finitely generated abelian group. The Birch Every step along the way has evoked wonder— and Swinnerton-Dyer Conjecture, one of the Clay the enigmatic letter, the Lost Notebook, and the millennium prize problems, predicts that work of many minds. If the past is the road map ords 1(L(E, s)) Rank of E(Q), to the future, then the yet unwritten acts promise = = where L(E, s) is the Hasse-Weil L-function forays, by intrepid mathematicians of today and for E. There is no known procedure for tomorrow, into new worlds presently populated with seemingly unattainable mathematical truths. computing ords 1(L(E, s)). Determining when = ords 1(L(E, s)) 1 for elliptic curves in a family of quadratic= twists≤ already requires the deep the- Acknowledgments orems of Kohnen [31] and Waldspurger [41], and This article is based on the author’s AMS Invited of Gross and Zagier [24]. These results, however, Address at the 2009 Joint Mathematics Meetings involve very disparate criteria for deducing the in Washington, DC, and his lectures at the 2008 analytic behavior at s 1. Harvard-MIT Current Developments in Mathemat- Using generalized Borcherds= products [17], Bru- ics Conference. The author thanks the National inier and the author have produced a single device Science Foundation, the Hilldale Foundation, and that encompasses these criteria. We present a spe- the Manasse family for their generous support. The cial case of these results. Suppose that E has prime author thanks Scott Ahlgren, Claudia Alfes, George conductor, and suppose further that the sign of E. Andrews, Bruce C. Berndt, Matt Boylan, Kathrin the functional equation of L(E, s) is 1. If is Bringmann, Jan Hendrik Bruinier, Amanda Folsom, a fundamental discriminant of a quadratic− field, Andrew Granville, Andreas Malmendier, David Pen- then let E( ) be the quadratic twist elliptic curve niston, Bjorn Poonen, Ken Ribet, Jean-Pierre Serre, Kannan Soundararajan, Heather Swan-Rosenthal, E( ) : y 2 x3 Ax B. = + + Marie Taris, Christelle Vincent, and Sander Zwegers Using harmonic Maass forms and their generalized for their comments and criticisms. Borcherds products, the author and Bruinier show that the coefficients of certain harmonic Maass References forms encode the vanishing of central derivatives [1] S. Ahlgren and K. Ono, Congruence properties for (resp. values) of the L-functions for the elliptic the partition function, Proc. Natl. Acad. Sci., USA 98 curves E( ). (2001), no. 23, 978–984. [2] G. E. Andrews, An introduction to Ramanujan’s lost Theorem 2.6 (Th. 1.1 of [17]). Assuming the hy- notebook, Amer. Math. Monthly 86 (1979), 89–108. potheses above, there is a weight 1/2 harmonic [3] , The fifth and seventh order mock theta Maass form functions, Trans. Amer. Math. Soc. 293 (1986), no. 1, n n 113–134. ME(τ) cM+ (n)q cM− (n) (1/2; 4π n y)q , [4] G. E. Andrews and B. C. Berndt, Ramanujan’s Lost = n +n<0 | | ≫−∞ Notebook, Parts I, II, Springer Verlag, New York, and a nonzero constant α(E) thatΓ satisfies: 2005, 2009.

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