HECKE OPERATORS ON CERTAIN SUBSPACES OF INTEGRAL WEIGHT MODULAR FORMS.

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1 HECKE OPERATORS ON CERTAIN SUBSPACES OF INTEGRAL WEIGHT MODULAR FORMS. MATTHEW BOYLAN AND KENNY BROWN Abstract. Recent works of Garvan [2] and Y. Yang [7], [8] concern a certain family of half-integral weight Hecke-invariant subspaces which arise as multiples of fixed odd powers of the Dedekind eta-function multiplied by SL 2 Z-forms of fixed weight. In this paper, we study the image of Hecke operators on subspaces which arise as multiples of fixed even powers of eta multiplied by SL 2 Z-forms of fixed weight. 1. Statement of Results. We let h denote the complex upper half-plane, and we recall the Dedekind eta-function, ηz := q 1/24 n=1 1 q n, z h, q := e 2πiz. In this paper, we study certain subspaces of modular forms constructed using the etafunction. To explain, we first give notation. Let N and k be integers with N 1, and let χ be a Dirichlet character modulo N. We let M k Γ 0 N, χ denote the complex vector space of holomorphic modular forms of weight k on Γ 0 N with nebentypus χ, and we let S k Γ 0 N, χ denote the subspace of cusp forms. When χ = 1 N, the trivial character modulo N, we may omit it; when N = 1, we abbreviate notation with S k M k. We extend notation to half-integral weights in the usual way: When 4 N and λ 0 is an integer, we denote by M λ+1/2 Γ 0 N, χ the space of holomorphic forms of weight λ + 1/2 which transform on Γ 0 N with theta-multiplier and nebentypus χ. For details on modular forms with integral weights, see Section 2 below and the references therein; for details on modular forms with half-integral weights, see [5] and [6]. Furthermore, when t Z is square-free, we let D t be the discriminant of Q t, and we define the Dirichlet character χ t := D t. We now precisely describe our setting. We let r be an integer with 1 r 23, and we define δ r := 24. We note that δ gcd24,r r is the least positive integer u for which 24 ru. Next, we let ψ r be the Dirichlet character modulo δ r defined by χ r/2 1 r even, 1.1 ψ r := χ 3 gcdr, 6 = 1, 1 δr r {3, 9, 15, 21}. Standard facts on the eta-function imply that { ηδ r z r S r/2 Γ 0 δ 2 r, ψ r, S r/2 Γ 0 δr, 2 ψ r, Date: February 9, Mathematics Subject Classification. 11F03, 11F11. 1 r even, r odd.

2 2 MATTHEW BOYLAN AND KENNY BROWN For all non-negative even integers s, we define the subspaces { 1.2 S r,s := {ηδ r z r S s+r/2 Γ 0 δ 2 F δ r z : F z M s } r, ψ r, S s+r/2 Γ 0 δr, 2 ψ r, r even, r odd. We extend the definition to all even s Z by defining S r,s = {0} when s < 0, and we note that M 2 = {0} implies that S r,2 = {0}. Recent works of Garvan [2] and Y. Yang [7] prove the Hecke invariance of S r,s when r is odd. These subspaces consist of half-integral weight modular forms. We recall that the Hecke operators with prime index are trivial on half-integral weight spaces. Furthermore, we mention that Yang [8] recently proved, for gcdr, 6 = 1, that S r,s is Hecke-module isomorphic via the Shimura correspondence to the χ 3 -twist of the subspace of newforms in S r+2s 1 Γ 0 6 with specified eigenvalues for the Atkin-Lehner involutions W 2 and W 3. Yang proved a similar result for r 0 mod 3. In contrast, r even implies that the subspaces S r,s consist of forms of integer weight. Therefore, for all primes p, the Hecke operators with index p are generally non-trivial. For all even integers 1 r 23 and all non-negative integers m, our main results describe the image of Hecke operators with index p m on S r,s. For example, we prove, for all even m 0, that the Hecke operators with index p m preserve S r,s. The m = 2 case is the analogue of the result of Garvan and Yang for even r. Hecke invariance results of this type can be very useful for explicit computation since S r,s has much smaller C-dimension than the ambient space of cusp forms in which it lies. In particular, we see that dim C S r,s = dim C M s, which is roughly s/12. For odd r, Garvan and Yang used Hecke invariance of S r,s in this way to prove explicit congruences for functions from the theory of partitions such as pn, the ordinary partition function, and sptn, Andrews function which counts the number of smallest parts in all partitions of n. We require further notation. Let N, k, and χ be as above. For all primes p, we denote the Hecke operator with index p on M k Γ 0 N, χ by T p,k,χ. When the context is clear, we use T p for T p,k,χ. For all integers 1 v 11, we define pv jv, p := pv 12, 12 the least positive residue of pv modulo 12, and we define We now state our main theorem. tv, p, k := k + v jv, p. Theorem 1.1. Let 1 v 11 be an integer, let s 0 be an even integer, and let p be prime. In the notation above, we have T p : S,s S 2jv,p,tv,p,s. Example. Let v = 7, and let s 0 be even. The theorem implies that S 14,s, p 1 mod 12, S 22,s 4, p 5 mod 12, T p : S 14,s S 2,s+6, p 7 mod 12, S 10,s+2, p 11 mod 12.

3 HECKE OPERATORS ON CERTAIN SUBSPACES OF INTEGRAL WEIGHT MODULAR FORMS. 3 Remark. When p δ, the operator T p agrees with Atkin s U p -operator, and we have T p : S,s {0} since forms in S,s have support on exponents coprime to δ, as in table 3.3 in Section 3 below. We also observe that T p : S,s {0} for v, p, s as in the table: v p s 1, 3, 5 3 mod 4 0, 4, 8 1, 4, 7 2 mod 3 0, mod mod 3 0, 4, 6, 10 All cases but v, p, s {5, 3 mod 4, 8, 7, 2 mod 3, 6} have tv, p, s < 0 or tv, p, s = 2, which give S 2jv,p,tv,p,s = {0}. When v, p, s = 5, 7 mod 12, 8, we have tv, p, s = 2, so to verify that T p : S 10,8 {0} for primes p 3 mod 4, it suffices to consider v, p, s = 5, 11 mod 12, 8. In this case, the theorem implies that T p : S 10,8 S 14,6. One may verify that η12z 26 S 2,12 has η12z 26 T 5 0 in S 10,8, a one-dimensional space. It follows that S 2,12 T 5 = S 10,8. We apply T p with p 11 mod 12 to obtain S 10,8 T p = S 2,12 T 5 T p = S 2,12 T p T 5 = 0, where the second equality results from commutativity of the Hecke operators and the third results from the third row of the table. A similar argument using η12z 26 T 7 0 shows, for primes p 2 mod 3, that T p : S 14,6 {0}. To conclude the remark, we recall that a q-series anq n is lacunary if and only if a density one subset of its coefficients vanish. For positive integers r, Serre [4] proved that η r is lacunary if and only if for all primes p 11 mod 12, we have η r T p = 0. Serre used this result to conclude that η r is lacunary with r > 0 and even if and only if r {2, 4, 6, 8, 10, 14, 26}. We observe that table entries v, p, s above with s = 0 imply the lacunarity of η r with r {2, 4, 6, 8, 10, 14}. The lacunarity of η 26 follows from the third row of the table. We now deduce some corollaries. Corollary 1.2. In the statement of the theorem, suppose that p is prime with p 1 mod δ. Then we have T p : S,s S,s. Proof. The hypothesis on p implies that there exists an integer t with p = 12t + 1. It follows that gcdv,12 v pv = 12t + v v mod 12. gcdv, 12 24t + 1 = gcd,24 Since 1 v 11, we see that v = jp, v is the least positive residue of pv modulo 12, which proves the statement. For all m 0 and primes p, we define T m p : = T p T p m times. With N, k, and χ as above, and with m 2, we recall that the Hecke operator with index p m on M k Γ 0 N, χ is 1.3 T p m := T p T p m 1 χpp k 1 T p m 2.

4 4 MATTHEW BOYLAN AND KENNY BROWN The m = 2 case of the following corollary is the integer weight analogue of the result of Garvan and Yang. Corollary 1.3. Let 1 v 11, let m and s be non-negative integers with s even, and let p be prime. In the notation above, we have { S,s, m is even, T p m : S,s S 2jv,p,tv,p,s, m is odd. Proof. We proceed by induction on m. When m = 0, the map T p m = T 0 is the identity; the case m = 1 is Theorem 1.1. We fix m 2, and we suppose that the statement of the corollary holds for 0 i m 1. When m is odd, the induction hypothesis implies that T p m 1 fixes S,s while T p and T p m 2 map S,s to S 2jv,p,tv,p,s. The statement of the corollary follows from 1.3. When m is even, the induction hypothesis implies that T p m 2 fixes S,s, and T p m 1 maps S,s to S 2jv,p,tv,p,s. Therefore, in view of 1.3, it suffices to show that T p maps S 2jv,p,tv,p,s to S,s. When p δ, the first part of the remark following Theorem 1.1 shows that T p : S,s {0}. Hence, for primes p δ, we verify that jjv, p, p = v. Since jv, p is the least positive residue of pv modulo 12, then jjv, p, p is the least positive residue of ppv = p 2 v mod 12. We suppose first that p 5. We observe that p 2 1 mod 12 and that 1 v 11 to conclude that jjv, p, p = v. When 2 δ, we have v {4, 8}; when 3 δ, we have v {3, 6, 9}. In these cases, one also sees that jjv, p, p = v. Next, we let n 1 with prime factorization n = p e i i. We recall that the Hecke operator with index n on M k Γ 0 N, χ is T n := T e p i. The following corollary subsumes Corollary i 1.2 and the even part of Corollary 1.3. Corollary 1.4. Let 1 v 11, let s 0 be an even integer, and let n 1 mod δ. In the notation above, we have T n : S,s S,s. The corollary follows from the definition of T n together with Corollary 1.3. When v has δ = 12, we also use the observation that n 1 mod 12 if and only if the multiplicity of all prime factors of n which are congruent to i mod 12 has the same parity for i {5, 7, 11}. For brevity, we omit the details. Remarks on eigenforms and one-dimensional subspaces. When s 4 is an even integer, we recall the Eisenstein series of weight s on SL 2 Z, E s z := 1 2s B s d s 1 q n M s, n=1 where B s is the sth Bernoulli number. We let E 0 z := 1, and we observe that M s is onedimensional if and only if s {0, 4, 6, 8, 10, 14}, in which case we have M s = CE s z. Hence, for such s and for all integers 1 v 11, we deduce that S,s = Cf,s z, with 1.4 f,s z := ηδ z E s δ z. For all n 0, we define a,s n Z by f,s z = a,s nq n. The remark following Theorem 1.1 together with Corollary 1.2 implies for all primes p, that f,s is an eigenform d n

5 HECKE OPERATORS ON CERTAIN SUBSPACES OF INTEGRAL WEIGHT MODULAR FORMS. 5 for T p for all, s in the set {2, 0, 4, 0, 4, 4, 4, 6, 4, 10, 6, 0, 6, 4, 6, 8, 8, 0, 8, 6, 12, 0, 12, 4, 12, 6, 12, 8, 12, 10, 12, 14}. v This fact is well-known; moreover, these forms are normalized newforms. With b v := gcd12,v as in 3.1, Corollary 1.4 together with 2.4 implies, for all s {0, 4, 6, 8, 10, 14} and for all n 1 mod δ, that f,s z as in 1.4 is an eigenform for T n with eigenvalue λ n,v,s = ψ dd v+s 1 a bvn,s d, where the sum is over d 2 gcdbv, n. The remaining sections of the paper proceed as follows. In Section 2, we provide further necessary facts on modular forms, and in Section 3, we prove Theorem 1.1. Acknowledgment. The authors thank Scott Ahlgren for helpful discussions in the preparation of this paper. 2. Background on modular forms. The proof of Theorem 1.1 requires certain facts from the theory of modular forms. For details, see for example [1] and [3]. a b We first discuss operators on spaces of modular forms. Let γ = GL c d + 2 Q, let N and k be non-negative integers with N 1, let χ be a Dirichlet character modulo N, and let fz = anq n M k Γ 0 N, χ. We define the slash operator on fz by We define W N := 0 1 N 0 f k γz := detγ k/2 cz + d k fγz. ; when χ is real, the Fricke involution 2.1 f f k W N = N k/2 z k f 1 Nz maps M k Γ 0 N, χ to itself. Next, for all primes p, we define the Hecke operator T p,k,χ. For convenience, we use T p when the context is clear. These operators map M k Γ 0 N, χ to itself, and they preserve the subspace of cusp forms. On q-expansions, we have 2.2 with a n p anq n T p,k,χ = apn + χpp k 1 a n p q n, = 0 when p n. When p N and χ is real, we observe the commutation 2.3 f k W N T p = χpf T p k W N. For all positive integers m, we recall the definition of T n as in the paragraph before Corollary 1.4. One can show that T n acts on q-expansions in M k Γ 0 N, χ by 2.4 amq m T n = mn d k 1 χda q m. d 2 d gcdm,n Let 1 v 11 and let s 0 be an even integer. In view of the definition 1.2 of our distinguished subspaces S,s, we record transformation formulas for SL 2 Z on the etafunction and on modular forms in M s. For all γ = SL a b c d 2 Z, there exists a 24th

6 6 MATTHEW BOYLAN AND KENNY BROWN root of unity, ɛ a,b,c,d, for which az + b η = ɛ a,b,c,d cz + d 1/2 ηz. cz + d We always take the branch of the square root having non-negative real part. As such, one can view the eta-function as a modular form of weight 1/2 on SL 2 Z with multiplier system given by {ɛ a,b,c,d }. Special cases of this transformation include 2.5 ηz + 1 = e 2πi 1 z 24, η = z i ηz. We also recall, for all F z M s, that 2.6 F z s γ = F z. 3. Proof of Theorem 1.1. We let 1 v 11 be an integer. We keep notation from Section 1, and we define 3.1 b v := δ 24 We note that 3.2 ηδ z = q δ 24 = v gcd12, v. 1 q δn = q bv + S v Γ 0 δ, 2 ψ n=1 has order of vanish at infinity equal to b v, and it has support on exponents congruent to b v mod δ. The following table gives values of v, b v, and δ. 3.3 v b v δ 1, 5, 7, 11 v 12 2, 10 v/2 6 3, 9 v/3 4 4, 8 v/ Let s 0 be a positive integer, and let F z M s. Then for all 1 v 11, we see that F δ z has support on exponents divisible by δ. Hence, we observe that forms in the subspace S,s have support on exponents congruent to b v mod δ. Lemma 3.1. Let v and s be non-negative integers with 1 v 11 and s even, and let δ, b v, and S,s be as above. Suppose that fz S,s, that p δ is prime, and that fz T p 0. Then fz T p has support on exponents congruent to pb v mod δ. Furthermore, fz T p has order of vanish at infinity greater than or equal to the least positive residue of pb v mod δ. Proof. For all integers m, we define a f m by fz = a f mq m S,s. The discussion preceding the lemma shows that a f m 0 implies that m b v mod δ. Since p δ and δ 12, we have p 5, p = 2 and v {4, 8}, or p = 3 and v {3, 6, 9}. In all cases we find that p 2 1 mod δ. Since fz T p 0, formula 2.2 shows that there exists m m 0 with a f pm 0 or p m and a f 0 If we suppose that a p f pm 0, then we have pm b v mod δ. Multiplying by p and using p 2 1 mod δ gives m pb v

7 HECKE OPERATORS ON CERTAIN SUBSPACES OF INTEGRAL WEIGHT MODULAR FORMS. 7 mod δ. m Similarly, if we suppose that p m and a f p 0, then we have m p b v mod δ. Multiplying by p gives m pb v mod δ. The second statement follows from the first since f T p 0. Proposition 3.2. Let 1 v 11 be an integer, and let δ and b v be as above. Suppose that p δ is prime and that j 1 is an integer. We have gcd12, v j and 2jδ pb 24 v mod δ with 1 2jδ δ 24 if and only if j pv mod 12 and 1 j 12. In other words, 2jδ is the least positive residue of pb 24 v modulo δ if and only if j is the least positive residue of pv modulo 12. Proof. First, we suppose that gcd12, v j and 2jδ pb 24 v mod δ. From the definitions, the congruence is equivalent to p v mod 12 j. We multiply by gcd12,v gcd12,v gcd12,v gcd12, v to obtain j pv mod 12. Conversely, we suppose that j pv mod 12. Then there exists t Z with 12t + pv = j. It follows that gcd12, v j. We divide by gcd12, v in the congruence to obtain j p v pb 12 j gcd12,v gcd12,v v mod. Since = 2j 12 = 2jδ, we find that gcd12,v gcd12,v 24 gcd12,v 24 2jδ pb 24 v mod δ. To conclude, we observe that 1 j 12 holds if and only if 1 2jδ 24 = j gcd12, v 12 gcd12, v = δ. We now prove Theorem 1.1. Let v and s be non-negative integers with 1 v 11 and s even, and let δ and b v be as above. In view of the remark following the statement of the theorem, we suppose that p δ is prime. We further recall that jv, p is the least positive residue of pv modulo 12 and that tv, p, s = s + v jv, p. For all F z M s, we will show that there exists G v,p,f z M tv,p,s such that For convenience, we define ηδ z F δ z T p ηδ z 2jv,p = G v,p,f δ z. H v,p,f z := ηδ z F δ z T p ηδ z 2jv,p. From Lemma 3.1, we see that the function in the numerator has support on exponents congruent to pb v mod δ. The definition of jv, p together with Proposition 3.2 imply that the denominator also has support on exponents in this progression. It follows, for all integers m, that there exists a v,p,f m with H v,p,f z = a v,p,f mq δm. We claim that G v,p,f z := H v,p,f z δ = a v,p,f mq m M tv,p,s. To see this, we first show that G v,p,f z transforms with weight s + v jv, p on SL 2 Z. From its q-series, we see that G v,p,f z = G v,p,f z + 1. It remains to prove that G v,p,f 1 = H v,p,f 1 z = z s+v jv,p H v,p,f = z s+v jv,p G v,p,f z. z δ z δ In particular, it suffices to prove that 3.4 H v,p,f 1 = δ 2 z s+v jv,p H v,p,f z.

8 8 MATTHEW BOYLAN AND KENNY BROWN The proof of 3.4 requires some basic facts. First, since F z M s, we replace z by δ z in 2.6 to obtain 3.5 F 1 = δ z s F δ 2s z. δ z Next, for all integers a 1, we replace z by δ z in 2.5 to get 3.6 η 1 2a = iδ z a ηδ z 2a. δ z We now use 2.1, 3.5, and 3.6 to compute ηδ z F δ z v+s W δ 2 = δ2 v+s 2 z v+s η 1 F 1 δ z δ z = δ z v+s iδ z v ηδ z δ z s F δ z = i v ηδ z F δ z = i v ηδ z F δ z. For convenience, we let g v,f z := ηδ z F δ z. In this notation, the previous calculation reads as 3.7 g v,f z v+s W δ 2 = i v g v,f z We study the function g v,f T p z that occurs in the numerator of H v,p,f z. Using 2.1, we observe that g v,f T p v+s W δ 2 = δ 2 v+s 2 z v+s g v,f T p 1 = δ 2 z v+s g v,f T p 1. 2 It follows that g v,f T p 1 = δ 2 z v+s g v,f T p v+s W δ 2 = δ z v+s ψ pg v,f v+s W δ 2 T p { δ z v+s i v g v,f z T p v even, 3.8 = δ z v+s i p 1 v g v,f z T p v odd. Since p δ, the commutation 2.3 gives the second equality; we used 1.1 and 3.7 for the third equality, observing that 1 p 1 2 = χ 1 p. We turn to the proof of 3.4. We suppose that 1 v 11 is odd, and we use 3.6 and 3.8 to compute g v,f T p 1 H v,p,f 1 = 2 η 1 δ z δ 2 z 2jv,p = ip 1 v δ z v+s g v,f T p iδ z jv,p ηδ z 2jv,p = ip 1 v δ z v+s jv,p g v,f T p i jv,p ηδ z 2jv,p = i p 1 v+jv,p δ z v+s jv,p H v,p,f z. Since v is odd and p δ, we see that p is odd. We recall that jv, p pv mod 12 to find that p 1 v + jv, p p 1 v + pv v + 1p 1 0 mod 4.

9 HECKE OPERATORS ON CERTAIN SUBSPACES OF INTEGRAL WEIGHT MODULAR FORMS. 9 Similarly, when 1 v 11 is even, we compute H v,p,f 1 = i v+jv,p δ 2 z v+s jv,p H v,p,f z. In this case, we have v + jv, p v + pv vp 1 mod 4. When p is odd, we have p 1 is even; then v and p 1 even imply that v + jv, p 0 mod 4. When p = 2, we have v {4, 8} implies that v + jv, p 0 mod 4. Hence, we conclude 3.4 in all cases. We replace z by z δ in 3.4 to see that G v,p,f 1/z = z v+s jv,p G v,p,f z. Since z z + 1 and z 1 generate SL z 2Z, it follows that G v,p,f z transforms with weight tv, p, s = v + s jv, p on SL 2 Z. To finish the proof of the lemma, we show that G v,p,f z is holomorphic on h and at the cusp z infinity. Since G v,p,f z = H v,p,f δ, it suffices to prove the same statement with G v,p,f z replaced by H v,p,f z. Since ηδ z F δ z T p is holomorphic on h and ηz 0 on h, we conclude that H v,p,f z is holomorphic on h. When ηδ z F δ z T p 0, Lemma 3.1 implies that it has order of vanish at infinity greater than or equal to the least positive residue of pb v mod δ. By Proposition 3.2 and the definition of jv, p, this least residue has value 2jv,pδ. Since ηδ 24 z 2jv,p has order of vanish at infinity equal to 2jv,pδ, we 24 deduce that H v,p,f z is holomorphic at infinity. References [1] F. Diamond and J. Shurman, A first course in modular forms, Graduate Texts in Mathematics, 228, Springer, New York, MR f:11045 [2] F. G. Garvan, Congruences for Andrews smallest parts partition function and new congruences for Dyson s rank, Int. J. Number Theory , no. 2, MR j:05032 [3] H. Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics, 17, Amer. Math. Soc., Providence, RI, MR e:11051 [4] J.-P. Serre, Sur la lacunarité des puissances de η, Glasgow Math. J , [5] G. Shimura, On modular forms of half integral weight, Ann. of Math , MR #10989 [6] G. Shimura, Modular forms: basics and beyond, Springer Monographs in Mathematics, Springer, New York, MR [7] Y. Yang, Congruences of the partition function, Int. Math. Res. Not. IMRN 2011, no. 14, MR e:11177 [8] Y. Yang, Modular forms for half-integral weights on SL 2 Z, Nagoya Math. J., to appear. Department of Mathematics, University of South Carolina, Columbia, SC address: boylan@math.sc.edu Cyan Inc., Petaluma, CA address: brownka5@mailbox.sc.edu