Equivariant functions and integrals of elliptic functions

Transcription

1 DOI 0.007/s ORIGINAL PAPER Equivariant functions and integrals of elliptic functions Abdellah Sebbar Ahmed Sebbar Received: 8 August 0 / Accepted: 3 December 0 Springer Science+Business Media B.V. 0 Abstract In this paper, we introduce the theory of equivariant functions by studying their analytic, geometric and algebraic properties. We also determine the necessary and sufficient conditions under which an equivariant form arises from modular forms. This study was motivated by observing examples of functions for which the Schwarzian derivative is a modular form on a discrete group. We also investigate the Fourier expansions of normalized equivariant functions, and a strong emphasis is made on the connections to elliptic functions and their integrals. Keywords Equivariant functions Schwarz derivative Cross-ratio Modular forms Platonic solids Integrals of elliptic functions Mathematics Subject Classification 000 F03 33E05 Introduction Though the problem we are studying is analytic and geometric in its nature, it can be given a general algebraic formulation as follows: Let G be a group acting on two sets X, Y and let S be a set of functions from X to Y on which G acts in the following way g. f x = g. f g.x, for all g G, f S, x X. An equivariant function f : X Y also sometimes called concomitant is a function such that g. f = f,thatis A. Sebbar Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON KN 6N5, Canada asebbar@uottawa.ca A. Sebbar B Institut de Mathématiques de Bordeaux, Université Bordeaux, 35 cours de la Libération, Talence cedex, France ahmed.sebbar@math.u-bordeaux.fr

2 f g.x = g. f x, g G, x X. If G acts trivially on Y,then f is called an invariant. We propose to illustrate this problem in the following setting which encompasses arithmetic, analytic and geometric flavors. A subgroup Ɣ of the modular group SL Z acts on the upper half-plane H ={z C, Iz > 0}, by linear fractional transformation γ z = az + b a b cz + d, z H,γ= Ɣ. c d We would like to investigate the class of meromorphic functions h on H which commute with this action. In other words, h satisfies the equivariance relation az + b h = ahz + b a b cz + d chz + d, z H, Ɣ.. c d These functions first appeared in the works [5] and[34] in the 960s. Our interest in these functions began with the study of the modular properties of the Schwarz derivative. If f is a modular function on a Fuchsian group G of the first kind, then its Schwarz derivative { f, z} is a weight 4 modular form on a group which is larger than G. In particular if G has genus 0, then this larger group is the normalizer of G in PSL R. One formulates the converse to this statement as follows: If a meromorphic function f on the upper half-plane H, with some favorable conditions at the cusps, is such that its Schwarz derivative { f, z} is a weight 4 modular form on a certain group G, what can we say about f itself in terms of invariance? Is f a modular function under a nontrivial subgroup of G? How is the size of this subgroup dependent on the analytic properties of f? This problem is equivalent to the following: Let f be a meromorphic function on H. Let G be a discrete subgroup of PSL R such that for all z H and for all γ G we have f γ z = γ f z, where γ is a certain matrix in PGL C. It is clear that is a group homomorphism. The problem is to determine the kernel of depending on analytic properties of f and/or geometric properties of G. When the homomorphism is the identity, then the function f satisfies the relation. for the group G. If a function h satisfies the relation. and such that hz z is meromorphic at the cusps, then h will be called an equivariant form. The group Ɣ canbetakentobeanydiscrete subgroup of SL R, but for the time being, only the modular group will be under our consideration. It turns out that to each meromorphic modular form f of weight k for Ɣ, one can associate an equivariant form h as follows hz = z + k f z f z.. In fact, we determine the necessary and sufficient conditions for an equivariant form to arise as in. and this will be called a rational equivariant form. We will also exhibit equivariant forms that are not rational. Further investigations of an equivariant function h is carried out by looking at the Fourier coefficients of the periodic function hz z, the Lambert series of or at the infinite hz z

3 product of the modular form attached to h in the sense of Borcherds. This will be made explicit for equivariant functions attached to classical modular forms such as the Eisenstein series. The same examples are used in studying the Fourier coefficients of the reciprocal of the classical Eisenstein series as was carried out by Hardy, Ramanujan, and more recently by Berndt and Bialek among others. Beside the modular aspect in the structure of equivariant functions, there is also a fascinating elliptic aspect to them. Indeed, the fundamental example of the equivariant function attached to the weight cusp form given by h z = z + 6 = z + iπ E z, where E is the weight Eisenstein series, is closely related to the Weierstrass ζ -function where ζ = and is the classical Weierstrass elliptic function. In fact, as was observed by Heins [5], if L = Zω + Zω is a lattice with τ = ω /ω H, andifη and η are the pseudo-periods of ζ,thenω η, as a function of τ, is an equivariant function for SL Z by way of the Legendre relation. It turns out that this equivariant function is nothing else but the fundamental example h above. We will show that, with two exceptions, the integral of each n, n Z, is an equivariant function. From the differential algebra point of view, we have the important feature that each equivariant form satisfies a differential equation of degree at most 6; a fact that one expects from a function that satisfies sufficiently many functional equations. To justify this property, one goes back to the differential ring of modular forms and their derivatives, also known as the ring of quasi-modular forms, which is simply C[E, E 4, E 6 ], and thus has transcendence degree 3. One more time, when we specify to explicit examples of equivariant forms coming from the Eisenstein series, we find important differential properties of the reciprocal of E, E 4 and E 6. Namely, using a theorem of Maillet, we show that, and satisfy E E 4 E 6 algebraic differential equations over Q. The fact that the equivariant functions are differentially algebraic enables us to use theorems, well known in transcendence theory, such as those of Maillet and Popken, to control gaps or growth coefficients, in the expansion of these functions in q-series. The same argument is also valid for the reciprocal of E, E 4 and E 6 completing, in some sense, the previous work of Hardy, Ramanujan and more recently of Berndt and Bialek. Beside the modular, the elliptic and the differential aspects of equivariant functions, the bulk of this work basically articulates on three main axes: the link between the cross-ratio and the Schwarz derivative, then between the equivariance and the Schwarz derivative and finally between equivariance and the cross-ratio. This scheme is a consequence of the following intriguing facts: The Schwarz derivative is simply the infinitesimal counterpart of the cross-ratio. The Schwarz derivative of an equivariant function is a weight four modular form. The Riccati equation is closely related to the Schwarz differential equation. The cross-ratio of four solutions to the Riccati equation is a constant in the field C. The cross-ration of four equivariant functions is a modular function, that is in the function field of a compact Riemann surface. All these connections make the equivariant functions extremely rich objects to study. Though the study of equivariant functions was undertaken since the 960s by M. Heins and M. Brady in the framework of elliptic functions, we learned only recently from D. Zagier that the fundamental example h was known by W. Nahm in connection with some

4 physical problems. We cite in this context, in Sect., some examples of a special kind of equivariance, named platonic, that appeared in the Physics literature. SL R, the Riccati equation, the Schwarz derivative and the cross-ratio The Riccati equation is naturally related to the Schwarz differential equation through a change of function. In this section, we will exhibit similar relations between the Riccati equation and the functional equations satisfied by equivariant functions. Many properties of these nonlinear equations have their origins in the projective differential geometry of the special linear group SL R that we recall now. The associated Lie algebra sl is three dimensional with basis A = 0 0, A = 0 0 0, A 3 = The associated infinitesimal generators are x, x x, x x, which have the same commutation rules as A, A, A 3. The group SL R acts on the real line as the projective group x R ax + b a b cx + d R, SL R. c d To give a general idea about the link between the group SL R and nonlinear differential equations, we first observe that if f t : R R is a smooth function with f 0 x = x then f t x = x + txx + Ot so that X is the infinitesimal generator vector field associated to f t. In the particular case where f t is the projective transformation associated with + ta tb Mt = SL tc + td R, we have + tax + tb f t x = tcx + + td = x + t b + a dx cx + Ot and the associated vector field is X = b + a dx cx. We note that for the linear x differential system ut at bt ut = vt ct dt vt the quotient x = u v satisfies the Riccati equation ẋt = bt + at dtxt ctxt which can be considered as a differential equation for the integral curve of the vector field X = b + a dx cx. It is important to notice here that the Riccati equation is x the only first order nonlinear ordinary differential equation which possesses the Painlevé property, that is of not having removable singularities. Its link with the cross-ratio, and hence projective geometry, was shown by Lie []. We recall that the cross-ratio of four different complex numbers is defined as [z, z, z 3, z 4 ]= z z z 4 z 3 z 3 z z 4 z.

5 If u, u, u 3, u are four solutions of the Riccati equation, then their cross-ratio is constant u u u u 3 u u u 3 u = k, leading to what is called the superposition formula that will be considered later: u = u 3u u + ku u 3 u.. u u + ku u 3 The Riccati equation is also the only ordinary nonlinear differential equation of first order which possesses such nonlinear superposition formula. As we will see later, this superposition formula is similar to one for the equivariant functions once we substitute the field of complex numbers by the function field of a Riemann surface. We now introduce the Schwartz derivative or Schwarzian. This is a differential operator introduced by Schwarz in [3] in the context of differential equations and quadratic differentials. It is defined for meromorphic functions over regions of the complex plane by f f { f, z} = f f = f f f 3 f.. If w is a function of z, then the Schwarzian satisfies the chain rule: { f, z} ={f,w}dw/dz +{w, z}. Moreover, If f is a linear fractional transformation of z,then{ f, z} =0. As a consequence, if w z 0 = 0 for some point z 0, then in a neighborhood of this point, the inverse function zw satisfies {z,w}= {w, z}dz/dw. More significant properties of the Schwarzian follow from its close relationship with second order differential equations. Indeed, let y and y be two linearly independent solutions to y + Rzy = 0,.3 4 where Rz is a meromorphic function on a certain domain. Then the quotient f = y /y satisfies { f, z} =Rz. Conversely, if f is a locally univalent function satisfying { f, z} =Rz, then y = f/ f and y = / f are two linearly independent solutions to.3. As a consequence, Proposition. We have { f, z} =0 if and only if f is a linear fractional transformation. { f, z} ={g, z} if and only if each function is a linear fraction of the other. Asacorollaryoftheabove,wehave { { f, z} = f, az + b } ad bc cz + d cz + d 4..4 Finally, from the definition of the Schwarzian, we have Proposition. If f is a meromorphic function, then { f, z} has double poles at the critical points of f and is holomorphic elsewhere.

6 The cross-ratio is projectively invariant, that is if f : z az + b, a, b, c, d C, ad bc = cz + d 0 is a Möbius transformation, then [ f z, f z, f z 3, f z 4 ]=[z, z, z 3, z 4 ]. For a general smooth function f, the Schwarz derivative measures the distortion of the crossratio in a certain sense. The following result is well known and is fundamental in Cartan [8]. Proposition.3 Let a, b, c and d be four distinct complex numbers and f a twice-differentiable function whose second derivative is continuous in an open set D C.Forz D and small t we define the cross-ratio then Sfz, t =[f z + ta, f z + tb, f z + tc, f z + td], Sfz, t =[a, b, c, d] + 6 a bc dsfzt + ot. The proof of Proposition.3 basically uses the following expansion, valid for all x and y close to z with x = z + h, y = z + k, log f x f y x y = log f z + f z f h + k z f z + 6 f z f z 8 f z f z 4 f hk +. z h + k f z 6 f z Remark.4 Taking the second derivative with respect to x and y gives another interpretation of the Schwarzian derivative, as a bi-differential dfxdfy f x f y = dxdy x y + f z 6 f z f z 4 f dxdy + ψh, kdxdy, z where the term ψh, k vanishes for x = y = z. 3 The case of modular functions Let H ={z C, Iz > 0} be the upper half of the complex plane. The group SL R of Möbius transformations acts on H in the usual manner. We restrict our attention to the modular group SL Z and its subgroups, but the general picture involves all the discrete subgroups of SL R.LetG be such a group, and let f be a modular form on G of weight k k 0, that is, a meromorphic function on H satisfying az + b a b f = cz + d k f z, z H, G, 3. cz + d c d with some growth conditions at the cusps. When k = 0, f is called a modular function. If f is holomorphic, it will be mentioned explicitly. The effect of the Schwarzian on modular functions is illustrated as follows. Using.4 and the fact that the derivative of a modular function is a modular form of weight, we have

7 Proposition 3. If f is a modular function for G, then {z, f } is a modular function and { f, z} is a modular form of weight 4 on G. Suppose now that G is a discrete group of genus 0, that is the compactification of the quotient G\H is a Riemann surface of genus 0. Any analytic embedding of this surface in the extended complex plane induces a modular function for G defined on H. It is called a Hauptmodul, and it generates the function field of the Riemann surface. Let f be a Hauptmodul for G. Then the modular function {z, f } is a rational function of f.asfor{ f, z},wehaveadeeperresult: Proposition 3. [5] Let G be a genus 0 discrete group and f a Hauptmodul for G. Then { f, z} is a weight 4 modular form on the normalizer of G in SL R and this normalizer is the maximal group with this property. To illustrate this proposition, Let Ɣ be the principal congruence group of level. A Hauptmodul is given by the classical Klein λ function. Since Ɣ has no elliptic elements, and thus λ has no critical point, we see that {λ, z} is a holomorphic weight 4 modular form on the normalizer of Ɣ in SL R which is SL Z. However, the space of weight 4 holomorphic modular forms for SL Z is one-dimensional and is generated by the weight 4 Eisenstein series E 4. In fact we have {λ, z} =π E 4 z. 3. A deeper study of this type of relations can be found in [5]. Here ηz 8 λz =, η4z where the eta function is defined by ηz = q 4 q n, q = e πiz, n and the Eisenstein series E 4 is defined by E 4 z = + 40 σ 3 nq n, q = e πiz, n where σ 3 n is the sum of the cubes of the positive divisors on n. Because the normalizer of congruence subgroups is often not a subgroup of the modular group, our focus should be on a more general class of discrete groups. Namely all the discrete subgroups that are commensurable with the modular group. These groups have finite index inside their normalizers. The next section deals with the converse to the previous proposition. 4 Modular Schwarzians In this section we look at the following question: Assume that f is a meromorphic function on H such that Fz ={f, z} is a modular form of weight 4 on a certain group G F.The invariance group of f on which f is a modular function is a subgroup G f of G F. What is the size of G f inside G F? Keeping in mind the relationship between G f and G F for a modular function f as it was seen in the previous section. We will make explicit some instances where we have complete answers. It turns out that these answers depend on the analytic properties of f andonthestructureofg F. So far, for all kind of plausible conditions, it seems that there are always examples that provide different answers.

8 We shall rephrase the problem differently. Since Fz ={f, z} is a modular form of weight 4, then for every γ = a b c d G F we have Fγ z = cz + d 4 Fz = cz + d 4 { f, z}. On the other hand, using.4, we have It follows that Fγ z ={f γ z, γ z} =cz + d 4 { f γ z, z}. { f, z} ={f γ z, z}. Therefore, using Proposition., there exists γ GL C such that This defines a group homomorphism f γ z = γ f z. 4. : G F GL C γ γ The invariance group G f of f is simply the kernel of. It is worth mentioning that M. Kaneko and M. Yoshida have considered a similar problem in [8] where is an epimorphism r : G G between two Fuchsian groups of the first kind. In particular, the authors were interested in the case G and G, the kernel and the co-kernel of r, are infinite groups. They have constructed the Kappa function defined by j κz = λz as an answer to this problem. We are mainly interested in the case where the co-kernel is rather finite. We now look at a case where there is a precise answer to the above question. Proposition 4. Let f be a meromorphic function on H such that f z + = f z.if{ f, z} is a weight 4 modular form on SL Z, then f is a modular function for SL Z. Proof Let U = 0. The condition f z + = f z means that U Ker, andthat U ={U n, n Z} Ker. SinceKer is normal in SL Z, it contains {L UL, L SL Z}, the normal closure of U. Now if we set V = 0 0, then one can show that V = UV UVU. Hence V belongs to the normal closure of U, and so does V.Since U and V generate SL Z, it follows that Ker = SL Z. Remark 4. The condition f z + = f z means that f has a Fourier expansion in q = expπiz. The above theorem provides an example where f is a modular function on the same group on which its Schwarzian is a modular form. This is equivalent to saying that the homomorphism is constant. This proposition can be generalized as follows. Theorem 4.3 Let n be an integer such that n 5. Suppose that f is a meromorphic function on H such that f z +n = f z and suppose that { f, z} is a modular form of weight 4 for SL Z. Then f is a modular function for a finite index normal subgroup of SL Z. Proof The case n = is settled in the above proposition. For n 5, the group Ɣn is of genus 0 and has no elliptic elements. Hence it is generated by parabolic elements only. Using this fact, one can show that the normal closure n of U n is simply Ɣn. Thus f is a modular function for a finite index subgroup of SL Z.

9 As for n 6, the argument above is no longer valid. Indeed the normal closure n of U n has infinite index in SL Z. We now provide an example where the invariance group of f is not larger than U, although the function f has the same conditions as in the theorem. Let G be a genus 0 group having at least two cusps. Let g be a Hauptmodul for G normalized to have the value at and 0 at the other cusp. Let gz = log f z.then f z + = f z and for γ SL Z not in U we have f γ z = f z + πin Ɣ, n Ɣ Z, n Ɣ = 0. However, { f, z} is a weight 4 modular form on G. 5 Equivariant forms, a first example The principal motivation behind equivariant forms was to look for examples of meromorphic functions f where { f, z} is a modular form of weight 4 for a discrete group G but f is not invariant under any nontrivial matrix. That is, the kernel of defined in the previous section is trivial. This will be the case if, for instance, =Id. In other word, f would satisfy az + b f = afz + b a b, for all z H and G. cz + d cfz + d c d On the other hand, in the paper [3], we studied and solved a family of Riccati equations of the form k iπ u + u = E 4, k 6. While the cases k 6 were given in terms of modular forms and functions, the solution in the case k = turns out to satisfy the above functional equations for G = SL Z. This paper is devoted to construct and study these very special functions with surprising properties arising along the way. We will focus on the case of G = SL Z for most of the paper. Definition 5. A meromorphic function hz on H is called an equivariant form for SL Z if For all z H and for all γ SL Z,wehave The function hz z is meromorphic at the cusps. hγ z = γ hz. 5. Since hz + = hz +, the function hz z is -periodic and hence has a Fourier expansion in q = expπiz, the local parameter of SL Z at. To say that hz z is meromorphic at means that the Fourier expansion has finitely many negative powers of q. The trivial example is h 0 z = z which is equivariant for any group. This example is very particular in two ways: First, it is the only Möbius transformation that is equivariant, and second, as was shown by Heins [5], h 0 is the only equivariant function that maps H into itself. In particular, this means that the composition of maps does not provide the set of equivariant holomorphic functions with a group structure. This result has a certain connection with the iteration of holomorphic maps from H into H. We give a quick proof using the following theorem of Denjoy and Wolff [7].

10 Theorem 5. Let f : D D be a holomorphic map. We assume that f is neither an elliptic Möbius transformation nor the identity, then the successive iterates f on converge, uniformly on compact subsets of D, to a constant function z c 0 D. We recall that the elliptic Möbius transformations in H are all of the following form z cos θ z + sin θ sin θ z + cos θ, θ 0, π, and they are not equivariant, for any θ 0, π. Thus, if h is an equivariant holomorphic function sending H into itself, so does any iterate h on, n N which is also equivariant and holomorphic. By the Wolff-Denjoy theorem, h on should tend to c H which, by equivariance, verifies the contradictory conditions c = c, c = c +. Thus, the only equivariant holomorphic function in H which maps H into H is the identity map. We now provide the first nontrivial example of equivariant functions. Let E z be the classical Eisenstein series E z = 4 σ nq n = 4 n= n= nq n q n, 5. where q = expπiz, z H and σ n is the sum of the positive divisors of n. Theseries E z is a holomorphic function on H, andifγ = a b c d SL Z, wehave E γ z = cz + d E z + 6c cz + d, 5.3 iπ that is to say that E is a quasi-modular form of weight. Meanwhile, if we define E z = E z 3, y =Iz, 5.4 πy then E is a non-holomorphic modular form of weight for SL Z. Moreover, E is the logarithmic derivative of the discriminant function z, the classical cusp form of weight for SL Z In fact, we have z = ηz 4 = q q n 4, q = e πiz. n= E z = z iπ z. 5.5 Theorem 5.3 The function h given by is an equivariant form for SL Z. h z = z + 6 iπ E z 5.6

11 Proof It suffices to show that h is equivariant under the transformations z z + and z /z which generate SL Z. SinceE z + = E z, it is clear that h z + = h z +. On the other hand, using 5.3, one has: h z = 6i/π z z E z 6iz/π E z = ze z 6i/π = h z. Furthermore, one can see from 5. thath z z has a holomorphic q-expansion, and therefore h is an equivariant function for SL Z. The equivariant function h z has no fixed points in H and also at in the sense that lim h z z = 6 = 0. z i iπ In the following, we will show that h z is the unique equivariant function for SL Z with the property that it has no fixed points in H { }and that this example actually fits in a general construction of equivariant functions. Theorem 5.4 Let hz be an equivariant function for SL Z with no fixed points in H { }. Then hz = h z = z + 6 iπ E z. Proof Since hz z does not have zeros on H { }, then the function gz = hz z iπ 6 E z is holomorphic in H { }. Moreover, we have gz + = gz and using the equivariance of h and 5.3 weget g /z = zhz hz z iπ 6 z E z z zhz z + z = hz z = z gz. iπ 6 z E z z Therefore, gz is a weight holomorphic modular form and thus gz = 0 since the space of weight holomorphic modular forms for SL Z is trivial. The theorem follows. Remark 5.5 An alternative way to prove this theorem is to notice that under the assumptions of the theorem, the integral z i dw h w w

12 does not depend on the path of integration W z from i to z in H since h z z is holomorphic and non-vanishing. Therefore, the function z f z = exp i dw h w w is a well defined function that is holomorphic and non-vanishing on H. Using the equivariance of h, one can show that f z is a non-vanishing modular form of weight and thus is a scalar multiple of z. Taking the logarithmic derivative of f and using 5.5 yield the theorem. Proposition 5.6 If z 0 is an elliptic fixed point of SL Z, then h z 0 = z 0. Proof If z 0 H Q is fixed by an element γ of SL Z,thenh z 0 is also fixed by γ. Thus if z 0 is an elliptic fixed point, and since h does not have fixed points in H, wemusthave h z 0 = z 0. 6 Analytic properties of h and an application In this section we show that there are infinitely many poles of h and that they are all simple. Proposition 6. We have The poles of h z are located at the zeros of E and they are simple. The critical points of h are located at the zeros of E 4. Proof It is clear that the poles of h z are exactly the zeros of E. Now recall the following differential relation between E and E 4 due to Ramanujan, [7,8] d πi dz E z = E E It follows that if a zero of E is not simple, then it is also a zero of E 4 and such a zero lies in the SL Z-orbit of ρ, the cubic root of unity. This is impossible since Proposition 5.4 states that E does not vanish at the elliptic fixed points. Therefore, the poles of h z are simple. Furthermore, h z = 6 E iπ E Hence h z vanishes exactly at the zeros of E 4, = E E 4 E = E 4 E. As a consequence, and using Proposition., E4 {h, z} is a weight holomorphic modular form and thus it is a linear combination of and E4 3 which constitute a basis of the space of weight holomorphic forms for SL Z. Investigation of the first two coefficients of the Fourier expansion yields Proposition 6. We have {h, z} = 6 3 π E4. 6.

13 Remark 6.3 The above proposition can be established by direct computation using similar identities to 6., also known as the Ramanujan identities [7], namely d πi dz E 4z = 3 E E 4 E together with the identity πi Here, E 6 is the weight 6 Eisenstein series d dz E 6z = E E 6 E4, 6.4 E 3 4 E 6 = 7 3. E 6 z = 504 n σ 5 n q n, where σ 5 n is the sum of the fifth powers of the positive divisors of n. If z 0 is a pole of h then all the translates of z 0 by integers are also poles of h. It turns out that the only other poles of h that are SL Z-equivalent to z 0 are its translates by integers. Indeed, Lemma 6.4 If z 0 and z are two poles of h, and if there exists γ SL Z such that z = γ z 0 then γ = n 0 for some integer n. Proof If γ = ab cd and if z0 and z = γ z 0 are poles of h, then necessarily c = 0and a = d =±. Thus z = z 0 + n where n is an integer. Using this property, we can restrict ourselves to the half strip { D = z = x + iy : y > 0, < x }. We will denote by T and S the transformations Tz = z +, Sz = z. Proposition 6.5 There exists a pole z 0 of h on the purely imaginary axis {z = iy : y > 0}. Proof Recall that E z = 4 n σ nq n.weseethate iy, y > 0 is real and strictly decreasing for y 0,. It takes the value at and lim y 0 Eiy =. Therefore, there exists y 0 > 0 such that E iy 0 = 0 and thus iy 0 is a pole of h. Moreover, y 0 < sinceby5.3, we have E i = 3/π. Recall that the fundamental domain for SL Z is Since Imz 0 < we see that F ={z H : z >, / < Rz /}. Theorem 6.6 There are infinitely many poles for h in D. z 0 SF, Sz 0 F. 6.5

14 Proof Let z = Sz 0 = /z 0 where z 0 is as in Proposition 6.5. Thenh z = 0andz is not an elliptic fixed point by Proposition 5.6. Choose an open neighborhood U F of z on which h is holomorphic, not containing an elliptic fixed point and such that no two points of U are SL Z equivalent. Then U is mapped by h onto an open neighborhood V of 0. There are infinitely many rational numbers in the open set V.Ifx = h z x, z x U,issucha rational number, let γ x be such that γ x x =.Thenh γ x z x = γ x h z x = γ x x =. Therefore, γ x z x is a pole of h. In the meantime, no two such poles γ x z x and γ y z y are SL Z equivalent because z x and z y, which are in U, are not. Corollary 6.7 The Eisenstein series E has infinitely many non-equivalent zeros. Remark 6.8 As a corollary, the Eisenstein series E has infinitely many non-equivalent zeros; a result that has been established without the notion of equivariant functions in []. 7 Rational equivariant functions, the general case In this section, from each modular form we construct an equivariant function. Using 5.5, one can rewrite the equivariant function h as h z = z +. It turns out that this expression can be generalized as follows Theorem 7. [34] Let f be a modular form on SL Z of weight k. Then the function hz = z + k f z f 7. z is equivariant for SL Z. Proof Let γ = ab cd SL Z. Wehave f γ z = cz + d k f z, hence Therefore, f γ z = kccz + d k+ f z + cz + d k+ f z. hγ z = az + b cz + d + kcz + d k f z kccz + d k+ f z + cz + d k+ f z = kfzabz + bc + + az + bcz + d f z cz + dkcf z + cz + d f. z On the other hand, we have γ hz = az + b f z + akf z cz + d f z + kcf z. Since ad bc =, we have az + bc + = acz + d. The identity hγ z = γ hz follows. Proposition 7. If f is a modular form of weight k, then scalar multiples of f and integral powers of f give rise to the same equivariant function h. The modular functions correspond to the trivial equivariant function h 0 z = z.

15 Proof This is straightforward keeping in mind that, for an integer m, the weight of f m is km. In what follows, we will find sufficient conditions for an equivariant function to arise from a modular form. Proposition 7.3 If h is equivariant for SL Z, then the set of residues of the meromorphic function /hz z at the simple poles is finite. a b Proof Let γ = SL Z. Differentiating hγ z = γ hz yields c d h γ z cz + d = h z chz + d. Thus, if hz 0 = z 0,thenh γ z 0 = h z 0,thatis,h takes the same value at the orbit of a fixed point of h. Hence, the set of values h z 0 when z 0 describes the set of fixed points of h is completely determined if we restrict ourselves to the fundamental domain F. Moreover, since hz z is meromorphic at i, there is a neighborhood of i of the form {z H :Iz > y 0 } on which hz z does not vanish except possibly at i. Therefore, all the zeros of hz z in F are within the closure of {z F :Iz y 0 } which is compact and thus we have only finitely many zeros. In the meantime, the residue of /hz z at a simple z 0 of h is simply /h z 0. The proposition follows. Theorem 7.4 Let h be an equivariant function satisfying the following conditions: The poles of /hz z in H are simple and their residues are rational numbers. At we have: lim z hz z πiq. Then there exists a modular form f of integer weight k for SL Z such that hz = z + k f z f z. Proof Define the function f z by z f z = exp i kdz hz z, where k is a positive integer to be chosen conveniently. The path of integration z is chosen to lie in H \ S,whereS is the set of simple zeros of hz z. By assumption, the residues of /hz z are rational numbers, and using Proposition 7.3, these rational numbers have bounded denominator. Therefore, there exists k Z + such that for each such residue r, kr is an integer and z lim z k hz z πiz, and we also suppose k 0 mod 4. If we choose a different path of integration z from i to z lying in H \ S,then kdz hz z kdz hz z = πik Residues πiz, z

16 where the sum of the residues is taken over the finite number of poles within the closed path z z. Therefore, f z is well defined on H \ S.Weextend f to a meromorphic function on S in the following way. Let m an integer be the residue of k/hz z at z 0.Ifr > 0 we define f z 0 = 0tomake f holomorphic at z 0 and the order of f at z 0 is r. Ifr < 0 then z 0 is a pole of f of order r. Thus f is a well-defined meromorphic function of H. Furthermore, where f z + = exp z+ gz = exp i kdz hz z = f zgz z+ z kdz hz z. Since h is an equivariant function, it is clear that g z = 0 and hence g is constant. Taking the limit z i yields gz = since by assumption, and ka 0 πiz. Therefore, On the other hand, k hz z = ka 0 + a n q n, q = e πiz, n f /z = exp f z + = f z. = exp /z i z = f z exp i kdw hw w khtdt tht t z i kdt t = z k f z since k 0 mod 4. Thus f is a meromorphic modular form of weight k for SL Z. Motivated by the above theorem, we have Definition 7. An equivariant function that arises from a modular form as in 7. is called a rational equivariant function. Remark 7.5 If f is a weight k modular form, then the corresponding equivariant function hz = z + kfz/f z satisfies the conditions of the above theorem; that is to say that the two conditions are necessary and sufficient conditions for h to be of that form. Moreover the

17 conditions are optimal as we will see that there are indeed examples of equivariant functions that do not satisfy them and thus they are not rational. Indeed, if we take h z = z + 6E E 6 iπ E E 4 E 6 +, then one can show that h is equivariant but h z z has a double pole at the cubic root of unity ρ. Also, h z = z + 6E 4 iπe E 4 + E 6 is equivariant having poles only at the zeros of E 4 but the residue of / h z z at ρ is irrational. Finally, h 3 z = z + 6 iπ E + E 4 is equivariant, but lim z i h 3 z z = 0. These three examples show that one cannot remove the conditions of the converse theorem above. 8 Lambert series and Borcherds products For a rational equivariant function hz = z + kfz/f z, we will investigate the Fourier coefficients of the periodic function hz z, the Lambert series of. We also point hz z out a possible link with a theorem of Borcherds at least by considering some examples. We begin by recalling some classical analytic facts [36], p. 47. Lemma 8. Given any sequence a n n 0, a 0 = 0, we formally have n=0 a n q n = a 0 + n G n n!a0 n+ q n with a a a 3a a G n = 6a 3 5a 4a 3a n a n... n a 0 na n n a n... a Corollary 8. Let E q be the weight Eisenstein series 5. seen as a function of q and set E q = α n q n, 8. n=0

18 then α 0 = and σ σ 3σ 0 0 n!4 n α n = 6σ 3 5σ 4σ n σ n... n nσ n n σ n... σ Lemma 8.3 Consider a formal power series n b nx n with complex coefficients, then a sequence a n n can be found such that the following expansion in Lambert series holds b n x n = x n a n x n 8. n n with b n = a m, a m = m μ b d = μdb m d d, 8.3 m n d m d m μ being the Möbius function defined as usual by the inverse of the Riemann zeta function ζs = n n s, ζs = μn n s, Rs >. n We now consider an equivariant function h and the associated function g,definedby k iπ gz = hz z = b n e iπnz. n 0 It is a meromorphic periodic function on the upper half plane H. Let, with the notation of Lemma 8.3, an = μdb n n d and f z = e iπb 0z e n iπnz an. 8.4 d n Applying the theta differential operator θ = iπ we have θ f = b 0 dade iπnz = f n= d n d and using the Möbius inversion formula, dz k iπ g. a b We define the slash operator of weight k as usual: Let γ = SL, Z, then c d Accordingto[9] we have the following F k [γ ]z = cz + d k f γ z. Definition 8. Let Ɣ Ɣ containing I. A meromorphic function F : H C is said a generalized modular form of weight k and a character ν if

19 F satisfies the modular transformation law F k [M]z = νmfz for all M Ɣ, with νm independent of z H. F has a left-finite Fourier expansion at each parabolic cusp in a fundamental region R of Ɣ. The generalized modular forms differ from the classical modular forms with a character by the fact the multiplier system ν need not be unitary. Proposition 8.4 For h equivariant, the infinite product f z, given in 8.4, is a generalized modular form Proof We have θ f 0 [γ ] f 0 [γ ] = θ f f [γ ]= k iπ g [γ ]= k iπ g + k iπ c cz + d where the last equality is a consequence of the equivariance of h. Therefore, f 0 [γ ] f θ f 0 [γ ] f 0 [γ ] θ f f = k c iπ cz + d, 8.5 which is a fundamental equation to study the modular properties of f. It follows that θ f 0 [γ ] f = θ f 0[γ ] f 0 [γ ] θ f = k c θcz + dk = f iπ cz + d cz + d k. f k [γ ] f 0 [γ ] Hence, = is a non-zero constant νγ on H. Using the cocycle relation of the slash operator, it is easily seen that ν is actually a character of SL Z yielding f z f zcz + d k the multiplier system of the generalized modular form f of weight k. Let us give some examples. For the basic equivariant function h given by 5.6, the associated infinite product is given by the discriminant function z = q q n 4, q = e πiz. n= Accordingto5.5 E z = z iπ z, and thus the fundamental Eq. 8.5 reduces to 5.3. We can also consider, as in [4], the following infinite products j τ = q q n 3c 0n n>0 = q q 744 q 8056 q , E 6 τ = q n an n>0 = q 504 q q

20 and define the corresponding equivariant functions and associated Lambert series expansions. More generally, in [4], Borcherds gives a striking description of the exponents in the infinite product expansion of several modular forms in terms of the Fourier coefficients of some half integer meromorphic modular forms. The q-expansion of j starts as Jz = q q q +, q = e iπz 8.6 and we introduce a sequence of modular functions J 0 z =, J z = Jz 744 and for m, we define j m z = J z T 0 m where T 0 m is the normalized mth Hecke operator defined by gz T 0 m = d az + b g d Then we have [4] d m, ad=m b=0 E 4 z = + 40 σ 3 nq n = q n cn n= n= 8.7 where the cns denote the coefficients of a weight /-modular form on Ɣ 0 4 whose expansion starts as f z = q q q q q 8 + or more explicitly, cn = 8 + 3n d n n μ d j d ω, ω = + 3 ; a formula which can be compared with the identity J = E3 4. Borcherds theorem shows at once that the knowledge of the sequence of exponents cn with the use of Lemma 8. gives, in principal, the full Fourier coefficients of the periodic part hz z of an equivariant function h. However, this not an easy task in practice, even for basic example such as h.indeed, due to the lack of modularity, the Fourier series of has not been investigated E before, contrary to and. The Fourier series of has been studied by Hardy and E 4 E 6 E 6 Romanian [4], and very recently the Fourier series of has been studied by Bernie, Bialys E 4 [3]. The results are deep and for comparison and later discussion we quote them and give an immediate consequence using Lemma 8.. Theorem 8.5 Hardy Ramanujan Let C = 3π 4Ɣ Define the coefficients p n by E 6 q = n=0 p n q n.

21 Then, for n 0, p n = μ T μ n, where μ runs over all integers of the form μ = a r j= p a j j, 8.8 where a = 0 or, p j is a prime of the form 4m +, and a j is a nonnegative integer and where T n = C enπ, T n = n C 4 e nπ, and for for μ>, T μ n = e C nπ μ μ 4 c,d cos ac + bd nπ μ + c 8tan, d where the sum is over all pairs c, d satisfying μ = c + d and a, b is any solution to ad bc =. Comparing with Lemma 8., we obtain at once, with σ 5 k = d n σ σ 5 3σ n!504 n p n = 6σ 5 3 5σ 5 4σ n σ 5 n... n nσ 5 n n σ 5 n... σ 5 Inasimilarway,wehaveforE 4 Theorem 8.6 Berndt Bialek Let ρ = 3 + i and set E 4 q = β n q n and then n=0 G = E 6 ρ, β n = n 3 G λ Here λ runs over the integers of the form 8.8, h λ n λ 3 e nπ 3 λ. d 5 h n =, h 3 n =,

22 and, for λ h λ n = cos ad + bc ac bd + λ nπλ c 3 6tan, d c c,d where the sum is over all pairs c, d satisfying λ = c cd + d and a, b is any solution to ad bc =. Comparing with the Lemma 8. we obtain once again, with σ 3 k = d 3, d n σ σ 3 3σ n!40 n β n = 6σ 3 3 5σ 3 4σ n σ 3 n... n nσ 3 n n σ 3 n... σ 3 Remark 8.7 It would be interesting to have another formulation for the determinant in 8. similar to those given for the coefficients of E 4 q and despite of the lack of the E 6 q modularity for E. 9 n as an elliptic function and a differential algebra In this section, we give some properties of the meromorphic elliptic function n, n Z and with two exceptions, we will associate to each n an equivariant function. As was observed by Heins, the basic example of equivariant functions is related to the Weierstrass functions ζ = by way of the Legendre relation. Our main task is to find integrals of n, n Z. For later use, let us recall the essential idea. If ω,ω are two complex numbers with Iω /ω >0and = ω Z + ω Z, the Weierstrass functions are z; ω,ω = z = z + ζz; ω,ω = ζz = z + σz; ω,ω = σz = ω \{0} ω \{0} ω \{0} z ω z ω ω, z ω + ω + z ω, z exp ω + z ω Hence = ζ, σ = ζ. In some situations, it is better to use the odd Jacobi theta function σ θ z τ = i e iπnz e iπnτ. n +Z n The zero divisors of this function form the lattice Z + τ Z. The general principal that we follow and which goes back to Liouville and Hermite is that if the principal part of an elliptic meromorphic function at each of its poles is known, then this function is determined up to an additive constant. More precisely we have the decomposition theorem [36]..

23 Theorem 9. If a k, k n, is the set of poles of an elliptic function f, of periods ω, ω and if at a k the principal part is then there exists constant A such that f z = A + r k s= n k= s= c ks z a k s, r k Consequently, a primitive of f is given by f zdz = Az + B + [ n c k log σ + k= s s! c ksζ s z a k. r k s= ] s s! c ksζ s z a k, with B being an arbitrary constant. Moreover, with Jacobi theta function, we have f z = C + n k= s= r k s d s s! c ks dz s log θ πz πak ω ω ω and f zdz = Cz + D + n k= s= r k s d s s! c ks dz s log θ πz πak ω. ω ω The coefficient c k is the residue of f at the pole a k, hence n c k = 0. i= In particular c = 0 if there is only one pole. This theorem is very deep in the sense that it gives all the differential relations that will be considered below and also all the known relations between Weierstrass [ elliptic ] functions and Jacobi elliptic functions. In addition, it d essentially says that if D = C is the ring of differential operators with constant coefficients and M the C-vector space of elliptic meromorphic functions with a pole at 0, then dz M is a left D-module, that is M, dz d is a differential graded algebra M = C C C C n. As a D-module, M is generated by two elements,,henceweahavethefreeresolution 0 D φ D ψ M 0 where ψd, D = D. + D and φd = D, D..

24 Perhaps the most fascinating examples of applications of this theorem are the two following identities of Frobenius and Stickelberger nn n k! σz 0 + z + +z n 0 i< j n z i z j σ n+ z 0 σ n+ z n z 0 z 0 n z 0 z z n z = z n z n n z n k= and of Kiepert n z z n z σnu n k! σ k= n z = z z n z. n z n z n 3 z In the identity of Frobenius and Stickelberger, the left hand side considered as function of z 0, is an elliptic function having z 0 = 0 as a pole of order at most n. Its decomposition according to the theorem 9. is given by the right hand side. The coefficients of the decomposition are obtained by developing the determinant with respect to the elements of the first row. The identity of Kiepert can be obtained from the one of Frobenius and Stickelberger by a limiting process. The Weierstrass function is homogeneousof degree and is a generating function of the classical Eisenstein series z; ω,ω = z + k + G k+ z k, G k+ = ω k. k= ω \{0} For τ = ω ω fixed, and its derivative are elliptic functions for Z + τ Z. The zeros of in C/Z + τ Z occur at the points of order, namely, τ and + τ. On the other hand, the zeros of were described in [] and more recently in [0]. Since assumes every value in C { }exactly twice in C/Z + τ Z, it follows that has two zeros therein which can be written as ±z 0 since is even. Proposition 9. The zeros of the -function are given by ±z 0 where, by the Eichler Zagier formula, z 0 = m + + nτ ± log iπ i 6 σ τ σ d σ iπ E 6 σ 3 for all m, n Z, where the integral is to be taken over the vertical line σ = τ + ir + or by the Duke-Imamoglu formula c x 4 F 3, 3, ; 4 3, 4 5 x z 0 = + τ + F, 5 x, c = τ i 6 3π

25 where x = 78 and where the generalized hypergeometric series defined for x < j by Fa,...,a m ; b,...,b m x = and a n = aa + a + n. n=0 a n...a m n x n b n b m n n!. The pseudo-periods η,η of the Weierstrass ζ -function are defined by ζu + ω α = ζu + η α, ζω α = η α, α =,. TheLegendrerelationisη ω η ω = iπ. The periods and pseudo-periods of the Weierstrass -function are related to Ramanujan Eisenstein series by E z = π η ω 4 ω 6 ω, E 4 z = g, E 6 z = 6 g3. π π In particular if ω =, ω = τ, τ = ω,then ω 6 h τ = τ + iπ E τ = η. 9. η Thus the basic equivariant function is a quotient of pseudo-periods. This interpretation will reveal important differential properties. We would like to extend this construction to the powers n, n Z with two exceptions. Let L be the set of lattices in the R-vector space C and M ={ω,ω C : τ =Iω /ω >0}. Then L can be identified with the quotient M/SL Z. Moreover, C acts on L and on M yielding two more identifications Following Brady [5], we introduce M/C H, R/C H/PSL Z. Definition 9. A function f : C L P is called pseudo-periodic if it is meromorphic in z and for each ω = ω Z + ω Z,ω,ω M there is a constant ηω such that for each z C f z + ω,ω = f z,ω+ ηω. In [5], Brady observed first that if f is a homogeneous pseudo-periodic function with pseudoperiods ηω, then the map τ ητ τ is an equivariant function, provided that η τ τ η τ, τ ητ τ are meromorphic and η τ does not vanish identically. In this paper, we are studying a similar question, namely the zeta function associated to the elliptic function n, n Z. This function is an even elliptic function, homogeneous of degree n, of periods ω,ω. We look for a primitive of n giving rise to pseudo-periodic function. We recall the following definition [] Definition 9. Let k and m be two fixed integers. A function φ : H C P is called a meromorphic Jacobi form of weight k and index m if

26 i φ is meromorphic on C H, ii φ satisfies z φ γτ + δ, ατ + β = γ τ + δ k exp iπm γ z φz,τ, γτ + δ γτ + δ αβ for every SL Z. γδ iii φ has a meromorphic q-expansion of the form φz,τ = n h c n zq n, 0 < ξ < A, 0 < q < B ξ N, ξ = e iπz, A > 0, B > 0, N N,, where the coefficients c n z are in the function field Cξ. The function is an example of a Jacobi form of weight and index 0, that is a meromorphic function that satisfies z γτ + δ ; ατ + β = γτ + δ z; τ. γτ + δ Thus n, n N is a Jacobi form of weight n. The origin is the unique pole in a fundamental domain, of order n. In general, let z,τbe a meromorphic periodic function in z with respect to the lattice τ = Z + τ Z. Assume that, as function of τ, it satisfies z γτ + δ ; ατ + β = γ τ + δ m z,τ γτ + δ for every α β M = SL γ δ Z. Let g p τ be the p-th coefficient of the Taylor expansion of z; τat x 0 = xτ + y for some x, y R. Then for any M SL Z such that x, y = x, ym x, y Z,wehave ατ + β g p = γ τ + δ m+p g p τ. γτ + δ This means that for a fixed integer n = 0, the Taylor coefficient g n p, p N at the origin of the function n is a modular form on SL Z of weight p + n. As is well known, many analytic properties of the -function come from the differential equation = 4 3 g g 3, g = 4 π 4 3 E 4, g 3 = 8π 6 7 E From the decomposition theorem 9.orfrom9. we obtain = 6 + g, =, 3 = g + g 3,...,

27 and more generally, we have Proposition 9.3 For each positive integer, n = Bn 0 n! n + Bn B n 3! n 4 r n + + n r! n r + + B n n + B n n + Bn n, 9.3 where, for 0 r n + B n+ r = n rn r+ nn+ B r n + 4n+ n Bn r g + n+ n Bn r 3 g and B r n = 0 for r < 0 and r > n. As a consequence of 9.3, we obtain B n+ 0 B n+ r n = n + n +! n+ + + n + n r +! n r n + Bn+ n + n + Bn+ n. Bn n are constants and Bn, Bn,...,Bn n are homogeneous polynomials in g and g 3.In particular Bn n = 0. Lemma 9.4 For every n N \{0}, wehave: n+ = nn + n n + 4n + g n + n n + g 3 n, with taken as 0. The proof of this lemma is a straightforward computation from 9. and9.3. We conclude from 9.3 that lower order coefficients and pseudo-periods are given by the four-term relations B n+ n+ = n 4n + g B n n + n n + g 3 B n n, 9.6 B n n+ = n 4n + g B n n + n n + g 3 B n n 3, 9.7 η n+ = n 4n + g η n + n n + g 3η n. 9.8 By inversion of these relations or by taking successive derivations of 9. we obtain = 6 g, =, = g g 3, etc. More generally, we have for each n N n = P n+, n+ = Q n, 9.0

28 where P n+, Q n are polynomials in of degree n + andn, respectively and their coefficients are polynomials in g and g 3 with rational coefficients. For l < n +, the coefficient of l is a modular form of weight n + l. Looking at one of these polynomials A m + A m +, we notice that it must be a homogeneous polynomial of degree m in u,ω,ω so that its coefficients must be of the following form A = 0, A = ag, A 3 = bg 3, A 4 = cg, A 5 = dg g 3, A 6 = eg 3 + fg 3, where a, b,..., f are numerical constants. A precise analysis of the polynomials P n, Q n can also be done using the following results due to Feldman [] and which are very useful in transcendence theory. Lemma 9.5 The jth derivative j z of z can be expressed in the form ut, t, t z t z t z t where the summation is over all non-negative integers t, t, t with t + 3t + 4t = j + and ut, t, t, j, k denotes rational integers with absolute values at most 3 j j + 7!. Lemma 9.6 For any positive integer k, the jth derivative of z k can be expressed in the form ut, t, t, j, k z t z t z t where the summation is over all non-negative integers t, t, t with t + 3t + 4t = j + k and ut, t, t, j, k denote rational integers with absolute values at most j!48 j 7! 8 k. We recall that for a field k and a non-zero polynomial P k[x, X,, X n ], the weight wp is defined by wp = deg t P tx, t X,...,t n X n. The polynomial P is called isobaric of weight wp if for any monomial X i...x i n n PX,...,X n,wehave of n wp = ri r. It is a natural problem to investigate the polynomial in three indeterminate variables r=