ARITHMETIC OF MAASS FORMS OF HALF-INTEGRAL WEIGHT BY NICKOLAS ANDERSEN DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate College of the University of Illinois at Urbana-Champaign, 2016 Urbana, Illinois Doctoral Committee: Professor Bruce Berndt, Chair Professor Scott Ahlgren, Director of Research Professor Kevin Ford Assistant Professor Martin Luu Abstract We investigate the arithmetic properties of coefficients of Maass forms in three contexts. First, we discuss connections to invariants of real and imagi- nary quadratic fields, expanding on the work of Zagier and Duke-Imamog¯lu- To´th. Next, we examine the deep relationship between sums of Kloosterman sums and Maass cusp forms, motivated by work of Kuznetsov and Sarnak- Tsimerman, among others. Finally, we focus on the classical mock theta functions of Ramanujan, and give a simple proof of the mock theta conjec- tures using the modern theory of harmonic Maass forms, especially work of Zwegers and Bringmann-Ono, together with the theory of vector-valued modular forms. iii For Emily. iv Acknowledgments Thanks to the University of Illinois Mathematics Department for supporting me for five years, and to the number theory group for awarding me the Bateman Fellowship and Prize. Thanks also to the Graduate College for awarding me a Dissertation Completion Fellowship and to the Illinois chapter of the ARCS foundation, for providing me with additional financial support to complete my research. Thanks to my advisor, Scott Ahlgren, for spending more hours than I can count, carefully reading each line of every one of my papers; his pre-refereeing not only saved me from untold embarrassment, but also significantly improved the exposition of everything I wrote. I owe much of my success as a graduate student to his encouragement and support. Mathematics aside, Scott taught me valuable lessons about being a teacher, a mentor, and a father. Thanks to Emily, who sacrificed so much in order for me to fulfill my dream of earning a PhD. Her unending love and support has played a key role in my success ever since we met. Finally, thanks to Logan for (sometimes begrudgingly) allowing his buddy to go to school and do some math. v Table of Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Invariants of quadratic fields and mock modular forms . . . . 2 1.2 Kloosterman sums and Maass cusp forms of half-integral weight 15 1.3 The mock theta conjectures . . . . . . . . . . . . . . . . . . . 22 2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.1 Multiplier systems . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Harmonic Maass forms . . . . . . . . . . . . . . . . . . . . . . 28 2.3 Maass cusp forms . . . . . . . . . . . . . . . . . . . . . . . . . 30 3 Periods of the j-function along infinite geodesics . . . . . 39 3.1 Binary quadratic forms . . . . . . . . . . . . . . . . . . . . . . 39 3.2 Poincar´e series . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.3 Proof of Theorem 1.1.2 . . . . . . . . . . . . . . . . . . . . . . 44 4 Harmonic Maass forms of weight 5/2 . . . . . . . . . . . . 49 4.1 Poincar´e series and P (τ,s) . . . . . . . . . . . . . . . . . . . 49 v 4.2 Binary quadratic forms and P (τ,s) . . . . . . . . . . . . . . 53 v,Q 4.3 Kloosterman sums and the proof of Theorem 1.1.5 . . . . . . . 58 4.4 Proof of Theorem 1.1.4 . . . . . . . . . . . . . . . . . . . . . . 70 5 Kloosterman sums and Maass cusp forms . . . . . . . . . . 78 5.1 A mean value estimate for coefficients of Maass cusp forms . . 79 5.2 The Kuznetsov trace formula . . . . . . . . . . . . . . . . . . 96 5.3 A theta lift for Maass cusp forms . . . . . . . . . . . . . . . . 101 5.4 Estimates for a K-Bessel transform . . . . . . . . . . . . . . . 116 5.5 Application to sums of Kloosterman sums . . . . . . . . . . . 124 5.6 A second estimate for coefficients of Maass cusp forms . . . . . 126 5.7 Sums of Kloosterman sums and Rademacher’s formula . . . . 130 6 The mock theta conjectures . . . . . . . . . . . . . . . . . . 137 6.1 Definitions and transformations . . . . . . . . . . . . . . . . . 137 6.2 Vector-valued modular forms and the Weil representation . . . 145 6.3 Proof of the mock theta conjectures . . . . . . . . . . . . . . . 146 6.4 The six remaining identities . . . . . . . . . . . . . . . . . . . 147 6.5 Proof of Lemma 6.1.3 . . . . . . . . . . . . . . . . . . . . . . . 148 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 vi 1 Introduction A weak Maass form of weight k is a real analytic function on the upper-half plane H which satisfies a modular transformation in weight k with respect to some subgroup of the modular group SL (Z), is an eigenfunction of the 2 weight k hyperbolic Laplacian, and has at most linear exponential growth at the cusps.1 Weak Maass forms with Laplace eigenvalue 0 are called harmonic Maass forms, and those which are holomorphic on H are called weakly holomorphic modular forms. Each weak Maass form f has a Fourier expansion of the form (cid:88) f(τ) = a(n,y)e(nx). n∈Q n(cid:29)−∞ Here, and throughout, τ = x+iy ∈ H and e(x) := exp(2πix). Motivated by the general principle that important sequences in number theory are often related to the coefficients of modular forms, a natural problem is to determine what arithmetic data, if any, is encoded in the coefficients of weak Maass forms. In this thesis, we investigate the arithmetic properties of coefficients of Maass forms in three directions. First, we discuss connections to invariants of real and imaginary quadratic fields, expanding on the work of Zagier and Duke-Imamog¯lu-To´th. Next, we examine the deep relationship between sums of Kloosterman sums and Maass cusp forms, motivated by work of Kuznetsov and Sarnak-Tsimerman, among others. Finally, we focus on the classical mock theta functions of Ramanujan, and give a simple proof of the mock theta conjectures using the modern theory of harmonic Maass forms, especially work of Zwegers and Bringmann-Ono, together with the theory of vector-valued modular forms.2 1There are a few different conventions used in defining Maass forms, the Laplacian, etc., even in this thesis. In the introduction we will be intentionally vague and wait until later to sort this all out. 2The work on sums of Kloosterman sums (Chapter 5) and the mock theta conjectures 1 1.1 Invariants of quadratic fields and mock modular forms At the 1932 International Congress of Mathematicians in Zurich, David Hilbert said that “the theory of complex multiplication (of elliptic modular functions) which forms a powerful link between number theory and analysis, is not only the most beautiful part of mathematics but also of all science” (see [Sc]). The theory of complex multiplication involves special values of the modular j-invariant 1 j(τ) = +744+196884q +··· , q := e2πiτ, q which is a weakly holomorphic modular form of weight 0 on SL (Z). For any 2 triple of integers (a,b,c) for which a > 0 and d := b2 −4ac < 0, the values (cid:18) √ (cid:19) −b+ d j (1.1.1) 2a are algebraic integers called singular moduli. One of the most beautiful results in the theory of complex multiplication asserts that if d is a fundamental √ discriminant (i.e. the discriminant of Q( d)) then the singular modulus √ (1.1.1) generates the Hilbert class field of Q( d). More generally, every √ abelian extension of Q( d) can be generated by the singular moduli, together with roots of unity and special values of certain elliptic functions (see [Cox, §11] for an excellent treatment of this story). Hilbert’s twelfth problem asks for the generalization of this theory to abelian extensions of any number field. Despite heroic efforts, essentially nothing is known about the next (supposedly) simplest case, that of the real quadratic fields. However, many fascinating connections between real quadratic fields and modular forms exist; for instance, the groundbreaking work of Duke, Imamo¯glu, and T´oth [DIT] connects invariants of real and imaginary quadratic fields to coefficients of harmonic Maass forms. (Chapter 6) together represent the completion of the research proposed in the author’s Dissertation Completion Fellowship proposal. TheresultsofSection1.1, togetherwiththecorrespondingChapters3and4, appeared in the papers [A1] and [A2]. 2 Briefly, a harmonic Maass form is a weak Maass form which is annihilated by the hyperbolic Laplacian (i.e. has eigenvalue 0). Such forms have a canonical decomposition f = f+ + f− into the holomorphic part f+ (also called a mock modular form) and the nonholomorphic part f−. There is a differential operator ξ which sends harmonic Maass forms of weight k to k weakly holomorphic modular forms of weight 2−k, and the modular form ξ f = ξ f− is called the shadow of f+. k k For a nonzero integer d ≡ 0,1 (mod 4), let Q denote the set of integral d binary quadratic forms Q(x,y) = [a,b,c] = ax2+bxy+cy2 with discriminant d = b2 − 4ac which are positive definite if d < 0. The modular group Γ = PSL (Z) acts on these quadratic forms as 1 2 (cid:18) (cid:19) A B (γ.Q)(x,y) = Q(Dx−By,−Cx+Ay) for γ = ∈ Γ . 1 C D It is well-known that the set Γ \Q forms a finite abelian group under 1 d Gaussian composition. For Q ∈ Q , let Γ denote the stabilizer of Q in Γ . d Q 1 Suppose first that d < 0. If Q ∈ Q then Q(τ,1) has exactly one root τ d Q in H, namely √ −b+ d τ = . Q 2a Here Γ = {1} unless Q ∼ [a,0,a] or Q ∼ [a,a,a], in which case Γ has order Q Q 2 or 3, respectively. For f ∈ C[j], we define the modular trace of f by (cid:88) 1 Tr (f) := f(τ ). (1.1.2) d Q |Γ | Q Q∈Γ1\Qd A theorem of Zagier [Z1] states that, for j := j −744, the series 1 1 (cid:88) g (τ) := −2− Tr (j )q−d 1 d 1 q 0>d≡0,1(4) is in M! , the space of weakly holomorphic modular forms of weight 3/2 3/2 on Γ (4) which satisfy the plus space condition (see Section 3.2 for details). 0 Zagier further showed that g is the first member of a basis {g } for 1 D 0<D≡0,1(4) M! . Each function g is uniquely determined by having a Fourier expansion 3/2 D 3 of the form (cid:88) g (τ) = q−D − a(D,d)q−d. (1.1.3) D 0>d≡0,1(4) The coefficients a(D,d) with D a fundamental discriminant are given by a(D,d) = −Tr (j ), d,D 1 where Tr denotes the twisted trace d,D 1 (cid:88) χ (Q) Tr (f) := √ D f(τ ), (1.1.4) d,D Q D |Γ | Q Q∈Γ1\QdD and χ : Q → {±1} is defined in (3.1.2) below. D dD The coefficients of the forms in Zagier’s basis {g } also appear as coeffi- D cients of forms in the basis {f } given by Borcherds in Section 14 D 0≥D≡0,1(4) of [Bo]. Borcherds showed that the coefficients of the f are the exponents D in the infinite product expansions of certain meromorphic modular forms. Suppose now that Q has positive nonsquare discriminant; then Q(τ,1) has two irrational real roots. Let S denote the geodesic in H connecting the Q roots, oriented counter-clockwise if a > 0 and clockwise if a < 0. In this case the stabilizer Γ is infinite cyclic, and C := Γ \S defines a closed geodesic Q Q Q Q on the modular curve. In analogy with (1.1.4) we define, for dD > 0 not a square, (cid:90) 1 (cid:88) dτ Tr (f) := χ (Q) f(τ) . (1.1.5) d,D D 2π Q(τ,1) Q∈Γ1\QdD CQ Let M+ denote the space of mock modular forms of weight 1/2 on Γ (4) 1/2 0 satisfying the plus space condition (see Section 3.2 for definitions). A beautiful result of Duke, Imamo¯glu, and T´oth [DIT] shows that the twisted traces (1.1.4) and (1.1.5) appear as coefficients of mock modular forms in a basis {f } for M+ . When D < 0, the form f is a weakly holomorphic D D≡0,1(4) 1/2 D modular form, and is uniquely determined by having a Fourier expansion of the form (cid:88) f (τ) = qD + a(d,D)qd. D 0<d≡0,1(4) The coefficients a(d,D) are the same as those in (1.1.3). Therefore, when D is a fundamental discriminant, they are given in terms of twisted traces. 4 When D > 0 the mock modular form f is uniquely determined by being D holomorphic at ∞ and having shadow equal to 2g . Let D (cid:88) f (τ) = a(d,D)qd. D 0<d≡0,1(4) If D is a fundamental discriminant and dD is not a square, then Theorem 3 of [DIT] shows that a(d,D) = Tr (j ). d,D 1 In [DIT] the coefficients a(d,D) for square dD are defined as infinite series involving Kloosterman sums and the J-Bessel function. The authors leave an arithmetic or geometric interpretation of these coefficients as an open problem. When the discriminant of Q is a square, the stabilizer Γ is trivial. In Q this case the geodesic C connects two elements of P1(Q), but since any Q f ∈ C[j] has a pole at ∞ (which is Γ -equivalent to every element of P1(Q)), 1 the integral (cid:90) dτ f(τ) (1.1.6) Q(τ,1) CQ diverges. This is the obstruction to a geometric interpretation of the modular trace for square discriminants. In a recent paper, Bruinier, Funke, and Imamo¯glu [BFI] address this issue by regularizing the integral (1.1.6) and showing that the corresponding modular traces 1 (cid:88) (cid:90) reg dτ Tr (j ) = j (τ) d 1 1 2π Q(τ,1) Q∈Γ1\Qd CQ give the coefficients of f . Their proof is quite different than the argument 1 given in [DIT] for nonsquare discriminants. It involves a regularized theta lift and applies to a much more general class of modular functions (specifically, weak harmonic Maass forms of weight 0 on any congruence subgroup of Γ ). 1 Here we provide an alternate definition of Tr when dD is a square and d,D show that the corresponding coefficients of f are given in terms of convergent D integrals of functions j which are related to j . Furthermore, using this 1,Q 1 definition we show that a suitable modification of the proof of Theorem 3 of [DIT] for nonsquare discriminants works for all discriminants. We first define a sequence of modular functions {j } which forms a m m≥0 5