Mock Theta Functions

Transcription

1 Mock Theta Functions Mock Thetafuncties met een samenvatting in het Nederlands Proefschrift Ter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezag van de Rector Magnificus, Prof. dr. W.H. Gispen, ingevolge het besluit van het College voor Promoties in het openbaar te verdedigen op woensdag 30 oktober 00 des ochtends te uur door Sander Pieter Zwegers geboren op 16 april 1975, te Oosterhout

2 Promotor: Copromotor: Prof. dr. D.B. Zagier Max-Planck-Institut für Mathematik, Bonn oud-hoogleraar aan de Universiteit Utrecht Dr. R.W. Bruggeman Faculteit Wiskunde en Informatica Universiteit Utrecht 000 Mathematics Subject Classification: 11F37, 11F50, 11E45, 11F7 Zwegers, Sander Pieter Mock Theta Functions Proefschrift Universiteit Utrecht Met samenvatting in het Nederlands ISBN Printed by PrintPartners Ipskamp, Enschede

3 Contents Introduction 1 Mock ϑ-functions This thesis Lerch Sums Introduction The Mordell integral Lerch sums A real-analytic Jacobi form? Period integrals of weight 3/ unary theta functions Indefinite ϑ-functions 5.1 Introduction Definition of ϑ Properties of the ϑ-functions Transformation properties of ϑ with respect to O + A Z Some examples Fourier Coefficients of Meromorphic Jacobi Forms Introduction A building block Transformation properties Simple poles An example Mock ϑ-functions Introduction General results The seventh order mock ϑ-functions The fifth order mock ϑ-functions Other mock ϑ-functions

4 iv Contents Bibliography 85 Samenvatting 89 Dankwoord 91 Curriculum Vitae 9

5 Introduction Mock ϑ-functions Early in 190, three months before his death, S. Ramanujan wrote his last letter to G.H. Hardy. For the mathematical part of this letter see [1, pp ] also reproduced in [4]. In the course of it he said: I discovered very interesting functions recently which I call Mock ϑ-functions. Unlike the False ϑ-functions studied partially by Prof. Rogers in his interesting paper [] they enter into mathematics as beautifully as the ordinary ϑ-functions. I am sending you with this letter some examples. He then provided a long list of mock ϑ-functions, together with identities satisfied by them. The first three pages in which Ramanujan explained what he meant by a mock ϑ-function are very obscure. G.N. Watson wrote the first papers [6] and [7] to elucidate the mock ϑ-functions. The first of these is Watson s Presidential Address to the London Mathematical Society in He entitled it The Final Problem: An Account of the Mock Theta Functions. In it he writes: I make no apologies for my subject being what is now regarded as old-fashioned, because, as a friend remarked to me a few months ago, I am an old-fashioned mathematician. His methods may have been a bit old-fashioned, but looking at the number of articles on mock ϑ-functions that have appeared since 1935, or even in the last ten years, we must conclude that the subject is still up-to-date. In these two papers, Watson proves most of the assertions found in the letter of Ramanujan. The first paper considers only the third-order functions. It provides three new mock ϑ-functions not mentioned in the letter. The bulk of the paper is devoted to the modular transformation properties of these functions. To get these transformations, he first proves certain identities. For example, for the third order mock ϑ-function fq := 1 + q 1 + q + q q 1 + q + q q 1 + q 1 + q 3 + he finds: fq = q n Z 1 n q 3 n + 1 n 1 + q n, 1

6 Introduction with q = e πiτ, τ H := {τ C Imτ > 0}, and q = n=1 1 qn = q 1 4 ητ, where η is the Dedekind eta-function. In Watson s second paper on mock ϑ-functions, he moves on to the fifth order functions. He manages to prove all of the identities given by Ramanujan in his letter. However, he is unable to find the modular transformation properties, simply because he is unable to find identities like 1 for the fifth order functions. He even expressed his doubts about finding anything comparable to 1. However Andrews see [] was able to find comparable results for most of the fifth order functions. For example, for the fifth order function which Watson denotes by f 0 one finds f 0 q = 1 q which may be rewritten as f 0 q = 1 q 1 j q 5 n + 1 n j 1 q 4n+, n 0 j n n+j 0,n j 0 n+j<0,n j<0 1 j q 5 n + 1 n j. The seventh order functions were mostly neglected by Watson, perhaps because Ramanujan makes no positive assertions about them. However A. Selberg see [3] provides a full account of the behaviour of the seventh order functions near the unit circle. In [1, pp. 666] we find identities similar to for the seventh order mock ϑ-functions F 0, F 1 and F. For example slightly rewritten F 0 q = 1 1 r+s q 3 r +4rs+ 3 s + 1 r+ 1 s. 3 q r,s 0 r,s<0 In [11] L. Göttsche and D. Zagier consider sums like the ones in and 3. They call them theta-functions for indefinite lattices. For some special cases they find modular transformation properties for these functions. However, these results do not include the sums in and 3. In [0], a theorem about the modularity of a certain family of q-series associated with indefinite binary quadratic forms is given. Again, the results do not include the sums in and 3. This thesis This thesis is the result of my research on the following two questions, both posed by Don Zagier: 1. How do the mock ϑ-functions fit in the theory of modular forms?. Is there a theory of indefinite theta functions?

7 This thesis 3 Since most of the mock ϑ-functions had been related to sums like the one in 1, I first considered this type of sum. The result of this research is Chapter 1. In it we consider the series n Z 1 n e πin +nτ+πinv 1 e πinτ+πiu τ H, v C, u C \ Zτ + Z. This function was also studied by Lerch in [15] see [14] for an abstract. Therefore we call this a Lerch sum. This sum is of the same type as the sum in 1. The function does not transform like a Jacobi form. However, we find that on addition of a relatively easy correction term the function does transform like a Jacobi form. This correction term is real-analytic. In Chapter we consider certain indefinite ϑ-functions, in an attempt to give a partial answer to the second question. These indefinite ϑ-functions are modified versions of the sums considered by Göttsche and Zagier in [11]. We find elliptic and modular transformation properties for these functions. Because of the modifications the indefinite ϑ-functions are no longer holomorphic in general. Although the results in this chapter are more general than the results in [11], the second question is far from being solved. This is because we only consider indefinite quadratic forms of type r 1, 1. It remains a problem of considerable interest to develop a theory of theta-series for quadratic forms of arbitrary type. In [3] Andrews gives most of the fifth order mock theta functions as Fourier coefficients of meromorphic Jacobi forms, namely certain quotients of ordinary Jacobi theta-series. This is the motivation for the study of the modularity of Fourier coefficients of meromorphic Jacobi forms, in Chapter 3. We find that modularity follows on adding a real-analytic correction term to the Fourier coefficients. In Chapter 4 we use the results from Chapter, together with, 3 and similar identities for other mock ϑ-functions, to get the modular transformation properties of the seventh-order mock ϑ-functions and most of the fifth-order functions. The final result is that we can write each of these mock ϑ-functions as the sum of two functions H and G, where: H is a real-analytic modular form of weight 1/ and is an eigenfunction of the appropriate Casimir operator with eigenvalue 3/16 this is also the eigenvalue of holomorphic modular forms of this weight; for the theory of real-analytic modular forms see for example [16, Ch. IV]; and G is a theta series associated to a negative definite unary quadratic form, i.e. has the form sgnfβf ye πif τ πifb, where f ranges over a certain arithmetic progression az + b a, b Q, τ = x + iy H and βx = x u 1 e πu du. Moreover G is bounded as τ tends vertically to any rational limit. This decomposition is thus similar to the one found in [9] for an Eisenstein series of weight 3/, the holomorphic part of the series, in that case, having class numbers as Fourier coefficients.

8 4 Introduction Many of the results of Chapter 4 could also be deduced using the methods from Chapter 1 or Chapter 3 instead of Chapter, i.e. we have actually given 3 approaches to proving modularity properties of the mock ϑ-functions.

9 Chapter 1 Lerch Sums 1.1 Introduction In this chapter, we first study a function h, which is essentially the function ϕ studied by Mordell in [17] and [18]. We reproduce some of the results of Mordell. In the next section we study a function µ, which is essentially a function studied by Lerch in [15]. We find elliptic and modular transformation properties of this function. One of these transformations involves the function h: 1 u e πiu v /τ µ iτ τ, v τ ; 1 + µu, v; τ = 1 hu v; τ. τ i In section 4 we find a real-analytic function R with essentially the same elliptic and modular transformation properties as µ. Combining the properties of µ and R we find a real-analytic function µ, which transforms like a Jacobi form. In the last section we relate h to a period integral of a unary theta function of weight 3/. 1. The Mordell integral In this section, we present results of Mordell found in [17] and [18], in a form suitable for the purpose of this chapter. The function h defined in Definition 1.1 is essentially the function ϕ studied by Mordell: ϕx; τ = 1 τe πiτ/4+πix hx τ + 1 ; τ. The same function was used earlier by Riemann as described by Siegel [4] to prove the functional equation for the Riemann zeta function. Mordell was the first to analyze the behaviour of this integral relative to modular transformations. Consequently, we shall refer to this integral as the Mordell integral.

10 6 Chapter 1. Lerch Sums Definition 1.1 For z C and τ H set hz = hz; τ := R e πiτx πzx cosh πx dx. Proposition 1. The function h has the following properties: 1 hz + hz + 1 = iτ e πiz+ 1 /τ, hz + e πiz πiτ hz + τ = e πiz πiτ/4, 3 z hz; τ is the unique holomorphic function satisfying 1 and, 4 h is an even function of z, 5 h z τ ; 1 τ = iτ e πiz /τ hz; τ, 6 hz; τ = e πi 4 hz; τ e πi e πiz /τ+1 4 τ + 1 Proof: 1 We have hz + hz + 1 = R z h τ + 1 ; τ. τ + 1 e πiτx πzx cosh πx 1 + e πx dx = e πiτx πxz+ 1 dx. R This last integral is well known and equals 1 iτ e πiz+ 1 /τ. If we change x into x + i we find R+i e πiτx πzx cosh πx dx = Now using Cauchy s theorem we find hz + e πiz πiτ hz + τ = R e πiτx+i πzx+i cosh πx + i R R+i dx = e πiz πiτ hz + τ. e πiτx πzx dx cosh πx e πiτx πzx = πi Res = e πiz πiτ/4. x=i/ cosh πx 3 If we have two holomorphic functions h 1 and h both satisfying the two equations, their difference f = h 1 h is a holomorphic function satisfying: { fz + fz + 1 = 0 fz + e πiz πiτ fz + τ = 0. So fz 0 + mτ + n = 1 m+n e πim τ+πimz 0 fz 0. Letting z 0 vary over a fundamental parallelogram [0, 1Z + [0, 1 and m, n vary over Z, we see that fz is bounded and

11 1.3 Lerch sums 7 tends to 0 as Imz, so f 0 by Liouville s theorem. 4 Replace x by x in the integral. 5 Let gx = 1 cosh πx. We first compute the Fourier transform Fg of g: Using Cauchy s formula we get e πizx e πizx dx = πi Res cosh πx x=i/ cosh πx = e πz, but so we find R+i R R+i e πizx cosh πx dx = R Fgz := R e πizx+i dx = e πz cosh πx + i R e πizx cosh πx dx = e πz = gz. 1 + e πz e πizx cosh πx dx, We see that g is its own Fourier transform! Note the unusual plus sign in the definition of the Fourier transform. Let f τ x = e πiτx, τ H. The Fourier transform of f τ is given by We now see R e πiτx +πizx cosh πx Ff τ = 1 iτ f 1 τ. dx = Ff τ gz = Ff τ Fgz = 1 f 1 gz = 1 e iτ τ iτ R 1 πi τ z x cosh πx This identity holds for z R. Since both sides are analytic functions of z, the identity holds for all z C. If we replace z by iz we get the desired result. We may also prove the identity of part 5 by using 1 and to show that z 1 iτ e πiz /τ h z τ ; 1 τ also satisfies the two equations 1 and. By uniqueness we get the equation. 6 Using 1 and we can show that the right hand side, considered as a function of z, also satisfies 1 and. The equation now follows from 3. dx. 1.3 Lerch sums In this section we will study the function n Z 1 n e πin +nτ+πinv 1 e πinτ+πiu τ H, v C, u C \ Zτ + Z.

12 8 Chapter 1. Lerch Sums This function was also studied by Lerch. The original paper [15] is in Czech and is not very easy to obtain. See [14] for an abstract in German. We will prove elliptic and modular transformation properties of this function in Proposition 1.4 and 1.5 respectively. These results are equivalent to the results found by Lerch. It is more convenient to normalize the above sum by dividing by the classical Jacobi theta function ϑ. Lerch did this too. We will first give, without proof, some standard properties of ϑ. For the theory of ϑ-functions see [19]. Proposition 1.3 For z C and τ H define ϑz = ϑz; τ := Then ϑ satisfies: 1 ϑz + 1 = ϑz. ϑz + τ = e πiτ πiz ϑz. e πiν τ+πiνz+ 1. ν 1 +Z 3 Up to a multiplicative constant, z ϑz is the unique holomorphic function satisfying 1 and. 4 ϑ z = ϑz. 5 The zeros of ϑ are the points z = nτ + m, with n, m Z. These are simple zeros. 6 ϑz; τ + 1 = e πi 4 ϑz; τ. 7 ϑ z τ ; 1 τ = i iτe πiz /τ ϑz; τ. 8 ϑz; τ = iq 1 8 ζ 1 1 q n 1 ζq n 1 1 ζ 1 q n, with q = e πiτ, ζ = e πiz. n=1 This is the Jacobi triple product identity. 9 ϑ 0; τ + 1 = e πi 4 ϑ 0; τ and ϑ 0; 1 τ = iτ3/ ϑ 0; τ. 10 ϑ 0; τ = πητ 3, with η as in the introduction. We now turn to the normalized version of Lerch s function: Proposition 1.4 For u, v C \ Zτ + Z and τ H, define Then µ satisfies: µu, v = µu, v; τ := 1 µu + 1, v = µu, v, eπiu 1 n e πin +nτ+πinv ϑv; τ 1 e πinτ+πiu. n Z

13 1.3 Lerch sums 9 µu, v + 1 = µu, v, 3 µu, v + e πiu v πiτ µu + τ, v = ie πiu v πiτ/4, 4 µu + τ, v + τ = µu, v, 5 µ u, v = µu, v, 6 u µu, v is a meromorphic function, with simple poles in the points u = nτ +m n, m Z, and residue 1 1 in u = 0, πi ϑv 7 µu + z, v + z µu, v = 1 ϑ 0ϑu + v + zϑz πi ϑuϑvϑu + zϑv + z, for u, v, u + z, v + z Zτ + Z, 8 µv, u = µu, v. Proof: 1 is trivial and follows from 1 of Proposition The definition of ϑ gives the following: ie πiτ/4+πiv ϑv = n Z = n Z = e πiv n Z 1 n e πin nτ+πinv 1 n e πin nτ+πinv 1 e πinτ+πiu 1 e πinτ+πiu 1 n e πin +nτ+πinv e πiu 1 n e πin +nτ+πinv 1 e πinτ+πiu+τ 1 e πinτ+πiu. n Z Dividing both sides by e πiu ϑv, we get the desired result. 4 Part of Proposition 1.3 gives µu + τ, v + τ = eπiu+τ 1 n e πin +nτ+πinv+τ ϑv + τ 1 e πinτ+πiu+τ n Z = eπiu+τ+πiτ+πiv ϑv n Z Replace n by n 1 in the last sum to get the desired result. 5 If we replace n by n in the definition of µ we see We multiply by eπinτ πiu e πinτ πiu 1 n e πin +3nτ+πinv 1 e πin+1τ+πiu. µu, v = eπiu 1 n e πin nτ πinv ϑv 1 e πinτ+πiu. n Z to find µu, v = e πiu ϑv n Z 1 n e πin +nτ πinv 1 e πinτ πiu.

14 10 Chapter 1. Lerch Sums Now using 4 of Proposition 1.3 we find µu, v = µ u, v. 6 From the definition we see that u µu, v has a simple pole if 1 e πinτ+πiu = 0, for some n Z. So u µu, v has simple poles in the points u = nτ +m n, m Z. The pole in u = 0 comes from the term n = 0. We see 1 lim u µu, v = u 0 ϑv lim u 1 1 = u 0 1 eπiu πi ϑv. 7 Consider fz = ϑu + zϑv + z µu + z, v + z µu, v. Using 1, and 5 of Proposition 1.3, and 1,, 4 and 6 of this proposition, we see that f has no poles, a zero for z = 0, and satisfies { fz + 1 = fz fz + τ = e πiτ πiu+v+z fz. It follows that the quotient fz/ϑzϑu + v + z is a double periodic function with at most one simple pole in each fundamental parallelogram, and hence constant: fz = Cu, vϑzϑu + v + z. 1.1 To compute C we consider z = u. If we take z = u in 1.1 we find by 4 of Proposition 1.3. By definition we have f u = Cu, vϑ uϑv = Cu, vϑuϑv 1. f u = lim ϑu + zϑv + z µu + z, v + z µu, v z u = ϑv u lim z 0 ϑzµz, v u ϑz = ϑv u lim lim zµz, v u = 1 z 0 z z 0 πi ϑ 0, where we have used 6. Combining 1. and 1.3 gives the desired result. 8 Take z = u v in 7 and use 5 of Proposition 1.3 to find 1.3 µ v, u = µu, v. If we now use 5, we get the desired result. Proposition 1.5 Let µ be as in Proposition 1.4. Then µ satisfies the following modular transformation properties:

15 1.4 A real-analytic Jacobi form? 11 1 µu, v; τ + 1 = e πi 4 µu, v; τ, u 1 iτ e πiu v /τ µ τ, v τ ; 1 τ with h as in Definition µu, v; τ = 1 hu v; τ, i Proof: 1 Use 6 of Proposition 1.3. Replacing u, v, z, τ by u τ, v τ, z τ, 1 τ in 7 of Proposition 1.4 and using 7 and 9 of Proposition 1.3 we see that the left hand side depends only on u v, not on u and v separately. Call it 1 hu v; i τ. Using 1 and 3 of Proposition 1.4 we see that h satisfies the two identities 1 and of Proposition 1., so if we can prove that h is a holomorphic function, then we may conclude that h = h, as desired. The poles of both u µu, v and u µ u τ, v τ ; 1 τ are simple, and occur at u Zτ + Z, so the only poles of u hu v could be simple poles for u Zτ + Z. Since this is a function of u v it has no poles at all, and hence is holomorphic. Alternatively, we can check, using 6 of Proposition 1.4 and 7 of Proposition 1.3, that the residue at u = 0 vanishes. By 1 and 3 of Proposition 1.4 the residues vanish for all u Zτ + Z, hence h is holomorphic. 1.4 A real-analytic Jacobi form? Definition 1.6 For z C we define Ez = z This is an odd entire function of z. Lemma 1.7 For z R we have 0 e πu du = π n z n+1 n! n + 1/. n=0 Ez = sgnz 1 βz, where βx = x u 1 e πu du x R 0. Proof: Write z du as sgnz z e 0 e πu πu du and substitute u = v. 0 We consider for u C and τ H the series Ru; τ = ν 1 +Z with y = Imτ and a = Imu Imτ. { sgnν E ν + a y } 1 ν 1 e πiν τ πiνu,

16 1 Chapter 1. Lerch Sums Lemma 1.8 For all c, ɛ > 0, this series converges absolutely and uniformly on the set {u C, τ H a < c, y > ɛ}. The function R it defines is real-analytic and satisfies and τ Raτ b; τ = i e πa y y R u u; τ = y 1/ e πa y ϑu; τ ν 1 ν + ae πiν τ πiνaτ b. 1.5 ν 1 +Z Proof: We split sgnν Eν + a y into the sum of sgnν sgnν + a y and sgnν + a yβν + a y. We see that sgnν sgnν + a y is nonzero for only a finite number of values ν 1 + Z this number depends on a, but since a is bounded, so is this number. Hence the series { sgnν sgn ν + a } y 1 ν 1 e πiν τ πiνu ν 1 +Z converges absolutely and uniformly. We can easily see that 0 βx e πx for all x R 0, hence { sgn ν + a } y β ν + a y 1 ν 1 e πiν τ πiνu e πν+a y e πiν τ πiνu = e πν+a y πa y e πν+aɛ. We have the inequality ν + a 1 ν, for ν ν 0, for some ν 0 R which depends only on c a is bounded by c. Hence we see that the series { sgn ν + a } y β ν + a y 1 ν 1 e πiν τ πiνu ν 1 +Z converges absolutely and uniformly on the given set. Since R is the infinite sum of real-analytic functions, and the series converges absolutely and uniformly, it is real-analytic. We fix τ H, and determine u = aτ b by the coordinates a, b R. We see Raτ b; τ = a + τ b a + τ b ν 1 +Z { sgnν E ν + a } y 1 ν 1 e πiν τ πiνaτ b

17 1.4 A real-analytic Jacobi form? 13 = y ν 1 +Z E = y = ye πa y ν + a y 1 ν 1 e πiν τ πiνaτ b ν 1 +Z e πν+a ν 1 +Z 1 ν 1 = i ye πay ϑaτ b; τ, y 1 ν 1 e πiν τ πiνaτ b e πiν τ πiνaτ b with ϑ as in Proposition 1.3 and the term-by-term differentiation being easily justified. Since u = i y a + τ b, this gives the differential equation 1.4. Similarly τ = 1 Raτ b; τ x + i y ν 1 +Z { sgnν E ν + a } y 1 ν 1 e πiν τ πiνaτ b = i 1 ν + a E ν + a y 1 ν 1 e πiν τ πiνaτ b y ν 1 +Z = i y e πa y 1 ν 1 ν + a e πiν τ πiνaτ b, ν 1 +Z proving equation 1.5. Proposition 1.9 The function R has the following elliptic transformation properties: 1 Ru + 1 = Ru, Ru + e πiu πiτ Ru + τ = e πiu πiτ/4, 3 R u = Ru. Proof: Part 1 is trivial, and for 3 we replace ν by ν in the sum and use the fact that E is an odd function. To prove, we start with e πiu πiτ Ru + τ = e πiu πiτ = ν 1 +Z ν 1 +Z { sgnν E ν + a + 1 } y 1 ν 1 e πiν τ πiνu+τ { sgnν 1 E ν + a } y 1 ν 1 e πiν τ πiνu,

18 14 Chapter 1. Lerch Sums where we have replaced ν by ν 1. We now find Ru+e πiu πiτ Ru + τ = { } sgnν sgnν 1 1 ν 1 e πiν τ πiνu = e πiu πiτ/4, ν 1 +Z since sgnν sgnν 1 is zero for all ν 1 + Z except for ν = 1. Proposition 1.10 R has the following modular transformation properties: 1 Ru; τ + 1 = e πi 4 Ru; τ, 1 u e πiu /τ R iτ τ ; 1 + Ru; τ = hu; τ. τ Proof: Part 1 is trivial. The left hand side of we call hu; τ. Using 1 and of Proposition 1.9 we can see that h satisfies: { hu + hu + 1 = iτ e πiu+ 1 /τ, hu + e πiu πiτ hu + τ = e πiu πiτ/4. Part 3 of Proposition 1. determines h as the unique holomorphic function with these properties. This reduces the proof to showing that h is a holomorphic function of u. We fix τ H, and determine u = aτ b by the coordinates a, b R this implies a = Imu Imτ as in Lemma 1.8. Since u = i y a + τ haτ b; τ = 0. b a + τ b, we have to show that According to Lemma 1.8 we have a + τ Raτ b; τ = i ye πay ϑaτ b; τ 1.6 b We have a + τ aτ b R ; 1 = τ b τ τ b + 1 R a b τ a τ ; 1. τ Up to a factor τ this is the same as a + τ b Raτ b; τ, with a, b, τ replaced by b, a, 1 τ. Hence by 1.6 we find a + τ aτ b R ; 1 = iτ y b τ τ e πb y ϑ b τ + a; 1 τ = iτ y e πb y ϑ aτ b ; 1, τ τ

19 1.4 A real-analytic Jacobi form? 15 with y = Im 1 τ = y ττ. In the last step we have used 4 of Proposition 1.3. If we now use 7 of Proposition 1.3, with z = aτ b and τ replaced by τ, we see that this equals iτ y e πb y i iτe πiaτ b /τ ϑaτ b; τ = i y iτe πiaτ b /τ e πay ϑaτ b; τ. 1.7 Using 1.6 and 1.7 we find a + τ haτ b; τ b = 1 e πiaτ b /τ iτ a + τ b + a + τ Raτ b; τ = 0. b aτ b R ; 1 τ τ We have established the fact that h is holomorphic, and hence equals h. In the next theorem we combine the properties of µ and R to find a function µ which is no longer meromorphic, but has better elliptic and modular transformation properties than µ. Theorem 1.11 We set then µu, v; τ = µu, v; τ + i Ru v; τ, µu + kτ + l, v + mτ + n = 1 k+l+m+n e πik m τ+πik mu v µu, v, for k, l, m, n Z, u µ cτ + d, v cτ + d ; aτ + b = vγ 3 cτ + d 1 e πicu v /cτ+d µu, v; τ, cτ + d for γ = a b c d SL Z, with vγ = η aτ+b cτ+d cτ / + d 1 ητ 3 µ u, v = µv, u = µu, v, 4 µu + z, v + z µu, v = 1 ϑ 0ϑu + v + zϑz πi ϑuϑvϑu + zϑv + z, for u, v, u + z, v + z Zτ + Z, 5 u µu, v has singularities in the points u = nτ + m n, m Z. Furthermore we have lim u 0 u µu, v = 1 1 πi ϑv.

20 16 Chapter 1. Lerch Sums Remark 1.1 Parts 1 and of the theorem say that the function µ transforms like a two-variable Jacobi form of weight 1 and index for the theory of Jacobi forms, see [9], where, however, only Jacobi forms of one variable are considered. Furthermore we can find several differential equations satisfied by µ. Therefore we would like to call this function a real-analytic Jacobi form. However, in the literature I haven t been able to find a satisfying definition of a real-analytic Jacobi form. I intend to return to this problem in the future. Remark 1.13 All three function in 1.8 have a property that the other two do not have: µ transforms well like a Jacobi form, µ is meromorphic and u, v Ru v depends only on u v. Proof: 1 Using the first four parts of Proposition 1.4 and the first two of Proposition 1.9 we find µu + 1, v = µu, v, µu, v + 1 = µu, v, µu + τ, v = e πiu v+πiτ µu, v, µu, v + τ = e πiv u+πiτ µu, v. Combining these equations we get the desired result. Using Proposition 1.5 and Proposition 1.10 we find µu, v; τ + 1 = e πi 4 µu, v; τ u µ τ, v τ ; 1 = iτe πiu v /τ µu, v; τ τ Set mu, v; τ := ϑu v; τ µu, v; τ. Using 6 and 7 of Proposition 1.3 we see and so mu, v; τ + 1 = mu, v; τ u m τ, v τ ; 1 = τ mu, v; τ τ u m cτ + d, v cτ + d ; aτ + b = cτ + d mu, v; τ, cτ + d for all a b c d SL Z. Hence u µ cτ + d, v cτ + d ; aτ + b cτ + d ϑu v; τ = cτ + d µu, v; τ. 1.9 ϑ u v cτ+d ; aτ+b cτ+d From 6 and 7 of Proposition 1.3 we find z ϑ cτ + d ; aτ + b = χγ cτ + d e πicz /cτ+d ϑz; τ, 1.10 cτ + d

21 1.5 Period integrals of weight 3/ unary theta functions 17 with χγ some eighth root of unity. Applying Using 10 of Proposition 1.3 we find d dz z=0 on both sides gives ϑ 0; aτ + b = χγcτ + d 3 ϑ 0; τ. cτ + d χγ = vγ 3 If we combine this with 1.9 and 1.10 we get the desired result. 3 Using 5 of Proposition 1.4 and 3 of Proposition 1.9 we find µ u, v = µu, v Using 8 of Proposition 1.4 and 3 of Proposition 1.9 we find µv, u = µu, v 4 This follows directly from 7 of Proposition R has no singularities, so the singularities come from µ. The location and nature of these singularities is already given in 6 of Proposition Period integrals of weight 3/ unary theta functions In this section we will rewrite h in terms of the period integral of a unary theta function of weight 3/. To state the main result we need the following definition: Definition 1.14 Let a, b R and τ H then g a,b τ := νe πiντ+πiνb. ν a+z The function g a,b is a unary theta function. Proposition 1.15 g a,b satisfies: 1 g a+1,b τ = g a,b τ g a,b+1 τ = e πia g a,b τ 3 g a, b τ = g a,b τ 4 g a,b τ + 1 = e πiaa+1 g a,a+b+ 1 τ

22 18 Chapter 1. Lerch Sums 5 g a,b 1 τ = ieπiab iτ 3 g b, a τ I will not prove these relations, since they are all easy. For 5 use Poisson summation. From these relations it follows that if a and b are rational, the function g a,b is a modular form of weight 3/. Theorem 1.16 Let τ H, then 1 for a 1, 1 and b R i τ with R as in Lemma 1.8. for a, b 1, 1 i with h as in Definition g a+ 1,b+ 1 z iz + τ dz = e πia τ+πiab+ 1 Raτ b, g a+ 1,b+ 1 z iz + τ dz = e πia τ+πiab+ 1 haτ b, Remark 1.17 The left hand side in is 1-periodic as a function of a and, up to a factor, as a function of b, while the right hand side is not. For the proof we need the following two lemmas: Lemma 1.18 If r R, r 0, and τ H then e πiτw i e πir z dw = πr dz. w + ir 0 iz + τ Proof: Both sides define a holomorphic function of τ H. So we only have to prove the identity for τ = it, with t R >0. The identity then becomes e πtw e πr u dw = πir du, w + ir 0 u + t where we have substituted z = iu in the integral on the right. We can easily see that both sides, considered as functions of t, are solutions of t + πr ft = πir t : t + πr LHS = π w ire πtw dw = πir t + πr RHS = e πr t t e πrt RHS = πire πr t t Both sides have the same limit 0 as t, hence they are equal. e πtw dw = πir t, t e πr x x dx = πir t.

23 1.5 Period integrals of weight 3/ unary theta functions 19 Lemma 1.19 Let b 1, 1 and let z C, such that z 1 + Z i, then eπbz cosh πz = 1 π eπiνb+ 1 z iν ν 1 +Z Proof: We first show that the series on the right hand side converges it doesn t converge absolutely. Since Dirichlet s test for convergence see [8, pp. 17] is not directly applicable, we prove this by means of partial summation we could also do this by comparing the series with the series e πinb+ 1 : Define n Z 0 T ν := eπiν+1b+ 1 e πib+ 1 1, then T ν T ν 1 = e πiνb+ 1 and T ν = 1 e πib Hence ν 1 +Z ν 0 ν ν 1 = i e πiνb+ 1 z iν ν 1 +Z ν 0 ν ν 1 = ν 1 +Z ν 0 ν ν 1 T ν T ν 1 z iν T ν z iνz iν + 1 T ν 0 1 z iν 0 + T ν0 1 n. T ν1 z iν 1 + 1, with ν 0, ν Z. We have lim ν 0 z iν 0 = 0, because T ν0 is bounded, and T also lim ν1 ν1 z iν = 0. Since T ν 1+1 ν 1 +Z z iνz iν+1 converges it converges absolutely, so does ν 1 +Z eπiνb+ 1 z iν. For z iz we consider the 1-periodic function given by b e πbz for b 1, 1. An easy calculation shows that 1 1 e πbz e πinb db = sinh πz π 1 n z in The given function is continuous on the interval 1, 1, hence we get from the theory of Fourier series that for b 1, 1 e πbz = sinh πz π n Z 1 n sinh πz e πinb+ 1 z in eπinb = π z in, n Z where n Z means lim m m n= m. If we replace z by z 1 i, substitute ν = n + 1 and multiply both sides by e πib / cosh πz, we find eπbz cosh πz = 1 π eπiνb+ 1 z iν ν 1 +Z.

24 0 Chapter 1. Lerch Sums We have shown that 1 ν 1 +Z eπiνb+ z iν converges, hence we get the desired result. Proof of Theorem 1.16: Let a 1, 1, then g a+ 1,b+ 1 z = O e πν 0 Imz for Imz, for some ν 0 > 0 for a 1, 0 we can take ν 0 = a + 1, for a [0, 1 we can take ν 0 = 1 a. From this estimate, the absolute convergence of both i τ g a+ 1,b+ 1 z iz+τ 1 We see i τ dz and i 0 g a+ 1,b+ 1 z dz = iz + τ = i = i g a+ 1,b+ 1 z iz+τ i g a+ 1,b+ 1 iy u 1 νe πiν y ν a+ 1 +Z dz follows. z τ iz νe πiν τ+πiνb+ 1 ν a+ 1 +Z y = i sgnνe πiν ν a+ 1 +Z = i ν a+ 1 +Z = e πia τ+πiab+ 1 ν 1 +Z iu τ+πiνb+ 1 du τ+πiνb+ 1 g a+ 1 dz = i y u 1 e πν u du yν v 1 e πv dv { sgnν E ν } y e πiν τ+πiνb+ 1 { sgnν + a E,b+ 1 iu τ du u ν + a } y 1 ν 1 e πiν τ πiνaτ b If we use that for a 1, 1 we have sgnν + a = sgnν for all ν 1 + Z, we get the desired result. We see by means of Cauchy s theorem that for a 1, 1 hx; τ = R e πiτz πxz cosh πz dz = ia+r e πiτz πxz cosh πz dz, hence e πia τ+πiab+ 1 haτ b = e πia = e πia ia+r R e πiτz+ia +πbz+ia cosh πz e πiτz +πbz cosh πz ia dz. dz If we replace z by z ia in Lemma 1.19, multiply both sides by e πiab+ 1 and replace

25 1.5 Period integrals of weight 3/ unary theta functions 1 ν by ν a, we find e πia with b 1, 1. Hence we find e πia e πbz e πbz cosh πz ia = 1 π cosh πz ia = eπia cos πa + 1 π eπiνb+ 1 z iν ν a+ 1 +Z ν a+ 1 +Z e πiνb+ 1, 1 z iν + 1. iν This identity also follows from the theory of partial fraction decompositions given in [8, pp ]. Using it we see e πia τ+πiab+ 1 haτ b = e πiτz eπia cos πa + 1 π R = eπia cos πa iτ π ν a+ 1 +Z e πiνb+ 1 e πiτz R ν a+ 1 +Z 1 z iν + 1 dz iν +πiνb+ 1 1 z iν + 1 dz. iν We want to change the order of summation and integration. This is allowed if +πiνb+ 1 1 eπiτz z iν + 1 dz 1.11 iν R ν a+ 1 +Z converges. We have +πiνb+ 1 1 eπiτz z iν = e πyz iν z iν + 1 iν z ν e πyz with y = Imτ. Both dz and R z e πyz ν a+ 1 +Z 1 ν converge and hence the expression in 1.11 converges. Now making the change of order we find e πia τ+πiab+ 1 haτ b = eπia cos πa iτ π Using Lemma 1.18 we see R e πiνb+ 1 ν a+ 1 +Z 1 e πiτz z iν + 1 i dz = πν iν 0 R 1 e πiτz z iν + 1 dz. iν e πiν z iz + τ dz + 1 iν 1 iτ.,

26 Chapter 1. Lerch Sums Using partial integration we see that this equals so We have and 1 ν e πia τ+πiab+ 1 haτ b = eπia 1 1 cos πa iτ π e πiνb+ 1 ν ν a+ 1 +Z i 0 i 0 ν a+ 1 +Z e πiν z dz, iz + τ 3 i e πiνb+ 1 0 e πiν z iz + τ 3 = 1 e πiν z ν z + τ 3 e πiν z ν τ 3 dz = 1 π τ 3 ν e πiν z dz. iz + τ 3 z eπiν, ν τ 3 ν a+ 1 +Z 1 ν 3 <. Hence we can interchange the order of summation and integration in 1.1: e πia τ+πiab+ 1 haτ b = eπia 1 1 cos πa iτ π Using partial integration we find ν a+ 1 +Z ν ν 0 i e πia τ+πiab+ 1 haτ b = eπia 1 + i cos πa iτ π i π i iz + τ d dz 1 iz + τ 3 ν 1 e πiν z+πiνb+ 1 dz. ν a+ 1 +Z ν a+ 1 +Z ν 1 e πiν z+πiνb+ 1 = eπia 1 i 1 lim cos πa iτ π iτ z i0 i g a+ 1 +,b+ 1 z dz. 0 iz + τ iz + τ ν a+ 1 +Z ν 1 e πiν ν a+ 1 +Z ν 1 e πiν ν a+ 1 +Z ν ν 0 i 0 z+πiνb+ 1 dz z+πiνb Let z ir 0, and ν 0 a Z, ν 0 > 0. By partial summation we find with T ν as in the proof of Lemma 1.19 ν 1 e πiν z+πiνb+ 1 = eπiν 0 z T ν0 1 + e πiν z z T ν eπiν+1, ν 0 ν ν + 1

27 1.5 Period integrals of weight 3/ unary theta functions 3 so Hence ν a+ 1 +Z ν ν 0 eπiν 0 z ν 1 e πiν z+πiνb+ 1 1 ν 0 e πib = eπiν 0 z ν a+ 1 +Z ν ν 0 ν 0 e πib+ 1 e 1 ν πib ν a+ 1 +Z 1 e πiν z e πib+ 1 1 ν ν 1 e πiν z+πiνb+ 1 z eπiν+1 ν + 1 ν a+ 1 converges uniformly for z ir 0. If we replace ν by ν we find ν 1 e πiν z+πiνb+ 1 = e πia 1 ν a+ 1 +Z ν a+ 1 +Z ν 1 e πiν z+πiν b+ 1, ν a 1 ν a+ 1 so ν a+ 1 +Z ν 1 e πiν z+πiνb+ 1 and hence ν a 1 ν a+ 1 +Z ν 1 e πiν z+πiνb+ 1 converge uniformly for z ir 0. Since the summand is continuous on ir 0, so is z ν a+ 1 +Z ν 1 e πiν z+πiνb+ 1. Hence lim ν 1 e πiν z+πiνb+ 1 = ν 1 e πiνb+ 1 = πi eπia z i0 cos πa, ν a+ 1 +Z ν a+ 1 +Z by Lemma 1.19 with z = 0. If we put this into 1.13, we get the desired result.

28 4 Chapter 1. Lerch Sums Remark 1.0 We may also prove part by using of Proposition 1.10 together with part 1: We split the integral i into the sum of i 0 τ and τ. For the first 0 integral we have i τ g a+ 1,b+ 1 z iz + τ dz = e πia τ+πiab+ 1 Raτ b, 1.14 by 1. In the second integral we replace z by 1 z : τ 0 g a+ 1,b+ 1 z i dz = iz + τ 1 τ 1 z i 1 1 z + τ g a+ 1,b+ 1 iz dz = i iτ e πia 1 b+ 1 i 1 τ g b+ 1, a+ 1 z i z 1 τ dz, by 5 and of Proposition Using part 1 of this proposition with a, b, τ replaced by b, a, 1/τ we see that this equals 1 aτ b e πib /τ+πia R ; iτ τ τ Combining 1.14 and 1.15 we see i 0 g a+ 1,b+ 1 z iz + τ dz = e πia τ+πiab+ 1 Raτ b 1 aτ b e πib /τ+πia R ; 1 iτ τ τ = e πia τ+πiab+ 1 { Raτ b; τ + 1 aτ b e πiaτ b /τ R ; 1 } iτ τ τ = e πia τ+πiab+ 1 haτ b, by of Proposition 1.10.

29 Chapter Indefinite ϑ-functions.1 Introduction The classical theta series associated to a positive definite quadratic form Q : R r R and B : R r R r R, the associated bilinear form Bx, y = Qx+y Qx Qy, is the series Θz; τ := n Z r e πiqnτ+πibn,z..1 These theta series have well-known transformation properties. In particular Θ0; τ is a modular form of weight r/. In [11] Göttsche and Zagier define a theta function for the case when the type of Q is r 1, 1. The definition of these functions is almost the same as in.1, only here the sum doesn t run over Z r, but some appropriate subset. However, in general, these functions do not have nice modular transformation properties. In this chapter we give a modified definition. We find elliptic and modular transformation properties for these functions. The theta functions we define depend not only on Q, but also on two vectors c 1, c R r with Qc i 0, i = 1,. The case Qc 1 = Qc = 0 gives the same functions as in [11]. There is a connection between the indefinite ϑ-functions from this chapter and certain ϑ-functions considered by Siegel see [5]. However, I will not give this connection here.. Definition of ϑ Let A be a symmetric r r-matrix with integer coefficients, which is non-degenerate. We consider the quadratic form Q : C r C, Qx = 1 x, Ax and the associated bilinear form Bx, y = x, Ay = Qx + y Qx Qy. The type of Q is the pair r s, s, where s is the largest dimension of a linear subspace of R r on which Q is negative definite. The signature of Q is the number

30 6 Chapter. Indefinite ϑ-functions r s. From now on we assume that s = 1, i.e., that Q has type r 1, 1. Then the set of vectors c R r with Qc < 0 has two components. If Bc 1, c < 0 then c 1 and c belong to the same component, while if Bc 1, c > 0 then c 1 and c belong to opposite components. Let C Q be one of the two components. If c 0 is a vector in that component, then C Q is given by: We further set C Q := {c R r Qc < 0, Bc, c 0 < 0}. S Q := {c Z r c primitive, Qc = 0, Bc, c 0 < 0}. c primitive means that the greatest common divisor of the components of c is 1. The r 1-dimensional hyperbolic space C Q /R + is the natural domain of definition of automorphic forms with respect to O + A Z see section.4 for the definition, and S Q is a set of representatives for the corresponding set of cusps {c Q r Qc = 0, Bc, c 0 < 0}/Q +. Note that S Q is empty in some cases, for example if A = Further we put C Q := C Q S Q. This is a generalisation of the usual construction H = H P 1 Q, which is the special cone C Q = C Q S Q for the quadratic form Qa, b, c = 1 b 4ac. For c C Q put { R r if c C Q Rc := {a R r Bc, a Z} if c S Q and Dc := {z, τ C r H Imz/ Imτ Rc}. Definition.1 Let c 1, c C Q. We define the theta function of Q with characteristics a Rc 1 Rc and b R r, with respect to c 1, c by ϑ a,b τ = ϑ c1,c a,b τ := ρν; τe πiqντ+πibν,b, ν a+z r where ρν; τ is defined by ρν; τ = ρ c1,c A ν; τ := ν; τ ρ ρc1 c ν; τ, with Bc,ν ρ c E y 1/ if c C Q, ν; τ = Qc sgnbc, ν if c S Q,

31 . Definition of ϑ 7 with y = Imτ and E as in Definition 1.6. For z, τ Dc 1 Dc, we define the theta function of Q with respect to c 1, c by ϑz; τ = ϑ c1,c A z; τ := e πiqaτ πiba,b ϑ a,b τ = ρn + a; τe πiqnτ+πibn,z, n Z r with a, b R r defined by z = aτ + b, so a = Imz Imτ, b = Imzτ Imτ. Remark. The definition doesn t change if we replace c i by λc i, with λ R +. Hence we could replace the condition Qc i < 0 by Qc i = 1. This would simplify the definition of ρ. Remark.3 In some special cases ϑ a,b is holomorphic: if c 1, c S Q and, as we will see in section.5, also for some special values of c 1, c, a and b. In general however, the functions ϑ and ϑ a,b are not holomorphic. Because Q is indefinite, e πiqnτ isn t bounded. Therefore it s not immediately clear that the series defining ϑz; τ converges absolutely. However, using an estimate for the growth of ρ we shall find: Proposition.4 The series defining ϑz; τ converges absolutely. For the proof of this proposition, we need two lemmas Lemma.5 Let c C Q. The quadratic form Q c : R r R, Q c ν := Qν Bc,ν Qc is positive definite, and we have with Q c ν λ c,c0 Q c0 ν ν R r, λ c,c0 = Bc, c 0 QcQc 0 Bc, c 0 Bc, c 0 4QcQc 0 QcQc 0 Proof: If ν R r is linearly independent of c, the quadratic form Q has type 1, 1 on span{c, ν}; hence the matrix Qc Bc, ν > 0. Bc, ν Qν has determinant < 0, so 4QνQc Bc, ν < 0. Rewriting gives Qν Bc, ν Qc Bc, ν > 0. 4Qc

32 8 Chapter. Indefinite ϑ-functions If ν = λc, with λ 0, we get Qν Bc,ν Qc = Qcλ > 0, which proves that Q c is positive definite. For the second part we consider the restriction of Q c to the ellipsoid S = {ν R r Q c0 ν = 1}. Since S is compact, Q c S assumes its absolute minimum at some point ν 0. We compute that minimum with the method of Lagrange multipliers: There is a real number λ, such that or equivalently A ν 0 Bν 0, c c Qc Q c ν 0 = λ Q c0 ν 0, = λa ν 0 Bν 0, c 0 c 0.. Qc 0 Taking the inner product with c on both sides of. we find Bν 0, c = λ Bν 0, c Bν 0, c 0 Bc, c 0..3 Qc 0 Taking the inner product with c 0 on both sides of. we find Combining.3 and.4 we find or Bν 0, c 0 Bν 0, cbc, c 0 Qc = λbν 0, c 0..4 QcQc 0 λ + 1 = λbc, c 0.5 Bν 0, c = Bν 0, c 0 = 0. If Bν 0, c = Bν 0, c 0 = 0, then. reduces to Aν 0 = λaν 0, from which we find λ = 1. The roots of.5 are λ ± = Bc, c 0 QcQc 0 ± Bc, c 0 Bc, c 0 4QcQc 0. QcQc 0 Taking the inner product with ν 0 on both sides of. and dividing by, we find Q c ν 0 = λq c0 ν 0 = λ. Hence the absolute minimum of Q c S is the minimum of {1, λ, λ + } which is λ = λ c,c0. So we have Q c ν λ c,c0 ν S. Let ν R r ν, ν 0, then S, so Qc0 ν ν λ c,c0 Q c Qc0 ν = Q cν Q c0 ν. Multiplying both sides by Q c0 ν we get the desired result.

33 . Definition of ϑ 9 Lemma.6 Let c 1, c C Q be linearly independent. The quadratic form Q + : R r R, Q + Bc ν := Qν + 1,c 4Qc 1Qc Bc 1,c Bc 1, νbc, ν is positive definite. Proof: If ν R r is not a linear combination of c 1 and c, the quadratic form Q has type, 1 on span{c 1, c, ν}; so the matrix Qc 1 Bc 1, c Bc 1, ν Bc 1, c Qc Bc, ν.6 Bc 1, ν Bc, ν Qν has determinant < 0. Rewriting gives Q + ν > Qc Bc 1, ν + Qc 1 Bc, ν 4Qc 1 Qc Bc 1, c 0. If ν R r is a linear combination of c 1 and c the determinant of the matrix in.6 is zero. Hence Q + ν = Qc Bc 1, ν + Qc 1 Bc, ν 4Qc 1 Qc Bc 1, c. So if Q + ν = 0, we have Bc 1, ν = Bc, ν = 0, which implies ν = 0. Thus if ν 0 then Q + ν is strictly positive, so Q + is positive definite. Proof of Proposition.4: If c 1, c C Q, we write ρν; τ, using Lemma 1.7, as the sum of the three expressions sgn Bc 1, ν β sgn Bc, ν β Bc 1, ν Qc 1 Bc, ν Qc y,.7 y.8 and sgn Bc 1, ν sgn Bc, ν,.9 with β as in Lemma 1.7. If c 1 C Q and c S Q we get only the sum of the first and the last expression. If c 1 S Q and c C Q we get the sum of the last two expressions. If c 1, c S Q we have only the last expression. Hence the proof is reduced to showing that the series Bc, ν Bc, ν β y e πiqντ+πibν,b.10 Qc ν a+z r sgn converges absolutely for all c with Qc < 0, and that the series { } sgn Bc 1, ν sgn Bc, ν e πiqντ+πibν,b.11 ν a+z r

34 30 Chapter. Indefinite ϑ-functions converges absolutely for all c 1, c C Q. We will first show that the series.10 converges absolutely for all c with Qc < 0: We can easily see that 0 βx e πx for all x R 0 ; hence if Qc < 0 Bc, sgn Bc, ν ν β y e πiqντ+πibν,b Qc e π Bc,ν y Qc e πiqντ+πibν,b.1 = e π Qν Bc,ν Qc y. Using Lemma.5, we see that the series ν a+z r e π Qν Bc,ν y Qc converges, and so the series.10 converges absolutely if Qc < 0. We will now show that the series.11 converges absolutely for all c 1, c C Q : If c 1 and c are linearly dependent, we have ρ c1,c = 0. Hence we can assume that they are linearly independent. Case 1: c 1, c C Q. If we have Bc 1, νbc, ν > 0, then sgnbc 1, ν sgnbc, ν = 0. If we have Bc 1, νbc, ν 0, then note that 4Qc 1 Qc Bc 1, c < 0, as we saw before Qν Q + ν, with Q + as in Lemma.6. Hence we find { } sgn Bc 1, ν sgn Bc, ν e πiqντ+πibν,b = sgn Bc 1, ν sgn Bc, ν e πqνy e πq+ νy. Using Lemma.6, we see that the series e πq + νy ν a+z r.13 converges, and so the series.11 converges absolutely. Case : c 1 C Q and c S Q. We can assume that c 1 C Q Z r, since otherwise we pick any c 1 C Q Z r, write { } sgn Bc 1, ν sgn Bc, ν e πiqντ+πibν,b ν a+z r = ν a+z r + ν a+z r { } sgn Bc 1, ν sgn Bc 1, ν e πiqντ+πibν,b { sgn Bc 1, ν } sgn Bc, ν e πiqντ+πibν,b,

35 . Definition of ϑ 31 and use that ν a+z r { } sgn Bc 1, ν sgn Bc 1, ν e πiqντ+πibν,b converges absolutely. We write ν = µ + nc with µ a + Z r and n Z, such that Bc1,µ Bc 1,c [0, 1 ] we see n =. Then sgnbc, ν = sgnbc, µ use Bc, c = 0 and [ Bc1,ν Bc 1,c sgnbc 1, ν = sgnn + Bc1,µ Bc 1,c. Hence { } sgn Bc 1, ν sgn Bc, ν Using we see ν a+z r = n Z µ a+z r n Z Bc 1,µ Bc 1,c [0,1 e πiqντ+πibν,b { sgn Bc, µ + sgn n + Bc 1, µ Bc 1, c } e πiqµτ+πibc,µnτ+πibµ,b+πibc,bn { 1 1 x = n=0 xn if x < 1 1 n= xn if x > 1, { sgn Bc, µ + sgn n + Bc 1, µ } Bc 1, c = Here we used that 1 e πibc,µτ+πibc,b δbc 1, µ. e πibc,µnτ+πibc,bn Bc, µ Bc, µ > 0,.14 for all µ a + Z r, and for some µ a + Z r. This is guaranteed by the fact that z, τ Dc. Since c 1, c Z r we have { } µ a + Z r Bc 1, µ [0, 1 = µ 0 + c 1 Z, Bc 1, c µ 0 P 0 for a suitable finite set P 0, with c 1 Z := {ξ Zr Bc 1, ξ = 0}. So { } sgn Bc 1, ν sgn Bc, ν e πiqντ+πibν,b ν a+z r = µ 0 P 0 ξ c 1 { } 1 e δbc 1, µ πibc,ξ+µ0τ+πibc,b 0 e πiqξ+µ0τ+πibξ+µ0,b.

36 3 Chapter. Indefinite ϑ-functions This series converges absolutely, since Q is positive definite on c 1 Z, and the term 1 e πibc,ξ+µ0τ+πibc,b δbc 1, µ 0 is bounded use.14. Case 3: c 1 S Q and c C Q. Since ϑ c1,c = ϑ c,c1, this follows directly from the previous case. Case 4: c 1, c S Q. Since ϑ c1,c = ϑ c1,c3 + ϑ c3,c, for arbitrary c 3 C Q, this follows directly from case and 3..3 Properties of the ϑ-functions The theta functions in Definition.1 have some nice elliptic and modular transformation properties, similar to those of the theta functions associated to positive definite quadratic forms. Proposition.7 The function ϑ satisfies: 1 For c 1, c, c 3 C Q and z, τ Dc 1 Dc Dc 3 we have the cocycle conditions ϑ c1,c + ϑ c,c1 = 0 and ϑ c1,c + ϑ c,c3 + ϑ c3,c1 = 0. ϑz + λτ + µ; τ = e πiqλτ πibz,λ ϑz; τ for all λ Z r and µ A 1 Z r. 3 ϑ z; τ = ϑz; τ. 4 The function c 1, c ϑ c1,c is continuous on C Q C Q. 5 Let c 1, c 3 C Q, c S Q and z, τ Dc. Set ct = c + tc 3. Then ct C Q for all t 0, and lim t 0 ϑ c1,ct z; τ = ϑ c1,c z; τ. 6 ϑz; τ + 1 = ϑz + 1 A 1 A ; τ with A = A A rr T Z r, the vector of diagonal elements of A. In particular, ϑz; τ + = ϑz; τ and ϑz; τ + 1 = ϑz; τ if the matrix A is even. 7 Let D c := { z, τ Dc z τ, 1 τ Dc } = {aτ + b, τ τ H, a, b R r, Bc, a Z, Bc, b Z}. If z, τ D c 1 D c then z ϑ τ ; 1 i = iτ r/ τ det A p A 1 Z r /Z r e πiqz+pτ/τ ϑz + pτ; τ. Proof: 1 follows from the corresponding relations for ρ c1,c. The identity ϑz +µ; τ = ϑz; τ for µ A 1 Z r is easy, and we find ϑz +λτ; τ = e πiqλτ πibz,λ ϑz; τ for λ Z r when we replace n by n + λ in the definition.

37 .3 Properties of the ϑ-functions 33 For 3, replace n by n in the definition and use that E and sgn are odd functions. 4 We show that c 1 ϑ c1,c is continuous on C Q. The result then follows from 1. Using the decomposition of ρ as the sum of.7,.8 and.9 we see that it s sufficient to prove that c ν a+z r sgn Bc, ν Bc, ν β y e πiqντ+πibν,b,.15 Qc and c 1 ν a+z r { } sgn Bc 1, ν sgn Bc, ν e πiqντ+πibν,b.16 are continuous on C Q. Using Lemma.5 and.1 we see Bc, sgn Bc, ν ν β y e πiqντ+πibν,b e π Qν Bc,ν y Qc Qc e πλc,c 0 Qc 0 νy Since c λ c,c0 is continuous and λ c,c0 > 0 for all c C Q, we can find an neighbourhood N c of c such that λ c,c0 ɛ > 0 for all c N c. Hence on N c we find Bc, sgn Bc, ν ν β y e πiqντ+πibν,b e πɛqc 0 νy. Qc The series ν a+z r e πɛqc0 νy converges, and so the series in.15 converges uniformly for c in N c. Hence the function in.15 is continuous on N c. Since this holds for all c C Q, the function in.15 is continuous on C Q. In.13 we have seen that { } sgn Bc 1, ν sgn Bc, ν e πiqντ+πibν,b e πq + νy. The function Q + restricted to the sphere S = {ν R r ν = 1} assumes its absolute minimum λc 1 > 0. Hence Q + ν λc 1 ν S, and so Q + ν λc 1 ν ν R r.

38 34 Chapter. Indefinite ϑ-functions Since c 1 λc 1 is continuous and λc 1 > 0 for all c 1 C Q, we can find an neighbourhood N c1 of c 1 such that λc 1 ɛ > 0 for all c 1 N c1. Hence on N c1 we find { } sgn Bc 1, ν sgn Bc, ν e πiqντ+πibν,b e πɛ ν y. The series e πɛ ν y ν a+z r converges, and so the series in.16 converges uniformly for c 1 in N c1. Hence the function in.16 is continuous on N c1. Since this holds for all c 1 C Q, the function in.16 is continuous on C Q. 5 Note that ϑ c1,ct = ϑ c1,c3 + ϑ c3,ct. We can therefore assume c 3 to be equal to c 1. We have Qct = Qc + tc 1 = tbc 1, c + t Qc 1 < 0 and Bc 1, ct = Bc 1, c + tqc 1 < 0 for all t 0,, since Bc 1, c < 0 and Qc 1 < 0. Hence ct C Q for all t 0,. Using ϑ c1,ct = ϑ c1,c + ϑ c,ct and the decomposition of ρ as the sum of.7,.8 and.9 we see that it s sufficient to prove that and lim t 0 lim t 0 ν a+z r { sgn ν a+z r sgn It is easy to see that sgn Bc, ν } Bc, ν sgn Bct, ν e πiqντ+πibν,b = 0,.17 Bct, ν Bct, ν β y e πiqντ+πibν,b = Qct sgn sgn Bct, ν Bc 1, ν sgn Bc, ν for all ν a+z r and t 0, Both sides can take on the values 0,1 and. If the right hand side is 0, then sgnbc 1, ν = sgnbc, ν, so sgnbc, ν = sgnbct, ν. Hence the left hand side is also 0, and the equation holds. If the right hand side is 1, then either Bc 1, ν or Bc, ν is zero. If Bc, ν = 0 we get that the left hand side equals the right hand side. If Bc 1, ν = 0 the left hand side equals 0. Hence { } sgn Bc, ν sgn Bct, ν e πiqντ+πibν,b { } sgn Bc 1, ν sgn Bc, ν e πiqντ+πibν,b, for all ν a + Z r and t 0,. In the proof of Proposition.4 Case we have seen that.11 converges absolutely, i.e. { } sgn Bc 1, ν sgn Bc, ν e πiqντ+πibν,b ν a+z r

39 .3 Properties of the ϑ-functions 35 converges. Hence ν a+z r { } sgn Bc, ν sgn Bct, ν e πiqντ+πibν,b converges uniformly for t 0,. Using this we find lim t 0 ν a+z r = { } sgn Bc, ν sgn Bct, ν e πiqντ+πibν,b { lim t 0 ν a+z r sgn Bc, ν This proves.17. We will now prove.18: Using.1, we see that } sgn Bct, ν e πiqντ+πibν,b = 0. Bct, sgn Bct, ν ν β y e πiqντ+πibν,b e π Qν Bct,ν y Qct. Qct We write a + Z r as the union of P 1, P and P 3, with P 1 := {ν a + Z r sgnbc, ν = sgnbc 1, ν} P := {ν a + Z r Bc 1, νbc 1, c Bc 1, ν Qc 1 Bc, ν 0} P 3 := {ν a + Z r sgnbc, ν = sgnbc 1, c Bc 1, ν Qc 1 Bc, ν} Note that Bc, ν 0 for all ν a + Z r, which is guaranteed by the fact that z, τ Dc. On P 1 we use e π Qν Bct,ν y Qct e πqνy, for all t 0,. We have seen in the proof of Proposition.4 Case that the series in.11 converges absolutely. Hence the series converges. On P we have for all t 0,, which we get from ν P 1 e πqνy Bct, ν Qct Bc 1, ν Qc 1 Bc, ν + Bc, νbc 1, ν Bc 1, c Bc 1, ν t 0 Qc 1

40 36 Chapter. Indefinite ϑ-functions for all t 0,. Hence we find e π Qν Bct,ν Qct y π e Qν Bc 1,ν Qc 1 y, for all t 0,. Using Lemma.5 we see that the series converges. On P 3 we use Bct, ν Qct ν P e π Qν Bc 1,ν y Qc 1 Bc, ν Bc 1, c Bc 1, c Bc 1, ν Qc 1 Bc, ν,.19 for all t 0,, which we get from the inequality Bc, ν + Bc 1, ν + Qc 1 Bc 1, c Bc, ν t 0. Note that.19 holds also on P 1 and P, but we use it only on P 3. Using it we find for all t 0,, with e π Qν Bct,ν Qct y e π Qν, Qν := Qν Bc, ν Bc 1, c Bc 1, c Bc 1, ν Qc 1 Bc, ν..0 Write ν = ν c1 c 1 + ν c c + ν, with ν such that Bc 1, ν = Bc, ν = 0. We see so Hence Bc 1, ν = Qc 1 ν c1 + Bc 1, c ν c Bc, ν = Bc 1, c ν c1, 1 ν c1 = Bc 1, c Bc, ν 1 ν c = Bc 1, c Bc 1, ν Qc 1 Bc 1, c Bc, ν. Qν = Qν + Qc 1 νc 1 + Bc 1, c ν c1 ν c = Qν + Bc, ν Bc 1, c Bc 1, c Bc 1, ν Qc 1 Bc, ν

41 .3 Properties of the ϑ-functions 37 and Qν = Qν Bc, ν Bc 1, c Bc 1, c Bc 1, ν Qc 1 Bc, ν. The quadratic form Q has type r 1, 1: Q has type 1, 1 on c 1, c R and type r, 0 on c 1, c R. On c 1, c R we have Q = Q and on c 1, c R we have Q = Q. Hence Q has type 1, 1 on c 1, c R and type r, 0 on c 1, c R. Set c 1 = Bc1,c Qc 1 c 1 c and c = c then Q c 1 = Q Bc1, c Qc 1 c Bc1, c 1 c = Q Qc 1 c 1 c = Bc 1, c + Bc 1, c 4Qc 1 Qc 1 Q c = Q c = Qc = 0 = Bc 1, c 4Qc 1 < 0 B c 1, c = Q c 1 c + Q c 1 + Q c Bc1, c = Q Qc 1 c 1 + Bc 1, c 4Qc 1 = Bc 1, c Qc 1 < 0. If we choose C Q such that c 1 C Q then we see that c S Q. Using.0 we see so Bx, y = Qx + y Qx Qy = Bx, y Bc, xbc 1, y + Bc 1, xbc, y Bc 1, c Qc Bc 1, c Bc, xbc, y, Bc 1, ν = Bc 1, ν Bc, ν = Bc, ν. Since c 1 and c are linear combinations of c 1 and c, we have for all ν R r. B c 1, ν = B c 1, ν B c, ν = B c, ν

42 38 Chapter. Indefinite ϑ-functions We rewrite the set P 3 : P 3 = {ν a + Z r sgnbc, ν = sgnbc 1, c Bc 1, ν Qc 1 Bc, ν} = {ν a + Z r Bc1, c sgnb c, ν = sgn Qc 1 Bc 1, ν Bc, ν } = {ν a + Z r sgnb c, ν = sgnb c 1, ν} = {ν a + Z r sgn B c, ν = sgn B c 1, ν}. In the proof of Proposition.4 Case,.11, we have seen that { } πi Qντ+πi Bν,b sgn B c1, ν sgn B c, ν e ν a+z r converges absolutely, i.e. the series ν P 3 e π Qνy converges. On all three sets P 1, P and P 3, we have found a suitable majorant, independent of t 0,. Combining these results, we see that the series ν a+z r sgn converges uniformly for t 0,. Hence We have and lim t 0 ν a+z r sgn = ν a+z r lim Bct, ν Bct, ν β y e πiqντ+πibν,b Qct Bct, ν Bct, ν β Qct sgn Bct, ν β t 0 y e πiqντ+πibν,b Bct, ν y Qct Bct, ν lim y = t 0 Qct lim βx = 0. x e πiqντ+πibν,b. Hence we get equation Since ρa; τ depends only on Imτ, we have ρa; τ + 1 = ρa; τ. Hence ϑz; τ + 1 = n Z r ρn + a; τe πiqn e πiqnτ+πibn,z,.1