Notes for a course given at the Second EU/US Summer School on Automorphic Forms on SINGULAR MODULI AND MODULAR FORMS

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1 Notes for a course given at the Second EU/US Summer School on Automorphic Forms on SINGULAR MODULI AND MODULAR FORMS LECTURERS: WILLIAM DUKE, PAUL JENKINS ASSISTANT: YINGKUN LI. Introduction Singular moduli are the values of the modular j-function at the points in the upper halfplane that satisfy a quadratic equation. These values have been studied by number theorists since the 9 th century. They are algebraic and generate class fields of imaginary quadratic fields. Both their norms and traces are integers. The work of Gross and Zagier in the 980s gave explicit factorizations of the norms and led to the celebrated Gross-Zagier formula. The traces, which were studied by Zagier in 2003, turn out to be related to Fourier coefficients of modular forms. In this course, we will explore the connections between singular moduli and modular forms, Borcherds product and harmonic Maass forms. Contents: Lectures ( Singular moduli: basic results (2 Weakly holomorphic modular forms (3 Traces of singular moduli and Borcherds products (4 Harmonic Maass forms of weight one Exercises References Lecture : Singular moduli: basic results A weakly holomorphic modular form f(z of weight k 2Z for Γ = SL(2, Z is a holomorphic function on the upper half-plane H := {z C : Im(z > 0} such that ( az + b (f k γ(z := (cz + d k f = f(z cz + d for all γ = ( a b c d Γ and f(z = n n 0 a(nq n, q = e(z = e 2πiz Date: July 204 in Bristol.

2 2 LECTURERS: WILLIAM DUKE, PAUL JENKINS ASSISTANT: YINGKUN LI with n 0 = ord f. Let M k! denote the C vector space of all such forms. Any f M k! satisfies the valence formula k ( 2 = ord f + ord z f, w z z F where F is the fundamental domain and w z is the order of the subgroup of Γ/{± ( } stabilizing z and given by 2, z = i, (2 w z = 3, z = ρ,, otherwise. F i ρ 0 This is obtained by integrating F (z := f (z ik around F and applying Green s Theorem. f(z 2y Write k = 2l + k with l Z, k {0, 4, 6, 8, 0, 4}. Then equation ( implies that f(z = 0 whenever ord f > l. Let S k M k M k! be the subspaces of cusp forms and modular forms, i.e. f S k, resp. f M k, if ord f > 0, resp. ord f 0. To construct modular forms, one can start with Eisenstein series. For k 4 even, let (3 E k (z := 2 (c,d Z 2 gcd(c,d= (cz + d = + 2 k ζ( k σ k (nq n M k, where ζ(s is the Riemann zeta function and σ k (n := d n dk is the divisor sum function. The Fourier expansion of E k (z can be derived from the following trigonometric identity (4 z + ( z d + = π cot(πz = πi 2πi q m. z + d d= When k = 4 and 6, we have E 4 (z = + 240q + O(q 2, E 6 (z = 504q O(q 2, ord ρ E 4 = ord i E 6 =. n m=0

3 SINGULAR MODULI 3 Define the modular discriminant function (z M 2 and modular j-invariant j(z by (z := E3 4(z E 2 6(z 728 j(z := E3 4(z (z = q c(nq n. n = q 24q q 3 472q , Since ord (z =, equation ( implies that (z has no zeros in H and j(z M! 0. For each m 0, there is a unique j m M! 0 such that j m (z = q m + O(q = F m (j(z with F m (x Z[x] a monic polynomial. The existence is easily shown by induction, e.g. F 0 (x =, F (x = x 744, F 2 (x = x 2 488x The F m (x s are called Faber polynomials. Their uniqueness follows from equation ( and that {j m } m 0 is a basis for M! 0. So any f(z = a(nq n M! 0 having a(n Z can be expressed uniquely as f(z = P (j(z with P (x Z[x]. This is called the q-expansion principle. The generating function of j m (z is given by (5 j (z j(z j(τ = m 0 j m (τq m = m 0 F m (j(τq m, which comes from integrating jm(zj (z j(z j(τ truncated fundamental domain F Y. around F Y, the boundary of the closure of the iy F Y i ρ 0 Letting τ = i and ρ yields the q-expansion of various modular forms j m (iq m = j (z j(z 728 = E2 4(zE 6 (z (z (z E6(z = E2 4(z 2 E 6 (z, m 0 m 0 j m (ρq m = j (z j(z = E2 4(zE 6 (z (z (z E 3 4(z = E 6(z E 4 (z. Remark. The generating function generalizes to other nonzero weight k as well.

4 4 LECTURERS: WILLIAM DUKE, PAUL JENKINS ASSISTANT: YINGKUN LI A CM point is τ H such that Aτ 2 + Bτ + C = 0, A, B, C Z, A > 0. Thus, τ = B+ D 2A with D = B 2 4AC 0, (mod 4 negative and D not a square. Theorem 2. If τ is a CM point, then j(τ is an algebraic integer. Example 3. If D is even, then τ = D, otherwise τ = + D. 2 2 D τ 0 ρ i 7i 2 2i + i 2 3i + 5i 2 j(τ As preparation for the proof of Theorem 2, consider the following polynomial in x for m ( ( σ(m az + b x j = f r (zx r. d ad=m b d Now the set {( a 0 d b : ad = m, b d} is a complete set of representatives for Γ\Γ m, where {( } a b Γ m := M(2, Z : ad bc = m. c d Each f r (z is a symmetric function in {j(γz : γ Γ\Γ m }. Since γ Γ m (γ = Γ m for any fixed γ Γ and that f r (z has at most exponential growth at, we know that f r (z M 0! for each 0 r σ(m. For fixed a, d, consider (6 b d ( x j ( az + b = ( d d b d ( e ( b d = ( d q a + n d+ r=0 q a d + (744 x + n A n (xq na d, c(ne ( bn d na q d where A n (x R[x], R = Z[e ( d ]. From equation (6, we see that An (x is left invariant under all Galois automorphisms of Q(e 2πi/d, which implies that A n (x Z[x]. Also A n (x vanishes unless d n since the left hand side of equation (6 is invariant under z z +. Thus, we could write b d ( x j ( az + b = ( d q a + A nd (xq na. d n 0 It follows that f r (z = n a r(nq n with a r (n Z and that there exists P r (x Z[x] satisfying f r (z = P r (j(z

5 SINGULAR MODULI 5 by the q-expansion principle. In particular, P σ(m (x = and P σ(m (j(z = ( az + b j = 744σ(m + d (7 ad=m b d = 744σ(m + j m (z. ad=m b d ( az + b j d For example when m = 2, σ(m = 3 and ( ( z ( ( z + x j x j (x j(2z = x 3 + P 2 (j(zx 2 + P (j(zx + P 0 (j(z, 2 2 where ( ( z z + P 2 (j(z = j + j + j(2z 2 2 = (q O(q + ( q O(q + (q O(q Similarly, we have We thus have = q O(q = j(z 2 488j(z P (j(z = 488j(z j(z , P 0 (j(z = j(z j(z j(z ad=m b d ( x j ( az + b = Ψ m (x, j(z, d where Ψ m (x, y Z[x, y] has degree σ(m in x. For example, the calculations above tells us that Ψ 2 (x, y = x 2 y 2 + x 3 + y (x 2 y + xy (x 2 + y xy (x + y Remark 4. It turns out that Ψ m (x, y = Ψ m (y, x. transformations in Γ m, we will consider Ψ m (x, x. Since CM points are fixed points of Lemma 5. For m not a square, the polynomial ±Ψ m (x, x is monic of degree (8 G(m := max ( d, m d. d m

6 6 LECTURERS: WILLIAM DUKE, PAUL JENKINS ASSISTANT: YINGKUN LI Remark 6. When m = 2, we have which has degree G(2 = 4. Proof. By definition, we have Ψ 2 (x, x = (x (x 728(x 8000, Ψ m (j(z, j(z = ad=m b d = ad=m b d ( j(z j ( q e ( b d ( az + b d q a d + O(. Using the fact that ( ( b d x e b d y = x d y d, we obtain Ψ m (j(z, j(z = ( q d q a + lower order terms, ad=m which implies the lemma. Note that q d q a is nonzero since m is not a perfect square. Turning now to the proof of Theorem 2, let τ be a CM point with Aτ 2 + Bτ + C = 0, D = B 2 4AC < 0. To realize τ as a fixed point of M = ( a b c d Γ m, observe that This happens if M τ = τ cτ 2 + (d aτ b = 0. M = ( 2 (t Bu Cu (t + Bu Au 2 for m = t2 Du 2, t = TrM, u Z nonzero and t Du (mod 2. For a fixed D, choose some 4 t, u such that t Du (mod 2 and t 2 Du 2 > 0 is not a square. Since j(mτ = j(τ, we have Ψ m (j(τ, j(τ = 0, where m = t2 Du 2. Since ±Ψ 4 m (x, x is monic integral, we see that j(τ is an algebraic integer. For any D < 0, D 0, (mod 4, let Q D := {Q(x, y = Ax 2 + Bxy + Cy 2 : A > 0, B 2 4AC = D, gcd(a, B, C = }. The set Q D is acted on by γ = ( a b c d Γ via (γ Q(x, y = A x 2 + B xy + C y 2 such that Q(x, y = A x 2 + B xy + C y 2, ( ( ( x a b x :=. c d y y

7 SINGULAR MODULI 7 This translates into linear fractional action on the associated CM point τ Q = B + D H. 2A The number of equivalence classes is denoted by h(d and is finite. We may choose the representatives in F. The class polynomial is defined by (9 H D (x := (x j(τ Q. Q Γ\Q D The following result is proven using the theory of CM elliptic curves. Theorem 7. The polynomial H D (x Z[x] is monic and irreducible. As a consequence, j(τ Q has degree h(d over Q with conjugates j(τ Q, Q Q. A deeper fact is that the field H := Q( D, j(τ Q is abelian over K = Q( D. If D is fundamental, then H is the Hilbert class field of K, i.e. its maximal unramified abelian extension. Furthermore, the Galois group Gal(H/K is isomorphic to the class group Γ\Q D. For complete proofs see [6] or [6]. Lecture 2: Weakly holomorphic modular forms Recall that for any integer D < 0 such that D 0, (mod 4, we have the set Q D = {Q(x, y = Ax 2 + Bxy + Cy 2 : A > 0, B 2 4AC = D, (A, B, C = }, and that the quadratic form Q Q D has root τ Q H. We have the class polynomial H D (x = (x j(τ Q = x h(d j(τ Q x h(d +. Q Γ\Q D Q Γ\Q D We modify these definitions slightly now: let Q D be the set of all positive definite binary quadratic forms of discriminant D, not just the primitive forms, and let ω Q = 2 if the form Q ax 2 + ay 2, ω Q = 3 if Q ax 2 + axy + ay 2, and ω Q = otherwise. Modify the class polynomial to get H D (x = (x j(τ Q /ωq. Q Γ\Q D This is still a polynomial unless D/3 is a square (in which case it is a polynomial multiplied by x /3 or D is a square (in which case it is a polynomial multiplied by (x 728 /2. The class number h(d now becomes the Hurwitz class number H(D = Q Γ\Q D ω Q, where H(0 =. 2 Borcherds [3] proved the following theorem.

8 8 LECTURERS: WILLIAM DUKE, PAUL JENKINS ASSISTANT: YINGKUN LI Theorem 8. We have H D (j(z = q H(D n= ( qn a /2( D,n 2, where a /2 ( D, n 2 is the Fourier coefficient of q n2 of the unique weakly holomorphic modular form of weight /2 and level 4 in the Kohnen plus space with Fourier expansion beginning q D + O(q. A more general version of the theorem is as follows. Let f(z = c(nq n be a weakly holomorphic modular form with integer coefficients of weight /2 and level 4 in the Kohnen plus space M! /2 (4. Let h be the constant term of the product f(z ( n 0 H( nqn. Then F (z = q h n>0 ( q n c(n2 is a meromorphic modular form for some character of SL 2 (Z of weight c(0; additionally, F (z has integer coefficients, and all of its zeros and poles are at cusps or imaginary quadratic irrationals. Specifically, if τ H has discriminant D < 0, then ord τ (F = d>0 c(dd2. This mapping is an isomorphism between the additive group of weakly holomorphic modular forms with integer coefficients in the Kohnen plus space M /2! (4 and the multiplicative group of meromorphic modular forms with integer coefficients, leading coefficient, and Heegner divisor. Example. The Jacobi theta function θ(z = n Z qn2 = +2q +2q is a modular form in M /2 (4. We look at 2θ(z = q + 24q q and see that the isomorphism gives h = and F (z = q n ( q n 24 = (z. Example. The Eisenstein series E 4 (z has a zero at ρ = +i 3 2 and has integer coefficients and leading coefficient. Expanding its Fourier series as an infinite product, we get E 4 (z = + 240q + 260q q = ( q 240 ( q ( q Sure enough, there is a modular form of weight /2 with Fourier expansion q q q q q q This can also be done for E 6 (z, E 8 (z, E 0 (z, E 4 (z, and j(z, among others, since we know exactly where their zeros and poles are in the fundamental domain. Note that the product H D (j(z = (j(z j(τ Q satisfies the conditions of the more general theorem. Explicit bases for spaces of modular forms. To better understand this theorem, we now look at explicit bases for spaces of modular forms of integral and half integral weight. For each even integer weight k, write k = 2l+k, where k, l Z and k {0, 4, 6, 8, 0, 4}. For each integer m l, there is a unique modular form f k,m (z M k! with a Fourier expansion of the form f k,m (z = q m + O(q l+.

9 SINGULAR MODULI 9 We can construct such a form explicitly using, j, and E k, where E 0 = ; specifically, we have f k,m = l E k F (j, where F (x is a monic polynomial in x of degree l + m with integer coefficients. We write f k,m (z = q m + n>l a k(m, nq n, so that all of the Fourier coefficients a k (m, n of these basis elements are integers. In SAGE, this is the Victor Miller basis for spaces of holomorphic modular forms; see also [9]. A generating function for these basis elements is given by f k,m (τq m = f k, l(τf 2 k, l (z. j(z j(τ m l For half integral weight modular forms for Γ 0 (4, there is a similar canonical basis. For any integer s we let M s+/2! (4 be the vector space of holomorphic functions on H that transform like θ 2s+ under Γ 0 (4, may have poles at the cusps, and satisfy the Kohnen plus-space condition that their q-expansion is supported on integers n with ( s n 0, (mod 4. ( The automorphy factor for θ is ( c d ε d cz + d for a matrix ( a b c d. Recall that the Shimura correspondence gives an isomorphism between spaces of modular forms of integral and half integral weight; specifically, for positive integers s, the space of cusp forms of weight s + /2 and level 4 in the Kohnen plus space is isomorphic to the space of cuspforms of weight 2s and level. From this, we compute dimensions of spaces of cusp forms of half integral weight and level 4 in the Kohnen plus space. In level 4, the space of holomorphic modular forms (not necessarily in the plus space can be constructed explicitly as products of powers of θ(z and the weight 2, level 4 Eisenstein series given by E(z = σ(2n + q 2n+. n=0 Once we have a set of holomorphic forms spanning the space, we take linear combinations to get into the plus space, and construct weakly holomorphic modular forms by multiplying by powers of j(4z. To construct forms of negative weight, we can divide holomorphic forms of positive weight by powers of (4z M 2 (4. It turns out that for every integer s, there is a canonical basis {f s+/2,m (z} for M s+/2! (4 where the basis elements have integral coefficients and a Fourier expansion of the form q m + a s+/2 (m, nq n, with the maximum possible gap in the Fourier expansion after the q m term. This is all constructed in [0]. For example, to get the form of weight /2 that corresponds to E 4 (z under the Borcherds isomorphism, we begin with weight 25/2 and find the two forms f 25/2, 4 = E 4 θ 9 6E 5 θ 5 = q 4 + O(q 5 and f 25/2, = Eθ 2 42E 2 θ E 3 θ E 4 θ E 5 θ E 6 θ = q + O(q 5. Divide both by (4z to get forms in the plus space of weight /2 beginning

10 0 LECTURERS: WILLIAM DUKE, PAUL JENKINS ASSISTANT: YINGKUN LI q 3 + O( and + O(q, and take the appropriate linear combination to get q O(q, which must be the correct form by dimension considerations. Just as in the integral weight case, there is a generating function for these basis elements. It is given by fk,m (τq m = f k(τf2 k (z + f k (τf 2 k(z ; j(4z j(4τ here, f k and fk are the first two basis elements of weight k. Zagier gave this generating function for k = /2, 3/2 in [20], and the generalization can be found in [0]. Examples. The first few coefficients of the first basis elements of weight /2 are f /2,0 (z = + 2q + 2q 4 + 0q 5 + 0q 8 +, f /2,3 (z = q 3 248q q q q 8 +, f /2,4 (z = q q q q q 8 +, f /2,7 (z = q 7 49q q q q 8 +. The first few basis elements of weight 3/2 are f 3/2, (z = q q 3 492q q 7, f 3/2,4 (z = q q q q 7, f 3/2,5 (z = q q q q 7, f 3/2,8 (z = q q q q 7. Looking at these Fourier expansions, we see that the coefficients of one basis element f /2,m (z appear as the negatives of the coefficients of a particular power of q in each of the f 3/2. This theorem is true more generally; in fact, for all integers m, n and any integral or half integral weight k we have the Zagier duality [20] a k (m, n = a 2 k (n, m. This duality theorem can be proved from the generating function (replace k with 2 k and switch τ and z, or by noting that the product f k,m (zf 2 k,n (z has constant term equal to a k (m, n + a 2 k (n, m, and that this product (hit with the U 4 operator if the weight is half integral is of weight 2 for SL 2 (Z, and every such form is the derivative of a polynomial in j and thus has a constant term of 0. Lecture 3: Traces of singular moduli and Borcherds products Borcherds proved his theorem (of which even the more general theorem above is a special case as a consequence of work on denominator formulas of infinite dimensional Lie algebras, but asked whether a proof of the isomorphism existed that used only modular forms for SL 2 (Z and not automorphic forms on larger groups. Work of Zagier [20] answered this question in the affirmative. The proof has two parts: first, showing that traces of singular

11 SINGULAR MODULI moduli are coefficients of weight 3/2 modular forms for Γ 0 (4, and then using the duality theorem above. We define traces of singular moduli of discriminant D for a SL 2 (Z-invariant function F (z by Tr D (F = F (τ Q. ω Q Q Γ\Q D It is convenient to use the basis elements f 0,m (z; for instance, if F = f 0,0 (z =, then Tr D (F = H(D, and if F = f 0, (z = j(z 744, then we get the algebraic trace of j(τ Q 744. For instance, we have the following traces of f 0, (z = j(z 744: D H(D /3 /2 4/3 2 Tr D (j Given a discriminant D, then, the class polynomial H D (j(z can be written as H D (j(z = (j(z j(τ Q /ωq Q Γ\Q D = (q f 0, (τ Q + O(q /ωq = q H(D ( Tr D (f 0, q + O(q 2. Q Γ\Q D In fact, it turns out that the identity (0 H D (j(z = q H(D exp ( Tr D (f 0,m qm m holds for all D. Zagier showed that these traces appear as coefficients of half integral weight modular forms. Specifically, he showed that if the basis element f 3/2,m (z is hit with the weight 3/2 Hecke operator T l to give a modular form T l f 3/2,m (z = a 3/2,l (m, nq n, then the trace Tr D (f 0,l is equal to the coefficient a 3/2,l (, D. (Recall that the half integral weight Hecke operators T l, for a prime l not dividing the level, act on a form f(z = a(nq n of weight s + /2 to give T l f(z = ( a(l 2 n + m= ( n l s a(n + l 2s a(n/l 2 q n. l The image is a modular form of the same weight and level. As an example, the traces of f 0, (z = j(z 744 appear as Fourier coefficients of the form f 3/2, (z = q 2+248q 3 492q 4 +49q q q 53008q q 5.

12 2 LECTURERS: WILLIAM DUKE, PAUL JENKINS ASSISTANT: YINGKUN LI A straightforward calculation using explicit formulas for the action of the Hecke operators shows that if T l f /2,m (z = a /2,l (m, nq n, then the stronger duality ( a /2,l (m, n = a 3/2,l (n, m holds for positive m, n. (Similar results will hold for pairs of weights k and 2 k for k = 5/2, 7/2, 9/2, /2, and 5/2, but not for higher weights where there are nonzero cuspforms of weight 2k. Taking l = m, m = D, and n =, we see from the Hecke operator formulas that a /2,m ( D, = n m na /2 ( D, n 2. Thus, we have the following computations, proving Borcherds s theorem: ( H D (j(z = q H(D exp Tr D (f 0,m qm m m= ( = q H(D exp a 3/2,m (, D qm (by modularity m m= ( = q H(D exp a /2,m ( D, qm (by duality m m= = q H(D exp m=( na /2 ( D, n 2 qm m n m ( ( = q H(D exp a /2 ( D, n 2 q mn m n m= = q H(D n exp(a /2 ( D, n 2 log( q n = q H(D ( q n a /2( D,n 2. n= The more general isomorphism between spaces of modular forms can be built from this along with product formulas for (z and E k (z; note that meromorphic modular forms satisfying the appropriate conditions can be built as a product of a power of, an Eisenstein series, and an appropriate rational function in j (which will be a quotient of class polynomials. The idea behind Zagier s proof of modularity is to show that the coefficients a 3/2 (, D satisfy certain recursions, and then to show that the traces satisfy the same recursions; it is

13 SINGULAR MODULI 3 a generalization of Kronecker s class number relations (see exercises. However, this proof does not generalize so well. A different proof is given by Duke [7] and Bruinier-Jenkins-Ono [5]. The ideas of the proofs generalize better to broader situations and is as follows: (BJO Construct Poincaré series of weight 3/2 for Γ 0 (4 with a pole of order m at ; explicit formulas for the Fourier coefficients Project to the Kohnen plus space to get a Maass form F m(z + with slightly different coefficients, with a pole of order m at all cusps Compare the nonholomorphic part of F m(z + to the nonholomorphic part of Zagier s Eisenstein series G(z = H(nq n + NH of weight 3/2 Make the nonholomorphic parts cancel, and we ve constructed f 3/2,m (z Explicit formulas for coefficients a 3/2 (m, n in terms of Bessel functions and Kloosterman sums K(m, n; c = ( ( c ma + na ε a e a c a (mod c (a,c= (D Traces can be written in terms of Bessel functions and Salie sums ( 2mx S d (m, c = e c x 2 d (mod c Use an identity between Kloosterman and Salie sums to complete the proof Applications of traces of singular moduli. The coefficients of the class polynomial H D (x = (x j(τ Q are symmetric polynomials in the singular moduli j(τ Q, and the coefficient of x h(d is the trace Tr D (j 744 = Tr D (f 0,. It turns out that knowing the values of Tr D (f 0,m for 0 m H(D is enough to compute H D (x. To do this, first note that f 0,m (z is a polynomial in j(z; for instance, f 0,2 (z = j(z 2 488j(z Let z = τ Q and sum over all Q Γ\Q D to see that Tr D (f 0,2 = j(τ Q 2 488Tr D (f 0, ; this lets us find j(τ Q 2. Similar computations allow us to recursively compute j(τ Q m. The Newton-Girard formulas may then be used to change power sums into symmetric polynomials, allowing us to compute all coefficients of H D (x if we know enough coefficients of f 3/2, (z. Another interesting application is the fact that e π 63 =

14 4 LECTURERS: WILLIAM DUKE, PAUL JENKINS ASSISTANT: YINGKUN LI We know that Q( 63 has class number, and the quadratic form x 2 + xy + 4y 2 has a root τ = +i 63 in the fundamental domain. The trace of j(τ 744 has only one term 2 but must still be an integer; approximating j(τ with just the q term and noting that the tail of the series is small gives exp(π 63 (integer. Generalizations. So far, we have interpreted the coefficients a 3/2,m (, D as traces of singular moduli; there are many more coefficients to these modular forms. Zagier showed that all of the coefficients a 3/2,l (m, n can be interpreted as twisted traces of modular functions by multiplying by a genus character ψ(q (coming from ψ d for some discriminant d which is a factor of D in the sum over quadratic forms. To take traces of a function F over Γ\Q D, the function F must be invariant under SL 2 (Z, so taking traces of functions of nonzero weight doesn t seem to make sense. However, it turns out that for modular forms of nonpositive weight, such a trace can be defined. If f is a modular form of integer weight k, the nonholomorphic differentiation operator q d dq of weight k (here z = x + iy raises the weight of f by 2 and preserves modularity but not holomorphicity. If k = 2 2s is nonpositive, applying this operator a total of s times gives a weak Maass form of weight 0, and traces can be taken. This was done by Zagier for small negative weights and by [0] for all negative weights; in fact, it turns out that the map (the Zagier lift given by differentiating to weight 0, taking traces, and adding an appropriate principal part takes modular forms of weight 2 2s with integer coefficients to modular forms of weight s + /2 or 3/2 s with integer coefficients, and is compatible with Hecke operators of integral and half integral weight. It is a negative weight analogue of the classical Shintani lift, summing over traces instead of integrating over geodesics. Zagier touches on this question of higher level at the end of his paper [20], and proves some results for small levels N where Γ 0(N has genus zero (so that there is still a nice explicit basis. Miller and Pixton [8] defined the Zagier lift for any weakly holomorphic modular form of nonpositive weight and general level N that has poles only at the cusp at. Bruinier and Funke [4] obtained results giving such a lift for arbitrary genus. See also the more recent work of Alfes and Ehlen [2]. The integrality-preserving property is extended to small prime level by Green [3], and denominators for all levels are bounded in work of Alfes []. If the discriminant D > 0, the problem changes; rather than summing over values of functions at points of discriminant D, the appropriate thing to do is to integrate over a geodesic. Cycle integrals of the j-function appear in Fourier coefficients of certain mock modular forms. See the work of Duke-Imamoḡlu-Tóth [8]. k 4πy

15 SINGULAR MODULI 5 Lecture 4: Harmonic Maass Forms of Weight One In this lecture, we will consider the product Q Γ\Q D Q 2 Γ\Q D2 (j(τ Q j(τ Q2 4/(wQ wq2 for discriminants D, D 2 < 0. By the theory of complex multiplication, this product of algebraic integers is in fact a rational integer when D, D 2 < 4 are fundamental. Gross and Zagier studied these integers in [4] and gave an explicit factorization. Let k Z be a half integer. For a positive integer N, let Γ 2 0(N SL(2, Z be the congruence subgroup defined by Γ 0 (N := {( a c d b SL(2, Z : N c}. We say that f(z has weight k and level N if it transforms like θ 2k (z with respect to Γ 0 (N, i.e. f(γz f(z = θ2k (γz θ 2k (z for all γ Γ 0 (N. Define the weight k Laplacian k by k := ξ 2 k ξ k where ξ k := 2iy k. The operator ξ z k has the property that it commutes the slash operator by changing the weight from k to 2 k. We are interested in studying real-analytic functions F : H C annihilated by the differential operator k. Such function has a Fourier expansion of the form where F (z = n n 0 c + (nq n n 0 c(nβ k (n, yq n, e 4πnt t k dt n > 0, y y β k (n, y := k n = 0, k, k log y n = 0, k =. In this form, one has (ξ k F (z = n 0 c(nqn. The Fourier expansion n n 0 c + (nq n is called the holomorphic part of F (z. When k = 2, the non-holomorphic Eisenstein series Ê2(z := πim(z n σ(nqn is a harmonic Maass form. When k = 3/2, there is an Eisenstein series Ê3/2(z studied by Hirzebruch and Zagier where ξ 3/2 (Ê3/2 is the Jacobi theta function of weight /2. When k = /2, examples are Ramanujan s mock theta functions. When k =, Deligne and Serre attached irreducible, odd, 2-dimensional complex Galois representations to newforms g(z, whose L-functions are then Artin L-functions. These were

16 6 LECTURERS: WILLIAM DUKE, PAUL JENKINS ASSISTANT: YINGKUN LI already studied by Hecke in the dihedral setting, i.e. the Galois representation is induced from a character of the absolute Galois group of a quadratic field. Let Q Q D for a fundamental discriminant D < 0. Define the quantity r Q (n = #{±(x, y Z 2 : Q(x, y = n}. The theta function ϑ Q (z := r Q (nq n n 0 is a modular form of weight one on Γ 0 ( D with character ( D. One could obtain an eigenform g ψ (z by summing together [Q] Γ\Q D with a class group character ψ g ψ (z := ψ(qϑ Q (z. [Q] Γ\Q D When ψ is a genus character, g ψ (z is an Eisenstein series. Otherwise, g ψ (z is a dihedral cusp form. We are interested in studying harmonic Maass forms ĝ(z whose image under ξ is g ψ (z. When D = p < 3 with p 3 (mod 4 and ψ a trivial character, the modular form g ψ (z is the Eisenstein series given by E p (z := ϑ Q (z = h( p + R p (nq n. 2 [Q] Γ\Q p n The harmonic Maass form Êp(z was constructed by Kudla-Rapoport-Yang in [5] by taking a derivative in s of the real-analytic Eisenstein series E (z, s. It satisfies ξ (Êp(z = E p (z and its holomorphic part is given by n 0 R+ p (nq n where ( for all l satisfying l p R p + (l = 2 log l =. These coefficients R p + (n has the arithmetic interpretation as the degrees of certain special cycles on an arithmetic curve parametrizing CM elliptic curves. Zagier noticed that his result with Gross on the factorization of differences of singular moduli can be written as (2 Q Γ\Q p log (j(τ Q j(τ Q 2/wQ Q Γ\Q D = ( Dp m δ p (mr p m Z where D < 0 is any discriminant relatively prime to p and δ p (m is 2 if p m and otherwise. For the newform g ψ (z, there also exists harmonic Maass forms ĝ ψ (z with holomorphic part r + ψ (nqn n p+ 24,

17 SINGULAR MODULI 7 such that ξ (ĝ ψ (z = g ψ (z. Furthermore, one could use these coefficients give an analogue of equation (2 [] (3 ψ 2 (Q log (j(τ Q j(τ Q 2/wQ Q Γ\Q p Q Γ\Q D = ( Dp m δ p (mr + 2 ψ. 2 4 m Z Notice that the sum is finite and even though Dp m2 could be negative. Similar result holds 4 with p replaced by a fundamental discriminant [2]. One way to prove equation (3 is to construct modular functions by Green s function. Then its special value at CM points can be expressed as an infinite sum involving the following counting function { } ρ Q (k, p := # Q k Q D : cosh d(τ Q, τ Q =, pd for k N and Q Q p. Here for z, z 2 H, cosh d(z, z 2 is the hyperbolic cosine of the hyperbolic distance between two points defined by cosh d(z, z 2 := + z z 2 2 2y y 2, where y j = Im(z j. An elementary argument using quadratic forms relates this to the Fourier coefficient of ϑ Q 2 by ( k 2 + pd ρ Q (k, p = r Q 2. 4 The details are outlined in the exercise. To give an example of the individual coefficients r + ψ (n, let p = 23 and ψ is a non-trivial character of the class group of K := Q( 23, the newform g ψ (z has the Fourier expansion g ψ (z = q ( q n ( q 23n. n= Let H := K[X]/(X 3 X be the Hilbert class field of K and α > the real root of X 3 X. Then there exists a harmonic Maass form ĝ ψ (z with ξ (ĝ ψ (z = g ψ (z and holomorphic part r + ψ (nqn n ( n 23 n=

18 8 LECTURERS: WILLIAM DUKE, PAUL JENKINS ASSISTANT: YINGKUN LI and r + ψ ( = 3r+ ψ ( = 3 log(α. It turns out that this harmonic Maass form is unique and the Fourier coefficients r + ψ (n has the shape r + ψ (n = 2 [Q] Γ\Q 23 ψ 2 (Q log u(n, [Q] with u(n, [Q] H having compatible action under Gal(H/K = C( p. When [Q] = [Q 0 ] is the principal class, we give some numerical values of the coefficients u(n, [Q 0 ] here. n u(n, Q 0 ϖ 5 := 2α 2 α ϖ 7 := α 2 + α 2 5α 6 ϖ5 ϖ := 2α 2 α n u(n, Q 0 7α 4 ϖ7 5αϖ5 ϖ 7 := 2α 2 + 3α + 3 ϖ 9 := 3α 2 + α n u(n, Q 0 5α 7 7α 0 ϖ7 α 5 ϖ ϖ 23 := 0α2 +8α+ 23 Notice that for all prime l satisfying ( l 23 =, the table shows that Nm(ϖl = l 2. This and more general results about the individual coefficient r + ψ (n has been obtained in [2, 9] using the technique of theta-lifting. Furthermore, the valuations of the algebraic numbers u(n, Q 0 have the arithmetic interpretation as the degrees of special cycles on arithmetic curves [2]. Beyond this, there are a lot of unknowns about the Fourier coefficients of weight one harmonic Maass forms ĝ(z when ξ (ĝ is a holomorphic newform associated to non-dihedral complex Galois representations. In [], there is a numerical example of the octahedral case, where the Fourier coefficients seem to be logarithms of algebraic numbers in the number field determined by the adjoint of the associated Galois representation. We expect these Fourier coefficients to be related to special values of automorphic forms, as well as to have arithmetic interpretations. n Z m Exercises Exercise. Define the Eisenstein series of weight 2 by G 2 (z := (mz + n = 2ζ(2 + 2 (mz + n, 2 m Z m 0 n Z where Z m = Z\{0} if m = 0 and Z otherwise. Note that the order of the summation matters here since the sum does not converge absolutely. The Fourier expansion of E 2 (z := G 2ζ(2 2(z is analogous to those of E 4 (z and E 6 (z and given by E 2 (z = 24 σ(nq n. n=

19 ( Show that (2 Show that z 2 G 2( /z = n Z m Z n m 0 d Z SINGULAR MODULI 9 (mz + n 2 = 2ζ(2 + n Z (mz + d(mz + d + = 0 m 0 (mz + n 2. and subtract it from G 2 (z to show that G 2 (z has the absolutely convergent expression G 2 (z = 2ζ(2 + (mz + d 2 (mz + d +. m 0 n Z (3 Show that G 2 (z z 2 G 2( /z = n Z = lim m 0 N n= N m 0 = 2 z lim (4 Prove the identity (4 z + ( z d + z + d d= and use it to deduce that (mz + d (mz + d + N N m= (mz + d (mz + d + ( N/z + m + ( N/z m. = π cot(πz = πi 2πi (5 E 2 (z z 2 E 2( /z = 2 2πiz. 3 (5 Show that Ê(z := E(z satisfies the transformation πim(z (Ê2 2 γ (z = Ê2(z for all γ Γ. Exercise 2. Define the function : H C by (6 (z := q ( q n 24. n= q m. m=0

20 20 LECTURERS: WILLIAM DUKE, PAUL JENKINS ASSISTANT: YINGKUN LI ( Prove that d 2πi dz log (z = 24 σ(me 2πimz = E 2 (z. m= (2 Using equation (5, show that ( d ( /z 2πi dz log = 0. z2 (z (3 Use the previous part and the valence formula to show that (z = (z = E3 4 (z E2 6 (z 728 S 2. (4 Suppose that f(z M k! has integral Fourier coefficients and leading coefficient. Prove that if f(z 0 = 0 for some z 0 H, then j(z 0 is an algebraic integer. Exercise 3. The polynomial Ψ m (x, y is reducible when m is not squarefree. Instead, one could consider Φ m (x, j(z := (x j(γz, γ Γ\ m (7 {( } a b m := Γ c d m : gcd(a, b, c, d =. ( Show the following subset of m consists of coset representatives of Γ\ m {( } a b 0 d m : gcd(a, b, d =, 0 b d and that it has size (2 Prove that ψ(m := m p m prime ( +. p Ψ m (x, y = d 2 m Φ m/d 2(x, y. In particular, Ψ m (x, y = Φ m (x, y when m is squarefree. Furthermore, one has σ(m = ( m ψ, d 2 d 2 m ζ(sζ(s ζ(2s = m ψ(m m s when Re(s is large enough.

21 SINGULAR MODULI 2 (3 Show that as a polynomial in x, the coefficients Φ m (x, j(z are modular functions in M! 0 with integral Fourier coefficients. From this, deduce that Φ m (x, y Z[x, y]. (4 Show that Φ m (x, y is irreducible as a polynomial in x over the field C(y. (5 Prove that Φ m (x, y = Φ m (y, x. Exercise 4. Let D < 0 be a discriminant. In this exercise, we will prove the finiteness of the size of Γ\Q D. For convenience, we will use (A, B, C to denote a binary quadratic form Ax 2 + Bxy + Cy 2 Q D and to represent equivalence under the action of Γ. We say a form (A, B, C is reduced if B A C. ( For any reduced form (A, B, C of a fixed discriminant D, prove that B < D /3. From this, deduce that the number of reduced forms of a fixed discriminant D < 0 is finite. (2 Let γ = ( a c d b Γ and calculate γ (A, B, C. What is the result when a = 0? (3 If (A, B, C, (A, B, C Q D are equivalent, prove that {Ax 2 + Bxy + Cy 2 : x, y Z} = {A x 2 + B xy + C y 2 : x, y Z}. (4 Prove that two forms (A, B, C and (A 2, B 2, C 2 of the same discriminant are equivalent if and only if there exist a, c Z such that A a 2 + B ac + C c 2 = A 2, 2A a + (B + B 2 c 0 (mod 2A 2, (B B 2 a + 2C c 0 (mod 2A 2. (5 Find all pairs of distinct reduced forms that are equivalent to each other. (6 Show that every binary quadratic form (A, B, C Q D is equivalent to a reduced form and prove that h(d is finite. (7 Show that every quadratic form (A, B, C Q 63 is equivalent to (,, 4 and that the size of Γ\Q 23 is 3. Exercise 5. For a fixed fundamental discriminant D < 0, let C(D denote the set of coset representatives of Γ\Q D. We will define the composition law after Dirichlet in this exercise. Two forms Q j = (A j, B j, C j Q D, j =, 2 are called united if gcd(a, A 2, (B +B 2 /2 =. ( If Q, Q 2 Q D are united forms, show that there exists b, c Z such that Q (A, b, A 2 c and Q 2 (A 2, b, A c. In this way, we define the composition of Q and Q 2 to be Q Q 2 := (A A 2, b, c Q D. (2 Prove that the composition respects the equivalence relation, i.e. if Q, Q 2 Q D and Q, Q 2 Q D are two pairs of united forms such that Q j Q j for j =, 2, then Q Q 2 Q Q 2. This then defines a composition law on C(D.

22 22 LECTURERS: WILLIAM DUKE, PAUL JENKINS ASSISTANT: YINGKUN LI (3 Deduce that (, B, C (A, B, C (A, B, C, (A, B, C (A, B, C (A, B, C (C, B, A (AC, B,, and conclude that C(D is an abelian group under composition. (4 Show that h(d is even if D < 0 is the product of a positive and a negative discriminant. Exercise 6. For a discriminant D < 0, define the Hurwitz class number H(D by H(D :=. w Q Q Γ\Q D It is the degree of the modified class polynomial H D (x := Q Γ\Q D (x j(τ Q /w Q. (8 ( When m is not a perfect square, prove that Ψ m (x, x = ± H t 2 4m(x. t <2 m (2 When m is a perfect square, prove that Ψ m (x, y Ψ (x, y = ± m H t 2 4m(x x=y H t 2 4(x. (3 Set H(0 := 2 and prove the following result due to Hurwitz max ( d, m d = H(t 2 4m. d m t Z t 2 m (4 Make a table of the Hurwitz class numbers H(D up to D 24. Exercise 7. In this exercise, we will explore the connection between CM points and CM elliptic curves. For a fixed τ H, one could associate a lattice, or a Z-module of rank 2, L τ = {u + vτ : u, v Z}. The complex torus C/L τ is the C points of an elliptic curve, denoted by E τ. Let (z, L τ be the Weiestrass -function defined by (9 (z, L τ := z + ( 2 (z + l. 2 l 2 l L τ l 0 ( By considering the derivative (z, L τ in z, show that (z, L τ is a doubly periodic function, i.e. (z + τ, L τ = (z +, L τ = (z, L τ.

23 SINGULAR MODULI 23 (2 Prove that (z, L τ and its derivative (z, L τ in z satisfy the equation (z, L τ 2 = f τ ( (z, L τ where f τ (x is a cubic polynomial in x given by f τ (x = 4x 3 g 2 (τx g 3 (τ = 4(x e (x e 2 (x e 3, g 2 (τ = 60 2ζ(4 E 4 (τ, g 3 (τ = 40 2ζ(6 E 6 (τ, e = (/2, L τ, e 2 = (τ/2, L τ, e 3 = (( + τ/2, L τ. Thus, the following map gives an embedding of the torus C/L τ into projective space C/L τ P 2 C z ( (z, L τ, (z, L τ, 0 (0,, 0. (Hint: Show that the difference between the two sides is a bounded, entire function in z C, which must be constant.. (3 Recall the discriminant of a polynomial f(x = a 0 x n + + a n x + a n = a 0 (x e (x e 2... (x e n is given by f := a 2n 2 0 (e i e j 2, i<j which is nonzero precisely when f(x has distinct roots. Furthermore, f can be written in terms of a k for 0 k n. Calculate the discriminant of f τ (x = 4x 3 g 2 (τx g 3 (τ in terms of the Eisenstein series E 4 and E 6. (4 Show that j(τ gives a bijection between Γ\H and the P C \{point}. (5 Use the previous part to show that for any A, B C satisfying A 3 27B 2, there exists λ C and τ H such that λ 4 g 2 (L τ = A, λ 6 g 3 (L τ = B. (6 An endomorphism ϕ : C/L τ C/L τ lifts to a homomorphism ϕ : C C, which is given by multiplication by α := ϕ(l τ. Clearly, the endomorphisms of an elliptic curve contains Z. Prove that there is extra endomorphism if and only if τ is a CM point. Exercise 8. For any f : C C and γ = ( a c d b GL(2, R, define the slash operator f k γ of integral weight k by (f k γ(z := det(γk/2 (cz + d k f ( az + b. cz + d

24 24 LECTURERS: WILLIAM DUKE, PAUL JENKINS ASSISTANT: YINGKUN LI For a prime p and integer k, define the operators U p, V p, T p on f(z by p ( ( U p (f(z := p k/2 f λ k 0 p (z, λ=0 V p (f(z := p k/2 (f k ( p 0 0 (z, T p (f(z := U p (f(z + p k V p (f(z. The T p operator is called the Hecke operator. ( Check that (f k (γ γ 2 (z = ((f k γ k γ 2 (z. (2 If f(z has the Fourier expansion f(z = n Z a(n, ye(nx, write out the Fourier expansion of U p (f(z and V p (f(z. (3 Show that if (f k γ(z = f(z for all γ Γ 0 (N, then T p (f has the same property for all p N. (4 Prove that E(z := n=0 σ(2n + q2n+ satisfies (E 2 γ(z = E(z for all γ Γ 0 (4. (5 Let k {4, 6, 8, 0, 4} and f k,m (z = n m a k(m, nq n = q m + O(q M! k. Write T p f k,m (z as a sum of basis elements f k,m (z. Compare Fourier coefficients to show that if p n, then the Fourier coefficient a k (m, np is divisible by p k. Is a similar statement true for k = 0 or for other weights? What about in half integral weight? Exercise 9. For each even integer weight k, write k = 2l + k, where k, l Z and k {0, 4, 6, 8, 0, 4}. For each integer m l, there is a unique modular form f k,m (z M! k with a Fourier expansion of the form f k,m (z = q m + O(q l+. We write f k,m (z = q m + n>l a k(m, nq n, so that all of the Fourier coefficients a k (m, n of these basis elements are integers. ( Express f k,m (z in terms of (z, j(z and E k (z with k {0, 4, 6, 8, 0, 4} for (k, m = (6, 3, (24, 2, (30,. (2 Prove the generating functions m l Specialize to k = 0 to obtain j (z f k,m (τq m = f k, l(τf 2 k, l (z, j(z j(τ j(z j(τ = m 0 j m (τq m = m 0 F m (j(τq m.

25 (3 Prove the identity H D (j(z = q H(D exp SINGULAR MODULI 25 ( Tr D (f 0,m qm m holds for all D. Exercise 0. ( Find the image of E 6 in M /2! (4 under Borcherds s isomorphism. (2 Use E(z, θ(z and (4z to construct f 3/2, (z. Then apply the Hecke operators T 2, T 3 to this form and compute the class polynomials of discriminants D = 5, 20, 23. m= (3 Using the class polynomial for D = 23 to show that ( ( H + := Q j = Q[X]/(X 3 X Show that H + is not Galois over Q and the Galois closure of H + is given by H := H + ( 23. Exercise. For two points z, z 2 H, the hyperbolic cosine of the hyperbolic distance between them is defined by cosh d(z, z 2 := + z z 2 2 2y y 2, where y j = Im(z j. ( Show that for any z, z 2 H and γ SL(2, R cosh d(γz, γz 2 = cosh d(z, z 2. (2 For discriminants D, D 2 < 0 and Q j = (A j, B j, C j Q Dj, let τ Qj H be the corresponding CM points for j =, 2. Show that cosh d(τ Q, τ Q2 is given by cosh d(τ Q, τ Q2 := 2A C 2 + 2C A 2 B B 2 D D 2. (3 We say that a quadratic form (A, B, C Q D is primitive if gcd(a, B, C =. Prove that every primitive quadratic form Q Q D is equivalent to a quadratic form (A, B, AC Q D with gcd(a, B =. Using the definition of composition to show that Q 2 is equivalent to (A 2, B, C. (4 For a positive definite quadratic form Q Q D, k 0 an integer and D < 0 a discriminant, define the sets S Q (k, D := { Q = (A, B, C Q D : cosh d(τ Q, τ Q = T Q (m := {(x, y Z 2 : Q(x, y = m}. k }, DD

26 26 LECTURERS: WILLIAM DUKE, PAUL JENKINS ASSISTANT: YINGKUN LI Suppose Q = (A, B, AC Q D. Show that for any Q = (A, B, C Q D ( (C CA, BA AB T Q 2. k 2 DD 4 (5 When D = p such that p 3 (mod 4 is a prime. Show that the map ( S Q (m, D T Q 2 k 2 pd 4 (A, B, C (C CA, BA AB is a bijection when p k. (6 Let ρ Q (k, D := #S Q (k, D and r Q (m := #T 2 Q(m for any Q Q p. Prove that ( k ρ Q (k, D 2 pd = r Q 2 δ p (k, 4 where δ p (k = 2 if p k and otherwise. Exercise 2. Recall that the differential operator ξ k is defined by for any integer k Z. ξ k := 2iy k z. ( Show that ( ξ 2 k ξ k = k := y 2 2 x y 2 ξ k (f k γ = (ξ k f 2 k γ. ( iky x + i, y for any differentiable function f : C C and γ SL(2, R. (2 Prove that for any z H and n > 0 β k (n, yq n = O(e 2πny. (3 Let f : H C be a real-analytic function defined by the Fourier series f(z := n 0 a(nβ k (nq n. Calculate ξ k (f as a Fourier series. (4 Calculate ξ 2 (Ê2(z, where Ê2(z = 3 πim(z + 24 n σ(nqn.

27 SINGULAR MODULI 27 References. Alfes, C., Formulas for the coefficients of half-integral weight harmonic Maass forms, Math. Z., Volume 227, Issue 3 (204, Alfes, C., Ehlen, S., Twisted Traces of CM values of weak Maass forms, J. Number Theory 33 ( Borcherds, R. Automorphic forms on O s+2,2 (R and infinite products. Invent. Math. 20 (995, no., Bruinier, J., Funke, J., On two geometric theta lifts. Duke Math. J. 25 (2004, no., Bruinier, J., Jenkins, P., Ono, K., Hilbert class polynomials and traces of singular moduli, Math. Ann. 334 (2006, Cox, D. Primes of the form x 2 + ny 2. Fermat, class field theory and complex multiplication. A Wiley- Interscience Publication. John Wiley & Sons, Inc., New York, 989. xiv+35 pp. 7. Duke, W., Modular functions and the uniform distribution of CM points, Math. Ann. 334 (2006, no. 2, Duke, W., Imamoḡlu, Ö. ; Tóth, Á. Cycle integrals of the j-function and mock modular forms. Ann. of Math. (2 73 (20, no. 2, Duke, W., Jenkins, P., On the zeros and coefficients of certain weakly holomorphic modular forms, Pure and Applied Mathematics Quarterly 4 (2008, no. 4, Duke, W., Jenkins, P., Integral traces of singular values of weak Maass forms, Algebra and Number Theory 2 (2008, no. 5, Duke, W., Li, Y., Harmonic Maass Forms of Weight One, (202, Duke Math. J., to appear. 2. Ehlen, S., CM values of regularized theta lifts, PhD thesis TU Darmstadt ( Green, N., Jenkins, P., Integral traces of weak Maass forms of genus zero odd prime level, preprint (203, arxiv: [math.nt]. 4. Gross, Benedict H.; Zagier, Don B. On singular moduli. J. Reine Angew. Math. 355 (985, Kudla, Stephen S.; Rapoport, Michael; Yang, Tonghai On the derivative of an Eisenstein series of weight one. Internat. Math. Res. Notices 999, no. 7, Lang, S. Elliptic Functions. With an appendix by J. Tate. Second edition. GTM 2. Springer-Verlag, New York, Marcus, D. Number Fields. Universitext. Springer-Verlag, New York-Heidelberg, 977. viii+279 pp. 8. Miller, A., Pixton, A., Arithmetic Traces of non-holomorphic modular invariants, Int. J. Number Theory 06, 69 ( Viazovska, M., Petersson inner products of weight one modular forms, preprint ( Zagier, D., Traces of singular moduli, Motives, Polylogarithms and Hodge Theory (Eds. F. Bogomolov, L. Katzarkov, Lecture Series 3, International Press, Somerville (2002,