RANKIN-COHEN BRACKETS AND SERRE DERIVATIVES AS POINCARÉ SERIES BRANDON WILLIAMS Abstract. We give expressions for the Serre derivatives of Eisenstein and Poincaré series as well as their Rankin-Cohen brackets with arbitrary modular forms in terms of the Poincaré averaging construction, and derive several identities for the Ramanuan tau function as applications. 1. Introduction Let k Z, k 4. To any q-series φq φe πiτ n0 a nq n on the upper half-plane τ H whose coefficients grow slowly enough, one can construct a Poincaré series P k φ; τ φ k M 1 c,d n0 a n k πin e cτ+d that converges absolutely and uniformly on compact subsets and defines a modular form of weight k. Here, 1 1 the first sum is taken over cosets of Γ SL Z by the subgroup Γ generated by and the second 0 1 over all coprime integers c, d Z. A sufficient growth condition is a n On l with l k 1 ε for any ε > 0: a sketch for why this is sufficient is that one can bound aτ + b n + l 1 φ e πn 1 cτ+d 1 e π l cτ+d l n0 up to a constant multiple; and since 1 e x 1 < x ε for any fixed ε > 1 and all sufficiently small x > 0, one can maorize P k φ; τ by the series c,d k+l. More generally, this construction is useful even when φ is not holomorphic; see [7], section 8.3 for an overview of both holomorphic and nonholomorphic Poincaré series. A particular case of this construction is the Poincaré series of index N N and weight k: P k,n τ P k q N ; τ q N k M. These Poincaré series have the following nice property with respect to the Petersson scalar product f, g fτgτy k dx dy, f, g S k : for any cusp form f a nq n S k, Γ\H f, P k,n a N k! 4πN k 1. It follows that P k,n span M k ; and therefore that every modular form arises through the Poincaré series construction P k φ; τ although it is difficult to get general expressions for such a function φ using this argument. For most functions φ, the Fourier coefficients of the modular form P k φ; τ are rather complicated. This includes the series P k,n whose coefficients are infinite series over special values of Bessel functions and Kloosterman sums; cf. [4], section 3.. The most reliable way to produce manageable Poincaré series seems to be to start with input functions φ which already have some sort of modular behavior. 010 Mathematics Subect Classification. 11F11,11F5. 1
Example 1. When φ 1 which one can view as a modular form of weight 0, we obtain the normalized Eisenstein series: P k 1; τ E k τ 1 k σ k 1 nq n, q e πiτ, σ k 1 n d k 1, B k d n where B k is the k-th Bernoulli number. More generally, if φτ is a modular form of any weight k then expanding formally yields P k+l φ; τ aτ + b k l a b φ, M c d k l k φτ φτe l τ; although the expression P k+l φ; τ makes sense only when l is sufficiently large compared to the growth of the coefficients of φ. In recent work [8] the author has considered the Poincaré series P k ϑ; τ constructed from what are essentially weight 1/ theta functions ϑ, which seem to be useful for computing with vector-valued modular forms for Weil representations; the details are somewhat more involved but this is related to the example above. The motivation of this note was to consider the Poincaré series P k φ; τ when φ is a quasimodular form, a more general class of functions which includes modular forms, their derivatives of all orders, and the series E τ 1 4 σ 1 nq n cf. [11], section 5.3. We find that one obtains Rankin-Cohen brackets and Serre derivatives see section below for their definitions of Eisenstein series and Poincaré series essentially from such forms φ: Proposition. For any modular form f M k and l N, l 4, and m, N N 0, with l k + if f is not a cusp form, set m k + m 1 l + m 1 φτ q N N m r πi r f r τ; m r r then [f, P l,n ] m P k+l+m φ; τ. This expression for φ simplifies considerably when N 0 i.e. P l,n is the Eisenstein series; in this case, we find [f, E l ] m P k+l+m φ; τ for the function l + m 1 φτ πi m f m τ. m There may be particular interest in the case that f itself is an Eisenstein or Poincaré series as other expressions for Rankin-Cohen brackets of Poincaré series of a different nature e.g. [1], sect. 6 are known. Proposition 3. For any m, N N 0 and l N with l m +, set m m l + m 1! φτ q N r l + m r 1! E τ/1 r N m r ; then the m-th order Serre derivative of P l,n is ϑ [m] P l,n P l+m φ; τ.
As before, this simplifies when N 0: in this case, we find for the function φτ ϑ [m] E l P l+m φ; τ l + m 1! 1 m l 1! E τ m. It is interesting to compare this to proposition which suggests that the Serre derivative at least of the Eisenstein series is analogous to a Rankin-Cohen bracket with E. Similar observations have been made before e.g. [], section. By computing Rankin-Cohen brackets and Serre derivatives of P l,n 0 in weights l 10 we can obtain new proofs of Kumar s identity [6], eq. 14 τm 0m11 σ 1 nτm + n m 5/6 m + n 11 and Herrero s identity [3], eq. 1 τm 40m 11 σ 3 nτm + n m + n 11 that express the Ramanuan tau function in terms of special values of a shifted L-series introduced by Kohnen [5], as well as four additional identities of this form. Namely we find τm 14m8 σ 1 nτm + n m 7/1 m + n 8 16m9 σ 1 nτm + n m /3 m + n 9 18m10 m 3/4 40m 10 σ 1 nτm + n m + n 10 σ 3 nτm + n m + n 10. Here σ k n d n dk and τm is the coefficient of q m in q 1 qn 4. We can also compute the values of these series with m 0. Based on numerical computations it seems reasonable to guess that there are no other identities of this type. The details are worked out in section 4.. Background and notation Let H {τ x + iy : y > 0} be the upper half-plane and let Γ be the group Γ SL Z, which acts a b on H by τ c d cτ+d. A modular form of weight k is a holomorphic function f : H C which transforms under Γ by aτ + b f k a b fτ, M Γ, τ H c d and whose Fourier expansion involves only nonnegative exponents: fτ n0 a nq n, q e πiτ. We denote by M k the space of modular forms of weight k and by S k the subspace of cusp forms which in this context means a 0 0. The Rankin-Cohen brackets are bilinear maps [, ] n : M k M l M k+l+n, 3
[f, g] n πi n n 0 k + n 1 l + n 1 1 f τg n τ, n where f denotes the -th order derivative. If f n0 a nq n is the Fourier expansion of f then f πi n0 a n n q n ; and in particular, the Rankin-Cohen brackets preserve integrality of Fourier coefficients. For example, the first few brackets are [f, g] 0 fg, [f, g] 1 πi 1 kfg lgf, [f, g] πi kk + 1 fg k + 1l + 1f g ll + 1 + f g. These can be characterized as the unique up to scale bilinear differential operators of degree n that preserve modularity [10]. and The Serre derivatives ϑ [n] following [11], section 5.1 are maps M k M k+n defined recursively by ϑ [0] f f, ϑ [1] f ϑf 1 πi f k 1 E f, ϑ [n+1] f ϑϑ [n] nk + n 1 f 1 E 4 τfτ, n 1. In particular ϑ [n] is not simply the n-th iterate of ϑ. Here E τ is the nonmodular Eisenstein series E τ 1 4 σ 1 nq n 1 4q 7q... of weight two. These functions are given in closed form by n n k + n 1! ϑ [n] f r k + r 1! E /1 n r πi r f r. 3. Proofs Proof of proposition. The coefficients a n of any modular form of weight k satisfy the bound a n On k 1+ε for any ε > 0, while cusp forms satisfy the Deligne bound a n On k 1/+ε. Weaker, more elementary bounds could also be used here. In particular, the coefficients of m k + m 1 l + m 1 φτ q N πin m r f r τ n r r always satisfy the bound On k+m 1+ε, and our growth condition k + m 1 + ε k + l + m 1 ε becomes k l ε and therefore since k, l Z l k + ; while for cusp forms we instead require k 1 + m + ε k + l + m 1 ε, or equivalently 1 l ε which is always satisfied. Repeatedly differentiating the equation aτ + b f k fτ yields f m aτ + b m m k + m 1! r k + r 1! cm r k+m+r f r τ, 4
as one can prove by induction or derive directly by considering the action of SL Z on τ in the generating series f m τ wm fτ + w, m! m0 for w sufficiently small. By another induction argument one finds the similar formula d m m m k + m 1! dτ m k e πinτ r k + r 1! cm r πin r k m r e πinτ for any N N 0. Let a m a+m 1! a 1! a a + 1... a + m 1 denote the Pochhammer symbol. Then m k + m 1 l + m 1 ] [q N N m r πi r f r τ M m r r k+l+m m 0 πi m r k + m 1 l + m 1 m r r 0 m 1 f τ m r c m r l m r e N m r πi r c r k + r +r l m e [ k + m 1 πin r r πin cτ+d ], l + m 1 m r m r where we have replaced r by m r in the second equality. Since k + m 1 l + m r m r k + m r r m r k + m 1!l + m 1!m r!k + m r 1! r!k + m r 1!m r!l + r 1!!m r!k + 1! k + m 1 l + m 1 m l + r m r, m r πin k + m r as we see by replacing m r!k+m r 1! m r!k+m r 1! by m!l+m 1! m!l+m 1! in the above expression, this equals m k + m 1 l + m 1 πi m 0 [ 1 f τ m m m ] πin r l + r m r c m r l m r πin e cτ+d r }{{} m k + m 1 l + m 1 πi m 1 m 0 dm dτ m cτ+d l e πin cτ+d f τp m l,n τ [f, P l,n ] m. Proof of proposition 3. As before, the condition l m+ makes the Fourier coefficients of φ grow sufficiently slowly: the coefficients of E m can be bounded by On m 1+ε, so the growth condition is satisfied for all l m +. m 1 + ε l + m 1 ε cτ+d f τ 5
Using the transformation law aτ + b E it follows that for m N. Therefore, with we find 0 r φ M l+m E aτ + b φτ q N m m m m m l + m 1! 1 r N m r r l + m r 1! M m m m l + m 1! 1 r N m r r l + m r 1! m m m E τ 1 r m N r r 0 πi m m E τ + 6 c, πi m m+r c m r 1 m re τ r r πi m l + m 1! r l + m r 1! E τ/1 r N m r, r r c r r+ 1/πi r E τ e 0 r l + m 1! l + r 1! 1/πi r E τ M 1/πi m r M m r where in the last line we replaced r by m r. Since m l + m 1! m r 1 r m N r r l + r 1! l + m 1! l + m 1! πi/1 0 1/πi m r m r πin cτ+d [c r r+ l m e πin cτ+d c m r l m r πin e cτ+d, l + m 1! πin r, l + r 1! as one can see by expanding both sides of this equation, the expression above equals m m [ πi m E τ m l + m 1! m l + m 1! l + m 1! πi/1 r l + r 1! 0 ] c m r l m r πin r πin e cτ+d M m m l + m 1! πi E τ/1 m P m l,n τ l + m 1! ϑ [m] P l,n τ. 4. Examples involving Ramanuan s tau function In weight 1, the space S k of cusp forms is one-dimensional and therefore all Poincaré series are multiples of the discriminant τ τnqn ; we find this multiple by writing P k,m τ λ m τ and using 10! λ m,, P k,m τm 4πm, such that 11 P k,m 10! τm 4πm 11,. We can form the Poincaré series P 1 φ; τ from any q-series φτ n0 a nq n with a n On 5 ε. This includes the q-series E, E 4 and E 4. Applying propositions and 3 together with the vanishing of cusp forms in weight 10 gives identities involving τn. Similar arguments can be used to derive identities for the coefficients of the normalized cusp forms of weights 16, 18, 0,, 6. 6 ]
Example 4. By proposition 3, 0 ϑp 10,m P 1 [q m m 5 ] 6 E ; τ so we recover Kumar s identity Example 5. By proposition, which yields Herrero s identity Example 6. By proposition, which implies τm 0m11 m 5/6 m 5/6P 1,m + 5/6 4 0 P 8,m E 4 P 1 q m E 4 ; τ P 1,m 40 τm 40m 11 τm + nσ 1 n m + n 11. σ 3 np 1,m+n τ σ 3 nτm + n m + n 11. 0 [E 4, P 6,m ] 1 P 1 4mE 4 + 6 E 4 πi ; τ 4mP 1,m + 40 τm 60m 10 Together with the previous identity this implies τm 40m 10 Example 7. By proposition 3, σ 1 np 1,m+n τ, 4m + 6nσ 3 np 1,m+n, 4m + 6nσ 3 nτm + n m + n 11. m + nσ 3 nτm + n m + n 11 40m 10 0 ϑ [] P 8,m P 1 q m m 3/mE τ + 1/E τ ; τ σ 3 nτm + n m + n 10. where by Ramanuan s equation E πi 6 E E 4, the coefficient of q n in E τ is 40σ 3 n 88nσ 1 n; so we find and therefore 0 m 3 m + 1 P 1,m + m 3m + 1τm 4m 11 Combining this with the previous examples, we find and therefore τm 180m 9 τm 18m10 m 3/4, 36mσ 1 n + 10σ 3 n 144nσ 1 n P 1,m+n 3m 1nσ 1 n + 10σ 3 nτm + n m + n 11, m N. 7 nσ 1 nτm + n m + n 11 σ 1 nτm + n m + n 10.
Example 8. It is not valid to form the Poincaré series P 1 φ; τ with either φ E 3 or E 6, because their Fourier coefficients grow too quickly; however, their difference E 3 E 6 9 πi E 4 + 7 πi E has coefficients that satisfy the required bound On 5 ε. We use to obtain 0 ϑ [3] P 6,m + 7 36 P 6,mE 6 P 1 q m m 3 m E + 7/6mE 7/36E 3 E 6 ; τ m 3 m + 76 [ ] m P 1,m + 48m 336mn + 336n σ 1 n + 80m 40n P 1,m+n τm 70m 8 and combining this with the previous examples, τm 16m9 m /3 n σ 1 nτm + n m + n 11, σ 1 nτm + n m + n 9. Similarly, by expressing E in terms of powers of E and derivatives of modular forms one obtains the formula τm 14m8 σ 1 nτm + n m 7/1 m + n 8. Remark 9. In particular, for any m N the values of the L-series σ 1nτm+n m+n s at s 8, 9, 10, 11 and of σ 3nτm+n m+n at s 10, 11 are rational numbers, and Lehmer s conecture on the nonvanishing of s τ is equivalent to the nonvanishing of any of these L-values. Computing these L-series at other integers s numerically does not seem to yield rational numbers. In any case, the methods of this note do not apply to other values of s. We can also evaluate the values of these L-series with m 0 by a similar argument. Comparing ϑe 10 τ 5 6 4q... 5 6 E 1 + 38016 691 with the result of proposition 3, ϑe 10 τ 5 6 E 1τ + 0 σ 1 np 1,n τ we find i.e. τm 0 691 38016 10! τnτmσ 1 n, 4πn 11, τnσ 1 n 19 11 n 11 3 5 3 7 691 π11, 0.968. Here, the Petersson norm-square of computed to 18 decimal places is cf. [9], page 116., 1.0353605679 10 6 Similarly, comparing E 8 τe 4 τ 1 + 70q +... E 1 τ + 43000 691 with E 8 τe 4 τ P 1 E 4 ; τ E 1 τ + 40 σ 3 np 1,n τ, 8
we find i.e. and τm With similar arguments applied to 40 691 43000 10! τnτmσ 3 n, 4πn 11, τnσ 3 n 17 n 11 3 7 691 π11, 0.917. 3456 [E 4, E 6 ] 1 6P 1 πi 1 E 4; τ, 1 E 1 49344 691 ϑ[] E 8 1 P 1E; τ, 168 ϑ [3] E 6 + 7 36 E 6 7 36 PE 6 E 3 ; τ, 600 ϑ [4] E 4 35 864 E 4E 8 7 40 [E 4, E 4 ] + 35 43 [E 6, E 4 ] 1 35 3 P 1πi 3 E ; τ one can compute the values τnσ 3 n 16 n 10 3 3 5 3 7 π11, 0.845, τnσ 1 n 17 n 10 3 5 5 7 π11, 0.939, τnσ 1 n n 9 13 3 4 5 7 π11, 0.880, τnσ 1 n 14 n 8 3 3 5 7 π11, 0.754. Unlike the L-values of examples 4 through 8, none of these are expected to be rational. References [1] Nikolaos Diamantis and Cormac O Sullivan. Kernels for products of L-functions. Algebra Number Theory, 78:1883 1917, 013. ISSN 1937-065. URL https://doi.org/10.140/ant.013.7.1883. [] Fernando Gouvêa. Non-ordinary primes: a story. Experiment. Math., 63:195 05, 1997. ISSN 1058-6458. URL http://proecteuclid.org/euclid.em/10479040. [3] Sebastián Daniel Herrero. The adoint of some linear maps constructed with the Rankin-Cohen brackets. Ramanuan J., 363:59 536, 015. ISSN 138-4090. URL https://doi.org/10.1007/ s11139-013-9536-5. [4] Henryk Iwaniec. Topics in classical automorphic forms, volume 17 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1997. ISBN 0-818-0777-3. URL https://doi.org/ 10.1090/gsm/017. [5] Winfried Kohnen. Cusp forms and special values of certain Dirichlet series. Math. Z., 074:657 660, 1991. ISSN 005-5874. URL https://doi.org/10.1007/bf0571414. [6] Arvind Kumar. The adoint map of the Serre derivative and special values of shifted Dirichlet series. J. Number Theory, 177:516 57, 017. ISSN 00-314X. URL https://doi.org/10.1016/.nt.017. 01.011. [7] Ken Ono. Unearthing the visions of a master: harmonic Maass forms and number theory. In Current developments in mathematics, 008, pages 347 454. Int. Press, Somerville, MA, 009. [8] Brandon Williams. Poincaré square series for the Weil representation. Ramanuan J., in press. doi: 10.1007/s11139-017-9986-. URL https://doi.org/10.1007/s11139-017-9986-. [9] Don Zagier. Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields. pages 105 169. Lecture Notes in Math., Vol. 67, 1977. 9
[10] Don Zagier. Modular forms and differential operators. Proc. Indian Acad. Sci. Math. Sci., 1041:57 75, 1994. ISSN 053-414. URL https://doi.org/10.1007/bf0830874. K. G. Ramanathan memorial issue. [11] Don Zagier. Elliptic modular forms and their applications. In The 1--3 of modular forms, Universitext, pages 1 103. Springer, Berlin, 008. URL https://doi.org/10.1007/978-3-540-74119-0_1. Department of Mathematics, University of California, Berkeley, CA 9470 E-mail address: btw@math.berkeley.edu 10