PRODUCTS OF NEARLY HOLOMORPHIC EIGENFORMS JEFFREY BEYERL, KEVIN JAMES, CATHERINE TRENTACOSTE, AND HUI XUE Abstract. We prove that the product of two neary hoomorphic Hece eigenforms is again a Hece eigenform for ony finitey many choices of factors. 1. Introduction It is we nown that the moduar forms of a specific weight for the fu moduar group form a compex vector space, and the action of the agebra of Hece operators on these spaces has received much attention. For instance, we now that there is a basis for such spaces composed entirey of forms caed Hece eigenforms which are eigenvectors for a of the Hece operators simutaneousy. Since the set of a moduar forms of a weights for the fu moduar group can be viewed as a graded compex agebra, it is quite natura to as if the very specia property of being a Hece eigenform is preserved under mutipication. This probem was studied independenty by Ghate [3] and Due [1] and they found that it is indeed quite rare that the product of Hece eigenforms is again a Hece eigenform. In fact, they proved that there are ony a finite number of exampes of this phenomenon. Emmons and Lanphier [2] extended these resuts to an arbitrary number of Hece eigenforms. The more genera question of preservation of eigenforms through the Ranin-Cohen bracet operator a biinear form on the graded agebra of moduar forms was studied by Lanphier and Taoo-Bighash [5, 6] and ed to a simiar concusion. One can see [7] or [9] for more on these operators. The wor mentioned above focuses on eigenforms which are new everywhere. It seems natura to extend these resuts to eigenforms which are not new. In this paper, we consider moduar forms which are od at infinity in the sense that the form comes from a hoomorphic form of ower weight. More precisey, we show that the product of two neary hoomorphic eigenforms is an eigenform for ony a finite ist of exampes see Theorem 3.1. It woud aso be interesting to consider the anaogous question for forms which are od at one or more finite paces. 2. Neary Hoomorphic Moduar Forms Let Γ = SL 2 Z be the fu moduar group and et M Γ represent the space of eve Γ moduar forms of even weight. Let f M Γ and g M Γ. Throughout, wi be positive even integers and r, s wi be nonnegative integers. Definition 2.1. We define Maass-Shimura operator δ on f M Γ by 1 δ f = 2πi 2iIm z + f z. z Write δ r := δ +2r 2 δ +2 δ, with δ 0 = id. A function of the form δ r f is caed a neary hoomorphic moduar form of weight + 2r as in [5]. 1
Let M Γ denote the space generated by neary hoomorphic forms of weight and eve Γ. Note that the image of δ is contained in M +2 Γ. Aso, the notation δ r f wi ony be used when f is in fact a hoomorphic moduar form. We define the Hece operator T n : M Γ M Γ foowing [4], as T n f z = n d 1 nz + bd 1 d f. d n b=0 A moduar form or neary hoomorphic moduar form f M Γ is said to be an eigenform if it is an eigenvector for a the Hece operators {T n } n N. The Ranin-Cohen bracet operator [f, g] : M Γ M Γ M ++2 Γ is given by [f, g] := 1 2πi + 1 + 1 1 a b a a+b= where f a denotes the a th derivative of f. Proposition 2.2. Let f M Γ, g M Γ. Then s δ r s fδs g = 1 δ s ++2r+2 Proof. Note that, δ ++2r δ r fg =0 δ r+ f a zg b z fg. = δ r+1 fg + δ r fδ g, and use induction on s. Combining the previous proposition and the Ranin-Cohen bracet operator gives us the foowing expansion of a product of neary hoomorphic moduar forms. Proposition 2.3. Let f M Γ, g M Γ. Then r+s δ r 1 s s fδs g = 1 +m m =0 ++2 2 m=max r,0 Proof. Lanphier [6] gave the foowing formua: δ n fz gz = n =0 1 n ++2 2 +n 1 n ++n+ 1 n r+m +r+m 1 r+m ++r+m+ 1 r+m δ n Substituting this into the equation in Proposition 2.2, we obtain s [ r+m δ r s 1 r+m fδs g = 1 m δ s m ++2r+2m m m=0 =0 Rearranging this sum we obtain the proposition. ++2 2 ++2 [f, g] z. +r+m 1 r+m ++r+m+ 1 r+m δ r+s ++2 [f, g] z. δ r+m ++2 [f, g] z We wi aso use the foowing proposition which shows how δ and T n amost commute. Proposition 2.4. Let f M Γ. Then δ m T n f z = 1 T n m n where m 0. 2 δ m f z ].
nz + bd Proof. Write F z = f. Note that z F z = n f z δ T n f z = n 1 d n = n 1 d n Next one computes that T n δ f z = n n 1 d n from which we see Now induct on m. d d 1 b=0 d d 1 b=0 [ 1 2πi nz + bd 2iImz F z + F z z [ 1 nz + bd 2πi 2iImz f d d 1 b=0 1 2πi 2iImz f nz + bd δ T n f z = 1 n T n δ f z. ] + n f z + n f z, so that nz + bd ]. nz + bd We woud ie to show that a sum of eigenforms of distinct weight can ony be an eigenform if each form has the same set of eigenvaues. In order to prove this, we need to now the reationship between eigenforms and neary hoomorhpic eigenforms. Proposition 2.5. Let f M Γ. Then δ r f is an eigenform for T n if and ony if f is. In this case, if λ n denotes the eigenvaue of T n associated to f, then the eigenvaue of T n associated to δ r f is nr λ n. Proof. Assume f is an eigenform. So T n f z = λ n fz. Then appying δ r and appying Proposition 2.4 we obtain the foowing: T n δ r f z = n r λ n δ r f z. So δ r f is an eigenform. Now assume that δ r f is an eigenform. Then T n δ r f f z. Us- z. Since δ r is ing Proposition 2.4, we obtain δ r T n f z = λ n inective, Hence f is an eigenform. n r δr T n f z = λ n n r fz. fz = δr Now our resut on a sum of eigenforms with distinct weights foow. z = λ n λn n f r δ r to both sides Proposition 2.6. Suppose that {f i } i is a coection of moduar forms with distinct weights i. t Then a i δ n i 2 i f i a i C i n is an eigenform if and ony if every δ 2 f i is an i=1 eigenform and each function has the same set of eigenvaues. 3 i
Proof. By induction we ony need to consider t = 2. : If T n δ r f = λδ r f, and T n δ +r 2 g = λδ T n, T n δ r f + δ 2 +r 2 +r g = λ δ r f + δ 2 +r 2 +r g, then by inearity of g. : Suppose δ r f + δ g is an eigenform. Then by Proposition 2.5 and inearity of δ r, f + δ 2 g is aso an eigenform. Write T n f + δ 2 g = λ n f + δ 2 g. Appying inearity of T n and Proposition 2.4 this is T n f + n 2 δ 2 T n g = λ n f + λ n δ 2 g. Rearranging this we get T n f λ n f = δ 2 λ n g n 2 Tn g. Now note that the eft hand side is hoomorphic and of positive weight, and that the right hand side is either nonhoomorphic or zero, since the δ operator sends a nonzero moduar forms to so caed neary hoomorphic moduar forms. Hence both sides must be zero. Thus we have T n f = λ n f and T n g = λ n n 2 g. Therefore f is an eigenvector for T n with eigenvaue λ n, and g is an eigenvector for T n with eigenvaue λ n n 2. By Proposition 2.5 we have that δ 2 g is an eigenvector for T n with eigenvaue λ n. Therefore f and δ 2 g are eigenvectors for T n with eigenvaue λ n. So δ r f and δ +r 2 g must have the same eigenvaue with respect to T n as we. Hence for a n N, δ r f and δ +r 2 g must be eigenforms with the same eigenvaues. Using the above proposition we can show that when two hoomorphic eigenforms of different weights are mapped to the same space of neary hoomorphic moduar forms that different eigenvaues are obtained. Lemma 2.7. Let < and f M Γ, g M Γ both be eigenforms. Then δ g and f do not have the same eigenvaues. Proof. Suppose they do have the same eigenvaues. That is, say g has eigenvaues λ n g, then by Proposition 2.5 we are assuming that f has eigenvaues n 2 λ n g. We then have 4 2
from mutipicity one there are constants c, c 0 such that fz = cn 2 λn gq n + c 0 = = n=1 1 2πi /2 /2 z /2 cλ n gq n + c 0 n=1 1 2πi /2 /2 z /2 gz + c 0 which says that f is a derivative of g pus a possiby zero constant. However, from direct computation, this is not moduar. Hence we have a contradiction. We sha need a specia case of this emma. Coroary 2.8. Let > and f M Γ, g M Γ. Then δ have the same eigenvaues. 2 +r g and δ r f do not From [6] we now that for eigenforms f, g, that [f, g] is a eigenform ony finitey many times. Hypotheticay, however, it coud be zero. In particuar by the fact that [f, g] = 1 [g, f], f = g and odd gives [f, g] = 0. Hence we need the foowing emma, where E denotes the weight Eisenstein series normaized to have constant term 1. Lemma 2.9. Let δ r f M +2r Γ, δ s g M +2s Γ. In the foowing cases [f, g] 0: Case 1: f a cusp form, g not a cusp form. Case 2: f = g = E, even. Case 3: f = E, g = E,. Proof. Case 1: Write f = A q, g = B q. Then a direct computation of the q- =1 coefficient of [f, g] yieds + 1 A 1 B 0 1 0. Case 2: Using the same notation, a direct computation of the q coefficient yieds + 1 + 1 + 1 A 0 B 1 + A 1 B 0 = 2A 0 A 1 0. =0 Case 3: This is proven in [6] using L-series. We provide an eementary proof here. Without oss of generaity, et >. A direct computation of the q coefficient yieds A 0 B + 1 1 + A 1 B + 1 0. Using the fact that A0 = B 0 = 1, A 1 = /B, B 1 = /B, we obtain 2 + 1 + 1 2 + 1. B B If is even, then both of these terms are nonzero and of the same sign. If is odd, then we note that for > 4, B + 1 = + 1 + 1B! > + 1 + 1B! = B + 1 5
using the fact that B > B for > 4, even. For = 4, the inequaity hods so ong as > 1. For = 1 the above equation simpifies to B > B which is true for, 8, 4, with this remaining cases handed individuay. For = 0, the Ranin-Cohen bracet operator reduces to mutipication. We wi need the fact that a product is not an eigenform, given in the next emma. Lemma 2.10. Let δ r f M +2r Γ, δ s g M +2s Γ both be cuspida eigenforms. Then fδs g is not an eigenform. δ r Proof. By Proposition 2.3 we may write δ r fδs g as a inear combination of δ r+s [f, g]. Then from [6], [f, g] is never an eigenform. Hence by Proposition 2.5, δ r+s ++2 never an eigenform. Finay Proposition 2.6 tes us that the sum, and thus δ r not an eigenform. ++2 [f, g] is fδs g is Finay, this ast emma is the driving force in the main resut to come: one of the first two terms from Proposition 2.3 is nonzero. Lemma 2.11. Let δ r f M +2r Γ, δ s g M +2s Γ both be eigenforms, but not both cusp forms. Then in the expansion given in Proposition 2.3, either the term incuding [f, g] r+s is nonzero, or the term incuding [f, g] r+s 1 is nonzero. Proof. There are three cases. Case 1: f = g = E. If r + s is even, then via Lemma 2.9, [f, g] r+s 0 and it is cear from Proposition 2.3 that the coefficient of [f, g] r+s is nonzero so we are done. If r + s is odd, then [f, g] r+s 1 is nonzero. Now because wtf = wtg, the coefficient of [f, g] r+s 1 is nonzero. This is due to the fact that if it were zero, after simpification we woud have = r + s + 1 0, which cannot occur. Case 2: If f is a cusp form and g is not then by Lemma 2.9, [f, g] r+s, and thus the term incuding [f, g] r+s is nonzero. Case 3: If f = E, g = E,. Again by Lemma 2.9, [f, g] r+s, and thus the term incuding [f, g] r+s is nonzero. 3. Main Resut Reca that E is weight Eisenstein series, and et be the unique normaized cuspida form of weight for {12, 16, 18, 20, 22, 26}. We have the foowing theorem. Theorem 3.1. Let δ r f M +2r Γ, δ s g M +2s Γ both be eigenforms. Then δ r fδs g is not a eigenform aside from finitey many exceptions. In particuar δ r fδs g is a eigenform ony in the foowing cases: 1 The 16 hoomorphic cases presented in [3] and [1]: E 2 4 = E 8, E 4 E 6 = E 10, E 6 E 8 = E 4 E 10 = E 14, E 4 12 = 16, E 6 12 = 18, E 4 16 = E 8 12 = 20, E 4 18 = E 6 16 = E 10 12 = 22, E 4 22 = E 6 20 = E 8 18 = E 10 12 = E 14 12 = 26. 6
2 δ 4 E 4 E 4 = 1 2 δ 8 E 8 Proof. By Proposition 2.3 we may write δ r fδs g = r+s =0 α δ r+s ++2 [f, g]. Now, by Proposition 2.6 this sum is an eigenform if and ony if every summand is an eigenform with a singe common eigenvaue or is zero. Note that by Coroary 2.8, α δ r+s ++2 [f, g] are aways of different eigenvaues for different. Hence for δ r fδs g to be an eigenform, a but one term in the summation must be zero and the remaining term must be an eigenform. If both f, g are cusp forms, appy Lemma 2.10. Otherwise from Lemma 2.11 either the term incuding [f, g] r+s or the term incuding [f, g] r+s 1 is nonzero. By [6] this is an eigenform ony finitey many times. Hence there are ony finitey many f, g, r, s that yied the entire sum, δ r fδs g, an eigenform. Each of these finitey many quadrupes were enumerated and a eigenforms found. See the foowing comments for more detai. Remar 3.2. In genera 2δ E E = δ 2 E 2. However, for 4, this is not an eigenform. Once we now that δ r fδs g is in genera not an eigenform, we have to rue out the ast finitey many cases. In particuar consider each eigenform and zero as eading term [f, g] n in Proposition 2.3. From [6] we now that there are 29 cases with g a cusp form 12 with n = 0, 81 cases with f, g both Eisenstein series 4 with n = 0. By case we mean instance of [f, g] n that is an eigenform. We aso must consider the infinite cass with f = g = E and r + s odd, where [f, g] r+s = 0. For the infinite cass when f = g and r + s is odd we do have [f, g] r+s = 0. By Lemma 2.11 the [f, g] r+s 1 term is nonzero. If r + s 1 = 0, then this is covered in the n = 0 case. Otherwise r + s 1 2. This is an eigenform ony finitey many times. In each of these cases one computes that the [f, g] 0 term is nonzero. Thus because there are two nonzero terms, δ r fδs g is not an eigenform. The 16 cases with n = 0 are the 16 hoomorphic cases. Now consider the rest. In the ast finitey many cases we find computationay that there are two nonzero coefficients: the coefficient of [f, g] 0, and [f, g] r+s. Now [f, g] 0 0, [f, g] r+s 0 and so in these cases fδs g is not an eigenform. The typica case, however, wi invove many nonzero terms such as δ r δ 4 E 4 δ 4 E 4 = 1 45 [E 4, E 4 ] 2 + 0 δ 10 [E 4, E 4 ] 1 + 10 45 δ2 8 [E 4, E 4 ] 0 = 1 2 2 42 E 4 45 z E 2 4 49 z E 4 + 10 45 δ2 8 E 8, δ 6 E 6 E 8 = 1 14 [E 6, E 8 ] 1 + 3 7 δ 14 [E 6, E 8 ] 0 = 1 6E 6 14 z E 8 8E 8 z E 6 + 3 7 δ 14 E 6 E 8 which cannot be eigenforms because of the fact that there are mutipe terms of different hoomorphic weight. 7
References [1] W. Due. When is the product of two Hece eigenforms an eigenform? In Number theory in progress, Vo. 2 Zaopane-Kościeiso, 1997, pages 737 741. de Gruyter, Berin, 1999. [2] Brad A. Emmons and Dominic Lanphier. Products of an arbitrary number of Hece eigenforms. Acta Arith., 1304:311 319, 2007. [3] Enath Ghate. On monomia reations between Eisenstein series. J. Ramanuan Math. Soc., 152:71 79, 2000. [4] Serge Lang. Introduction to moduar forms, voume 222 of Grundehren der Mathematischen Wissenschaften [Fundamenta Principes of Mathematica Sciences]. Springer-Verag, Berin, 1995. With appendixes by D. Zagier and Water Feit, Corrected reprint of the 1976 origina. [5] Dominic Lanphier. Combinatorics of Maass-Shimura operators. J. Number Theory, 1288:2467 2487, 2008. [6] Dominic Lanphier and Ramin Taoo-Bighash. On Ranin-Cohen bracets of eigenforms. J. Ramanuan Math. Soc., 194:253 259, 2004. [7] Goro Shimura. The specia vaues of the zeta functions associated with cusp forms. Comm. Pure App. Math., 296:783 804, 1976. [8] Goro Shimura. Eementary Dirichet series and moduar forms. Springer Monographs in Mathematics. Springer, New Yor, 2007. [9] D. Zagier. Moduar forms whose Fourier coefficients invove zeta-functions of quadratic fieds. In Moduar functions of one variabe, VI Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976, pages 105 169. Lecture Notes in Math., Vo. 627. Springer, Berin, 1977. Jeffrey Beyer Department of Mathematica Sciences, Cemson University, Box 340975 Cemson, SC 29634-0975 E-mai address: beyer@cemson.edu Kevin James Department of Mathematica Sciences, Cemson University, Box 340975 Cemson, SC 29634-0975 E-mai address: eva@cemson.edu URL: www.math.cemson.edu/ eva Catherine Trentacoste Department of Mathematica Sciences, Cemson University, Box 340975 Cemson, SC 29634-0975 E-mai address: ctrenta@cemson.edu Hui Xue Department of Mathematica Sciences, Cemson University, Box 340975 Cemson, SC 29634-0975 E-mai address: huixue@cemson.edu 8