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Research in the Mathematical Sciences
, Volume 8 (1) – Feb 22, 2021

/lp/springer-journals/indefinite-zeta-functions-pPDPg8acOz

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- Springer Journals
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- Copyright © The Author(s) 2021
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- 2197-9847
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- 10.1007/s40687-021-00252-9
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gene.kopp@bristol.ac.uk School of Mathematics, We deﬁne generalised zeta functions associated with indeﬁnite quadratic forms of University of Bristol, Bristol, UK signature (g − 1, 1)—and more generally, to complex symmetric matrices whose Full list of author information is available at the end of the article imaginary part has signature (g − 1, 1)—and we investigate their properties. These indeﬁnite zeta functions are deﬁned as Mellin transforms of indeﬁnite theta functions in the sense of Zwegers, which are in turn generalised to the Siegel modular setting. We prove an analytic continuation and functional equation for indeﬁnite zeta functions. We also show that indeﬁnite zeta functions in dimension 2 specialise to diﬀerences of ray class zeta functions of real quadratic ﬁelds, whose leading Taylor coeﬃcients at s = 0 are predicted to be logarithms of algebraic units by the Stark conjectures. Keywords: Indeﬁnite quadratic form, Indeﬁnite theta function, Siegel modular form, Epstein zeta function, Real quadratic ﬁeld, Stark conjectures 1 Introduction The Dedekind zeta functions of imaginary quadratic ﬁelds are specialisations of real ana- lytic Eisenstein series. For example, for the Gaussian ﬁeld K = Q(i)and Re(s) > 1, 1 1 1 ζ (s) ζ (s):= = = E(i, s), (1.1) Q(i) s 2 2 s N(a) 4 (m + n ) 2 a≤Z[i] (m,n)∈Z a=0 (m,n)=(0,0) where E(τ,s) is the real analytic Eisenstein series given for Im(τ) > 0and Re(s) > 1by 1 Im(τ) E(τ,s):= . (1.2) 2s |mτ + n| (m,n)∈Z gcd(m,n)=1 Placing the discrete set of Dedekind zeta functions into the continuous family of real analytic Eisenstein series allows us to prove many interesting properties of Dedekind zeta functions—for instance, the ﬁrst Kronecker limit formula is seen to relate ζ (0) to the value of the Dedekind eta function η(τ) at a CM point. In this paper, we ﬁnd a new continuous family of functions, called indeﬁnite zeta func- tions, in which ray class zeta functions of real quadratic ﬁelds sit as a discrete subset. Moreover, we construct indeﬁnite zeta functions attached to quadratic forms of signature (g − 1, 1). In the case g = 2, norm forms of quadratic number ﬁelds give the specialisation of indeﬁnite zeta functions to ray class zeta functions of real quadratic ﬁelds. © The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 0123456789().,–: volV 17 Page 2 of 34 G. S. Kopp Res Math Sci (2021) 8:17 Indeﬁnite zeta functions have analytic continuation with a functional equation about the line s = . This is in contrast to many zeta functions deﬁned by a sum over a cone—such as multiple zeta functions and Shintani zeta functions—which have analytic continuation but no functional equation. Shintani zeta functions are used to give decompositions of ray class zeta functions attached to totally real number ﬁelds, which are then used to evaluate those ray class zeta functions (and the closely related Hecke L-functions) at non-positive integers [21] (see also Neukirch’s treatment in [17], Chapter VII §9). Our specialisation of indeﬁnite zeta functions to ray class zeta functions of real quadratic ﬁelds is an alternative to Shintani decomposition that gives a diﬀerent interpolation between zeta functions attached to real quadratic number ﬁelds. Indeﬁnite zeta functions also diﬀer from archetypical zeta functions in that they are not (generally) expressed as Dirichlet series. Indeﬁnite zeta functions are Mellin transforms of indeﬁnite theta functions. Indeﬁnite theta functions were ﬁrst described by Marie-France Vignéras, who constructed modular indeﬁnite theta series with terms weighted by a weight function satisfying a particular diﬀerential equation [22,23]. Sander Zwegers rediscovered indeﬁnite theta functions and used them to construct harmonic weak Maass forms whose holomorphic parts are essen- tially the mock theta functions of Ramanujan [27]. Zwegers’s work triggered an explosion of interest in mock modular forms, with applications to partition identities [4], quantum modular forms and false theta functions [8], period integrals of the j-invariant [6], spo- radic groups [7], and quantum black holes [5]. Readers looking for additional exposition on these topics may also be interested in the book [3] (especially section 8.2) and lecture notes [20,26]. The indeﬁnite theta functions in this paper are a generalisation of those introduced by Zwegers to the Siegel modular setting. Our generalised indeﬁnite zeta functions satisfy a modular transformation law with respect to the Siegel modular group Sp (Z). 1.1 Main theorems Given a positive deﬁnite quadratic form Q(x , ... x ) with real coeﬃcients, it is possible 1 g to associate a “deﬁnite zeta function” ζ (s), sometimes called the Epstein zeta function: ζ (s):= . (1.3) Q(n , ... ,n ) 1 g (n ,...,n )∈Z \{0} 1 g However, if Q is instead an indeﬁnite quadratic form, the series in Eq. (1.3) does not converge. One way to ﬁx this issue it to restrict the sum to a closed subcone C of the double cone of positivity {v ∈ R : Q(v) > 0}. This gives rise to a partial indeﬁnite zeta function ζ (s):= . (1.4) Q(n , ... ,n ) 1 g (n ,...,n )∈C∩Z 1 g However, unlike the Epstein zeta function, this partial zeta function does not satisfy a functional equation. Our family of completed indeﬁnite zeta functions do satisfy a functional equation, although they are not (generally) Dirichlet series. The completed indeﬁnite zeta func- c ,c 1 2 tion ζ (,s) is deﬁned in terms of the following parameters: p,q • a complex symmetric (not necessarily Hermitian) matrix = = iM + N,with M = Im() having signature (g − 1, 1); G. S. Kopp Res Math Sci (2021) 8:17 Page 3 of 34 17 • “characteristics” p, q ∈ R ; • “cone parameters” c ,c ∈ C satisfying the inequalities c Mc < 0; 1 2 j j • a complex variable s ∈ C. c ,c 1 2 Due to invariance properties, ζ (,s) remains well deﬁned with several of the param- p,q eters taken to be in quotient spaces: g g • the characteristics on a torus, p, q ∈ R /Z ; g −1 • the cone parameters in complex projective space, c ,c ∈ P (C). 1 2 The functional equation for the completed indeﬁnite zeta function is given by the following theorem. c ,c 1 2 Theorem 1.1 (Analytic continuation and functional equation) The function ζ (,s) a,b may be analytically continued to an entire function on C. It satisﬁes the functional equation g e(p q) c ,c c ,c −1 1 2 1 2 ζ , − s = ζ − ,s . (1.5) p,q −q,p det(−i) In the case of real cone parameters, the completed indeﬁnite zeta function has a series expansion that may be viewed as an analogue of the Dirichlet series expansion of a deﬁnite zeta function. It decomposes (up to -factors) as a sum of an incomplete indeﬁnite zeta c ,c j 1 2 function ζ (,s), which is a Dirichlet series, and correction terms ξ (,s) that depend p,q p,q only on the cone parameters c and c separately. 1 2 Theorem 1.2 (Series decomposition) For real cone parameters c ,c ∈ R ,and Re(s) > 1, 1 2 the completed indeﬁnite zeta function may be written as c ,c −s c ,c −(s+ ) c c 1 2 1 2 2 1 ζ (,s) = π (s)ζ (,s) − π s + ξ (,s) − ξ (,s) , (1.6) p,q p,q p,q p,q where M = Im(), c ,c −s 1 2 ζ (,s) = sgn(c Mn) − sgn(c Mn) e p n Q (n) , (1.7) −i p,q 1 2 n∈Z +q and −s 1 (c Mn) ξ (,s) = sgn(c Mn)e p n p,q 2 Q (c) ν∈Z +q 1 2Q (c)Q (n) M −i · F s, s + ,s + 1; . (1.8) 2 1 2 (c Mn) Here, for any complex symmetric matrix ,Q (v) = v v denotes the associated quadratic form; also, F denotes a hypergeometric function (see Eq. (6.1)). The sum- 2 1 mand in Eq. (1.7) should always be interpreted as 0 when sgn(c Mn) = sgn(c Mn); 1 2 whenever it is nonzero, Re(Q (n)) > 0, and the complex power is interpreted as −i −s Q (n) = exp −s log Q (n) where log is the principal branch of the logarithm ( ( )) −i −i with a branch cut along the negative real axis. 17 Page 4 of 34 G. S. Kopp Res Math Sci (2021) 8:17 c ,c 1 2 The series deﬁning the incomplete indeﬁnite zeta function ζ (,s) is a variant of the p,q partial indeﬁnite zeta function 1.4, which may be seen by writing it as ∗ ∗ c ,c −s −s 1 2 ζ (,s) = e p n Q (n) − e p n Q (n) , −i −i p,q + g − g n∈C ∩(Z +q) n∈C ∩(Z +q) (1.9) + g − g where C ={v ∈ R :sgn(c Mv) ≤ 0 ≤ sgn(c Mv)} and C ={v ∈ R :sgn(c Mv) ≤ 2 1 1 0 ≤ sgn(c Mv)} are subcones of the two components of the double cone of positivity of Q (v), and the notation means that points on the boundary of the cone are weighted by , except for n = 0, which is excluded. The indeﬁnite zeta function is deﬁned as a Mellin transform of an indeﬁnite theta function (literally, an indeﬁnite theta null with real characteristics, see Deﬁnition 5.1 and the deﬁnitions in Sect. 3). Indeﬁnite theta functions were introduced by Sander Zwegers in his PhD thesis [27]. The indeﬁnite theta functions introduced in this paper generalise Zwegers’s work to the Siegel modular setting. Our deﬁnition of indeﬁnite zeta functions is in part motivated by an application to the computation of Stark units over real quadratic ﬁelds, which will be covered more thoroughly in a companion paper [13]. In special cases, an important symmetry, which c c 1 2 we call P-stability,causesthe ξ and ξ terms in Eq. (1.6) to cancel, leaving a Dirichlet c ,c 1 2 series ζ (,s). In the 2-dimensional case (g = 2), this Dirichlet series is a diﬀerence of p,q two ray class zeta functions of an order in a real quadratic ﬁeld. Theorem 1.3 (Specialisation to a ray class zeta function) Let K be a real quadratic number ﬁeld, and let Cl denote the ray class group of O modulo c∞ ∞ (see Eq. (7.1)). For c∞ ∞ K 1 2 1 2 −1 each class A ∈ Cl and integral ideal b ∈ A , there exists a real symmetric matrix c∞ ∞ 1 2 M of signature (1, 1), along with c ,c ,q ∈ C ,suchthat 1 2 c ,c −s 1 2 (2πN(b)) (s)Z (s) = ζ (iM, s). (1.10) 0,q Here, Z (s) is the diﬀerenced ray class zeta function associated with A (see Deﬁnition 7.2). This paper is organised as follows. In Sect. 2, we review the theory of Riemann theta functions, which we extend to the indeﬁnite case in Sect. 3, generalising Chapter 2 of Zwegers’s PhD thesis [27]. In Sects. 4 and 5,wedeﬁnedeﬁniteandindeﬁnitezetafunctions, respectively, and prove their analytic continuations and functional equations; in particular, we prove Theorem 1.1. In Sect. 6, we prove a general series expansion for indeﬁnite zeta functions, which is Theorem 1.2. In Sect. 7, we prove that indeﬁnite zeta functions restrict to diﬀerences of ray class zeta functions of real quadratic ﬁelds, which is Theorem 1.3,and we work through an example computation of a Stark unit using indeﬁnite zeta functions. 1.2 Notation and conventions We list for reference the notational conventions used in this paper. • e(z):= exp(2πiz) is the complex exponential, and this notation is used for z ∈ C not necessarily real. • H :={τ ∈ C :Im(τ) > 0} is the complex upper half-plane. • Non-transposed vectors v ∈ C are always column vectors; the transpose v is a row vector. G. S. Kopp Res Math Sci (2021) 8:17 Page 5 of 34 17 •If is a g × g matrix, then is its transpose, and (when is invertible) is a −1 shorthand for • Q (v) denotes the quadratic form Q (v) = v v, where is a g × g matrix, and v is a g × 1 column vector. • f (c) := f (c ) − f (c ), where f is any function taking values in an additive group. 2 1 c=c v v 1 1 2 2 •If v = ∈ C and f is a function of C , we may write f (v) = f rather than v v 2 2 f . • Unless otherwise speciﬁed, the logarithm log(z) is the standard principal branch with log(1) = 0 and a branch cut along the negative real axis, and z means exp(a log(z)). • Throughout the paper, will be used to denote a g × g complex symmetric matrix. We will often need to express = iM + N where M, N are real symmetric matrices. Matrices denoted by M and N will always have real entries, even when we do not say so explicitly. 1.3 Applications and future work A paper in progress will prove a Kronecker limit formula for indeﬁnite zeta functions in dimension 2, which specialises to an analytic formula for Stark units [13]. This formula may be currently be found in the author’s PhD thesis [12]. While one application of indeﬁnite zeta functions (the new analytic formula for Stark units) is known, we are hopeful that others will be found. We formulate a few research questions to motivate future work. • Can one formulate a full modular transformation law for indeﬁnite theta functions c ,c 1 2 [f ](z, ) for some general class of test functions f ? • (How) can indeﬁnite theta functions of arbitrary signature, as introduced by Alexan- drov, Banerjee, Manschot, and Pioline [1], Nazaroglu [16], and Westerholt-Raum [25], be generalised to the Siegel modular setting? What do the Mellin transforms of indeﬁnite theta functions of arbitrary signature look like? (Note: Since this paper was posted, a preprint of Roehrig [19] has appeared that answers the ﬁrst question by providing a description of modular indeﬁnite Siegel theta series by means of a system of diﬀerential equations, in the manner of Vignéras.) • The symmetry property we call P-stability is not the only way an indeﬁnite theta function can degenerate to a holomorphic function of ; there is also the case when M = i, the quadratic form Q factors as a product of two linear forms, and the cone parameters are sent to the boundary of the cone of positivity. How do the associated indeﬁnite zeta functions degenerate in this case? • What can be learned by specialising indeﬁnite zeta functions at integer values of s besides s = 0and s = 1? 2 Deﬁnite theta functions In this expository section, we discuss some classical results on (deﬁnite) theta functions to provide context for the new results on indeﬁnite theta functions proved in Sect. 3.Most of the results in this section may be found in [14], [15], or [18]. The expert may skip most 17 Page 6 of 34 G. S. Kopp Res Math Sci (2021) 8:17 of this section but will need to refer to back Sect. 2.3 for conventions for square roots of determinants used in Sect. 3.2. Deﬁnite theta functions in arbitrary dimension were introduced by Riemann, building on Jacobi’s earlier work in one dimension. The work of many mathematicians, including Hecke, Siegel, Schoenberg, and Mumford, further developed the theory of theta functions (including their relationship to zeta functions) and contributed ideas and perspectives used in this exposition. The deﬁnite theta function—or Riemann theta function—of dimension (or genus) g is a function of an elliptic parameter z and a modular parameter . The elliptic parameter g g z lives in C , but may (almost) be treated as an element of a complex torus C / , which happens to be an abelian variety. The parameter is written as a complex g × g matrix and lives in the Siegel upper half-space H , whose deﬁnition imposes a condition on M = Im(). 2.1 Deﬁnitions and geometric context An abelian variety over a ﬁeld K is a connected projective algebraic group; it follows from this deﬁnition that the group law is abelian. (See [15] as a reference for all results mentioned in this discussion.) A principal polarisation on an abelian variety A is an isomorphism between A and the dual abelian variety A . Over K = C, every principally polarised g g g abelian variety of dimension g is a complex torus of the form A(C) = C /(Z + Z ), where is in the Siegel upper half-space (sometimes called the Siegel upper half-plane, g(g +1) although it is a complex manifold of dimension ). Deﬁnition 2.1 The Siegel upper half-space of genus g is deﬁned to be the following open subset of the space M (C) of symmetric g × g complex matrices. (0) H := H :={ ∈ M (C): = and Im() is positive deﬁnite}. (2.1) g g g When g = 1, we recover the usual upper half-plane H = H ={τ ∈ C :Im(τ) > 0}. Deﬁnition 2.2 The deﬁnite (Riemann) theta function is, for z ∈ C and ∈ H , (z; ):= e n n + n z . (2.2) n∈Z Deﬁnition 2.3 When g = 1, the deﬁnite theta function is called a Jacobi theta function and is denoted by ϑ(z, τ):= ([z]; [τ]) for z ∈ C and τ ∈ H. The complex structure on A(C) determines the algebraic structure on A over C; indeed, the map A → A(C) deﬁnes an equivalence of categories from the category of abelian varieties over C to the category of polarisable tori (see Theorem 2.9 in [15]). Concretely, theta functions realise the algebraic structure from the analytic. The functions (z + t; ) for representatives t ∈ C of 2-torsion points of A(C) may be used to deﬁne an explicit holomorphic embedding of A as an algebraic locus in complex projective space. These shifts t are called characteristics. More details may be found in Chapter VI of [14], in particular pages 104–108. The positive integer g is sometimes called the “genus” because the Jacobian Jac(C) of an algebraic curve of genus g is a principally polarised abelian variety of dimension g. Not all principally polarised abelian varieties are Jacobians of curves; the question of G. S. Kopp Res Math Sci (2021) 8:17 Page 7 of 34 17 characterising the locus of Jacobians of curves inside the moduli space of all principally polarised abelian varieties is known as the Schottky problem. 2.2 The modular parameter and the symplectic group action The Siegel upper half-space has a natural action of the real symplectic group. This group, and an important discrete subgroup, are deﬁned as follows. Deﬁnition 2.4 The real symplectic group is deﬁned as the set of 2g × 2g real matrices preserving a standard symplectic form. 0 −I 0 −I Sp (R):= G ∈ GL (R): G G = , (2.3) 2g 2g I 0 I 0 where I is the g × g identity matrix. The integer symplectic group is deﬁned by Sp (Z):= 2g Sp (R) ∩ GL (Z). 2g 2g The real symplectic group acts on the Siegel upper half-space by the fractional linear transformation action AB AB −1 · := (A + B)(C + D) for ∈ Sp (R). (2.4) CD CD We will show in Proposition 3.3 (speciﬁcally, by the case k = 0) that H is closed under this action. 2.3 A canonical square root On the Siegel upper half-space H ,det(−i) has a canonical square root. Lemma 2.5 Let ∈ H .Then 1 1 e x x dx = . (2.5) g 2 det(−i) x∈R Proof Equation (2.5) holds for diagonal and purely imaginary by reduction to the −πax 1 one-dimensional case e dx = . Consequently, Eq. (2.5) holds for any purely −∞ imaginary by a change of basis, using spectral decomposition. g(g +1) Consider the two sides of Eq. (2.5) as holomorphic functions in complex variables g(g +1) (the entries of ); they agree whenever those variables are real. Because they are holomorphic, it follows by analytic continuation that they agree everywhere. Deﬁnition 2.6 Lemma 2.5 provides a canonical square root of det(−i): −1 det(−i):= e x x dx . (2.6) g 2 x∈R Whenever we write “ det(−i)” for ∈ H , we will be referring to this square root. We will later need to use this square root to evaluate a shifted version of the integral that deﬁnes it. Corollary 2.7 Let ∈ H and c ∈ C .Then, 1 1 e (x + c) (x + c) dx = . (2.7) g 2 x∈R det(−i) 17 Page 8 of 34 G. S. Kopp Res Math Sci (2021) 8:17 Proof Fix . The left-hand side of Eq. (2.7) is constant for c ∈ R , by Lemma 2.5. Because the left-hand side is holomorphic in c,itisinfactconstantfor all c ∈ C . −1 Note that, if ∈ H , then is invertible and − ∈ H . The latter is true because g g 0 −I −1 − = · , where “·” is the fractional linear transformation action of Sp (R) 2g I 0 on H deﬁned by Eq. (2.4). The behaviour of our canonical square root under the modular transformation → −1 − is given by the following proposition. −1 Proposition 2.8 If ∈ H ,then det(−i) det(i ) = 1. Proof This follows from Deﬁnition 2.6 by plugging in = iI, because the function −1 given by → det(−i) det(i ) is continuous and takes values in {±1},and H is connected. 2.4 Transformation laws of deﬁnite theta functions Proposition 2.9 The deﬁnite theta function for z ∈ C and ∈ H satisﬁes the following g g transformation law with respect to the z variable, for a + b ∈ Z + Z : (z + a + b; ) = e − b b − b z (z; ). (2.8) Proof The proof is a straightforward calculation. It may be found (using slightly diﬀerent notation) as Theorem 4 on pages 8–9 of [18]. Theorem 2.10 The deﬁnite theta function for z ∈ C and ∈ H satisﬁes the following transformation laws with respect to the variable, where A ∈ GL (Z),B ∈ M (Z),B = B : g g (1) (z; A A) = (A z; ). (2) (z; + 2B) = (z; ). e z z −1 (3) (z; − ) = (z; ). −1 det(i ) Proof The proof of (1) and (2) is a straightforward calculation. A more powerful version of this theorem, combining (1)–(3) into a single transformation law, appears as Theorem A on pages 86–87 of [18]. To prove (3), we apply the Poisson summation formula directly to the theta series. The Fourier transforms of the terms are given as follows. e Q (n) + n z e −n ν dn = e Q (n) + n (z − ν) dn (2.9) −1 = e −Q −1(z − ν) e Q n + (z − ν) dn (2.10) e −Q −1(z − ν) = . (2.11) det(−i) G. S. Kopp Res Math Sci (2021) 8:17 Page 9 of 34 17 In the last line, we used Lemma 2.5 and Deﬁnition 2.6. Now, by the Poisson summation formula, e −Q −1(z − ν) (z; ) = (2.12) det(−i) ν∈Z e Q −1(z) −1 = e Q −1(ν) + ν z (2.13) det(−i) ν∈Z e Q (z) −1 −1 = e Q −1(ν) − ν z (sending ν →−ν) (2.14) det(−i) ν∈Z −1 e − z z 2 −1 −1 = − z, − . (2.15) det(−i) −1 If is replaced by − ,weobtain(3). As was mentioned, it is possible to combine all of the modular transformations into a single theorem describing the transformation of under the action of Sp (Z), 2g AB −1 · = (A + B)(C + D) . (2.16) CD This rule is already fairly complicated in dimension g = 1, where the transformation law involves Dedekind sums. The general case is done in Chapter III of [18], with the main theorems stated on pages 86–90. 2.5 Deﬁnite theta functions with characteristics There is another notation for theta functions, using “characteristics,” and it will be nec- essary to state the transformation laws using this notation as well. We replace z with z = p + q for real variables p, q ∈ R . The reader is cautioned that the literature on theta functions contains conﬂicting conventions, and some authors may use notation identical to this one to mean something slightly diﬀerent. Deﬁnition 2.11 Deﬁne the deﬁnite theta null with real characteristics p, q ∈ R , for ∈ H : ():= e q q + p q (p + q; ) . (2.17) p,q The transformation laws for () follow directly from those for (z; ). p,q Proposition 2.12 Let ∈ H and p, q ∈ R . The elliptic transformation law for the deﬁnite theta null with real characteristics is given by () = e a (q + b) (). (2.18) p+a,q+b p,q for a, b ∈ Z . Proposition 2.13 Let ∈ H and p, q ∈ R . The modular transformation laws for the deﬁnite theta null with real characteristics are given as follows, where A ∈ GL (Z), B ∈ M (Z),and B = B . (1) (A A) = (). p,q A p,Aq 17 Page 10 of 34 G. S. Kopp Res Math Sci (2021) 8:17 (2) ( + 2B) = e(−q Bq) (). p,q p+2Bq,q e p q ( ) −1 (3) − = (). p,q −q,p −1 det(i ) 3 Indeﬁnite theta functions If we allow Im() to be indeﬁnite, the series expansion in Eq. (2.2) no longer converges anywhere. We want to remedy this problem by inserting a variable coeﬃcient into each term of the sum. In Chapter 2 of his PhD thesis [27], Sander Zwegers found—in the case when is purely imaginary—a choice of coeﬃcients that preserves the transformation properties of the theta function. The results of this section generalise Zwegers’s work by replacing Zwegers’s indeﬁnite c ,c 1 2 c ,c 1 2 theta function ϑ (z, τ) by the indeﬁnite theta function [f ](z; ). The function has been generalised in the following ways. • Replacing τM for τ ∈ H and M ∈ M (R) real symmetric in of signature (g − 1, 1) by (1) g(g +1) ∈ H . (Adds − 1 real dimensions.) • Allowing c ,c to be complex. (Adds 2g − 2 real dimensions.) 1 2 • Allowing a test function f (u), which must be specialised to f (u) = 1 for all the modular transformation laws to hold. One motivation for introducing a test function f is to ﬁnd transformation laws for a more general class of test functions (e.g. polynomials). We may investigate the behaviour of test functions under modular transformations in future work. However, for the purpose of this paper, only the cases u → |u| will be relevant. 3.1 The Siegel intermediate half-space Deﬁnition 3.1 If M ∈ GL (R)and M = M ,the signature of M (or of the quadratic form Q ) is a pair (j, k), where j is the number of positive eigenvalues of M,and k is the number of negative eigenvalues (so j + k = g). Deﬁnition 3.2 For 0 ≤ k ≤ g,wedeﬁne the Siegel intermediate half-space of genus g and index k to be (k) H :={ ∈ M (C): = and Im() has signature (g − k, k)}. (3.1) g g (k) g g g We call a complex torus of the form T := C /(Z + Z ) for ∈ H , k = 0,g,an intermediate torus. Intermediate tori are usually not algebraic varieties. An example of intermediate tori in the literature are the intermediate Jacobians of Griﬃths [9–11]. Intermediate Jacobians generalise Jacobians of curves, which are abelian varieties, but those deﬁned by Griﬃths are usually not algebraic. (In contrast, the intermediate Jacobians deﬁned by Weil [24]are algebraic.) The symplectic group Sp (R)actsonthe setof g × g complex symmetric matrices by 2g the fractional linear transformation action, AB −1 · = (A + B)(C + D) . (3.2) CD G. S. Kopp Res Math Sci (2021) 8:17 Page 11 of 34 17 AB (k) (k) −1 Proposition 3.3 If ∈ H and ∈ Sp (R),then (A + B)(C + D) ∈ H . g g 2g CD (k) Moreover, the H are the open orbits of the Sp (R)-action on the set of g × gcomplex 2g symmetric matrices. IB A 0 0 −I Proof Trivial for .For , this is Sylvester’s law of inertia. For , −1 0 I 0 A I 0 −1 −1 −1 −1 1 −1 1 −1 −1 we have Im(− ) = (− + ) = (− + ) = Im() = 2i 2i −1 −1 −1 Im() ,soIm(− )and Im() have the same signature. These three types of matrices generate Sp (R). 2g (k) Now suppose , ∈ H . There exists a matrix A ∈ GL (R) such that A Im( )A = 1 2 g g 1 Im( ). For an appropriate choice of real symmetric B ∈ M (R), we thus have A A + 2 g 1 B = .Thatis, IB A 0 · · = , (3.3) 1 2 −1 0 I 0 A so and are in the same Sp (R)-orbit. 1 2 2g (k) Thus, the H are the open orbits of the Sp (R)-action on the set of g × g symmetric 2g complex matrices. 3.2 More canonical square roots From now on, we will focus on the case of index k = 1, which is signature (g − 1, 1). The construction of modular theta series for k ≥ 2 utilises higher-order error functions arising in string theory [1]. More research is needed to develop the higher index theory in the Siegel modular setting. Lemma 3.4 Let M be a real symmetric matrix of signature (g − 1, 1). On the region R ={z ∈ C : z Mz < 0}, there is a canonical choice of holomorphic function g(z) such that g(z) =−z Mz. Proof By Sylvester’s law of inertia, there is some P ∈ GL (R) (i.e. with det(P) > 0) such that M = P JP, where ⎛ ⎞ −100 ... 0 ⎜ ⎟ ⎜ ⎟ 01 0 ... 0 ⎜ ⎟ ⎜ ⎟ 00 1 ... 0 J := ⎜ ⎟ . (3.4) ⎜ ⎟ . . . . ⎜ ⎟ . . . . . . . . . ⎝ ⎠ 00 0 ... 1 g −1 2 2 The region S :={(z , ... ,z ) ∈ C : |z | + ···|z | < 1} is simply connected (as it 2 g 2 g g −1 2 2 is a solid ball) and does not intersect {(z , ... ,z ) ∈ C : z + ··· + z = 1} (because, 2 g 2 g 2 2 2 2 if it did, we’d have 1 = z + ··· + z ≤|z | + ···|z | < 1, a contradiction). Thus, 2 g there exists a unique continuous branch of the function 1 − z − ··· − z on S sending (0, ... , 0) → 1; this function is also holomorphic. For z ∈ R ,deﬁne 2 2 z z g (z):= z 1 − − ··· − . (3.5) J 1 z z 1 1 17 Page 12 of 34 G. S. Kopp Res Math Sci (2021) 8:17 This g is holomorphic and satisﬁes g (z) =−z Jz, g (αz) = αg (z), and g (e ) = 1 where J J J J J 1 ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ e := . (3.6) ⎜ ⎟ ⎝ ⎠ Conversely, if we have a continuous function g(z) satisfying g(z) =−z Jz and g(e ) = 1, it follows that g(αz) = αg(z), and thus g(z) = g (z). Now, we’d like to deﬁne g (z):= g (Pz), so that we have g (z) =−z Mz.Weneed M J M to check that this deﬁnition does not depend on the choice of P. Suppose M = P JP = −1 −1 −1 P JP for P ,P ∈ GL (R). So J = P P J P P ,thatis, P P ∈ O(g − 1, 1). 2 1 2 2 2 2 2 1 1 1 −1 −1 −1 But det(P P ) = det(P )det(P ) > 0, so, in fact, P P ∈ SO(g − 1, 1). 2 2 1 2 1 1 For any Q ∈ SO(g − 1, 1), we have g (Qe ) = 1. The function Q → g (Qe )must J 1 J 1 be either the constant 1 or the constant −1, because SO(g − 1, 1) is connected. Since g (e ) = 1(Q = I), we have g (Qe ) = 1 for all Q ∈ SO(1,g − 1). The function z → g (Qz) J 1 J 1 J −1 is a continuous square root of −z Jz sending e to 1, so g (Qz) = g (z). Taking Q = P P 1 J J 2 and replacing z with P z,wehave g (P z) = g (P z), as desired. 1 J 2 J 1 Deﬁnition 3.5 If M is a real symmetric matrix of signature (g − 1, 1), we will write −z Mz for the function g (z) in Lemma 3.4. We may also use similar notation, such 1 1 as − z Mz := −z Mz. Lemma 3.6 Suppose M is a real symmetric matrix of signature (g − 1, 1),and c ∈ C such −1 that c Mc < 0.Then, M +M Re − c Mc cc M is well deﬁned (that is, c Mc = 0) and positive deﬁnite. Proof Because M has signature (g − 1, 1) and c Mc < 0, 2 2 c Mc c Mc c Mc − c Mc = det < 0. (3.7) c Mcc Mc 2 −1 Thus, c Mc > c Mc > 0, so c Mc = 0and M + M Re − c Mc cc M is well deﬁned. Let −1 A : = M + M Re − c Mc cc M (3.8) −1 −1 = M − M c Mc cc M − M c Mc cc M. (3.9) On the (g − 1)-dimensional subspace W ={w ∈ C : c Mw = 0}, the sesquilinear form w → w Mw is positive deﬁnite; this follows from the fact that c Mc < 0, because M has signature (g − 1, 1). For nonzero w ∈ W , −1 −1 w Aw = w Mw − (c Mc) (w Mc)(c Mw) − (c Mc) (w Mc)(c Mw) (3.10) −1 −1 = w Mw − (c Mc) (0)(c Mw) − (c Mc) (w Mc)(0) (3.11) = w Mw > 0. (3.12) G. S. Kopp Res Math Sci (2021) 8:17 Page 13 of 34 17 Moreover, −1 −1 c Aw = c Mw − (c Mc) (c Mc)(c Mw) − (c Mc) (c Mc)(c Mw) (3.13) −1 = c Mw − c Mw − (c Mc) (c Mc)(0) (3.14) = 0, (3.15) and −1 −1 c Ac = c Mc − (c Mc) (c Mc)(c Mc) − (c Mc) (c Mc)(c Mc) (3.16) = c Mc − c Mc − c Mc (3.17) =−c Mc (3.18) =−c Mc > 0. (3.19) We have now shown that A is positive deﬁnite, as it is positive deﬁnite on subspaces W and Cc, and these subspaces span C and are perpendicular with respect to A. Lemma 3.7 Let = N + iM be an invertible complex symmetric g × g matrix. Consider c ∈ C such that c Mc < 0. The following identities hold: −1 −1 (1) M = Im − . −1 −1 (2) M − 2iM M = Im − . −1 c Im − c ( ) 2i (3) det −i − Mcc M = det(−i) . c Mc c Mc Proof Proof of (1): 1 1 −1 −1 −1 M = ( − ) = (I − ) (3.20) 2i 2i −1 −1 −1 = ( − ) = Im − . (3.21) 2i Proof of (2): −1 −1 M − 2iM M = M ( − 2iM) (3.22) −1 = Im − − ( − ) using (1) (3.23) −1 = Im − . (3.24) Proof of (3): Note that det(I + A) = 1 + Tr(A) for any rank 1 matrix A.Thus, 2i det −i − Mcc M c Mc 2i = det(−i)det I + (Mc)(Mc) (3.25) c Mc 2i = det(−i) 1 + Tr (Mc)(Mc) (3.26) c Mc 2i −1 = det(−i) 1 + c M Mc (3.27) c Mc −1 −c M − 2iM M c = det(−i) (3.28) −c Mc 17 Page 14 of 34 G. S. Kopp Res Math Sci (2021) 8:17 −1 −(c) Im − (c) = det(−i) , (3.29) −c Mc using (2) in the last step. (1) Deﬁnition 3.8 (Canonical square root) If ∈ H ,thenwedeﬁne det(−i) as follows. Write = N + iM for N, M ∈ M (R), and choose any c such that c Mc < 0. By Lemma −1 3.6,the matrix M + M Re − c Mc cc M is positive deﬁnite. We can also rewrite −1 1 2i this matrix as M + M Re − c Mc cc M = Im − Mcc M .Bypart(3) c Mc of Lemma 3.7, −1 2i −(c) Im − (c) det −i − Mcc M = det(−i) . (3.30) c Mc −c Mc We can thus deﬁne det(−i) as follows: 2i −c Mc det −i − Mcc M c Mc det(−i):= , (3.31) −1 −(c) Im − (c) where the square roots on the right-hand side are as deﬁned in Deﬁnitions 2.6 and 3.5. This deﬁnition does not depend on the choice of c, because {c ∈ C : c Mc < 0} is connected. 3.3 Deﬁnition of indeﬁnite theta functions Deﬁnition 3.9 For any complex number α and any entire test function f , deﬁne the incomplete Gaussian transform −πu E (α):= f (u)e du, (3.32) where the integral may be taken along any contour from 0 to α. In particular, for the constant functions 1(u) = 1, set α |α| 2 α 2 −πu −1/2 −π(α/|α|) t E(α):= E (α) = e du = t e dt. (3.33) 2|α| 0 0 When α is real, deﬁne E (α) for an arbitrary continuous test function f : −πu E (α):= f (u)e du. (3.34) Deﬁnition 3.10 Deﬁne the indeﬁnite theta function attached to the test function f to be ⎛ ⎞ c Im(n + z) 1 c ,c 1 2 ⎝ ⎠ [f ](z; ):= E e n n + n z , (3.35) n∈Z − c Im()c c=c (1) g g where ∈ H , z ∈ C , c ,c ∈ C , c Mc < 0, c Mc < 0, and f (ξ) is a continuous g 1 2 1 1 2 2 function of one variable satisfying the growth condition log f (ξ) = o |ξ | .Ifthe c are not both real, also assume that f is entire. c ,c c ,c 1 2 1 2 Also deﬁne the indeﬁnite theta function (z; ):= [1](z; ). G. S. Kopp Res Math Sci (2021) 8:17 Page 15 of 34 17 c ,c c ,c 1 2 1 2 The function (z; ) = [1](z; ) is the function we are most interested in, −1 because it will turn out to satisfy a symmetry in →− . We will also show that the c ,c r 1 2 functions [u →|u| ](z; ) are equal (up to a constant) for certain special values of the parameters. Before we can prove the transformation laws of our theta functions, we must show that the series deﬁning them converges. Proposition 3.11 The indeﬁnite theta series attached to f (Eq. (3.35)) converges absolutely g g and uniformly for z ∈ R + iK , where K is a compact subset of R (and for ﬁxed ,c ,c , 1 2 and f ). Proof Let M = Im . We may multiply c and c by any complex scalar without chang- 1 2 ing the terms of the series Eq. (3.35), so we assume without loss of generality that Re(c Mc ) < 0. 1 2 For λ ∈ [0, 1], deﬁne the vector c(λ) = (1 − λ)c + λc and the real symmetric matrix 1 2 −1 A(λ):= M + M Re − c(λ) Mc(λ) c(λ)c(λ) M. Note that c(λ) Mc(λ) = (1 − 2 2 λ) c Mc + 2λ(1 − λ)Re(c Mc ) + λ c Mc < 0 because each term is negative 1 1 1 2 2 2 (except when λ = 0 or 1, in which case one term is negative and the others are zero). By Lemma 3.6, A(λ) is well deﬁned and positive deﬁnite for each λ ∈ [0, 1]. Consider (x, λ) → x A(λ)x as a positive real-valued continuous function on the compact set that is the product of the unit ball {x x = 1} and the interval [0,1].Ithasaglobal minimum ε> 0. c(λ) (Mn+y) The parameterisation γ :[0, 1] → C, γ (λ):= , deﬁnes a contour from − c(λ) Mc(λ) c (Mn+y) c (Mn+y) 1 2 to , so that 1 1 − c Mc − c Mc 1 2 2 1 2 2 c (Mn + y) 2 −πu E = f (u)e du. (3.36) − c Mc c=c We give an upper bound for 2 1 −πγ (λ) max e e n n + n z λ∈[0,1] 2 −π −1 (c(λ) M(n+M y)) −1 1 −1 −1 πy M y −π n+M y M n+M y − c(λ) Mc(λ) ( ) ( ) = e max e e (3.37) λ∈[0,1] −1 −1 −1 πy M y −π n+M y A(λ) n+M y ( ) ( ) = e max e (3.38) λ∈[0,1] −1 −1 πy M y −πε n+M y ≤ e e , (3.39) where the vector norm is v := v v for v ∈ R .Thus, c (Mn + y) 1 E e n n + n z − c Mc c=c −1 −1 πy M y −πε n+M y ≤ f (u) e e du (3.40) −1 −πε n+M y ≤ p(n)e , (3.41) 17 Page 16 of 34 G. S. Kopp Res Math Sci (2021) 8:17 πε 2 −1 2 − n +M y ( ) where log p(n) = o n . Thus, the terms of the series are o e ,and so the series converges absolutely and uniformly for x ∈ R and y ∈ K . 3.4 Transformation laws of indeﬁnite theta functions We will now prove the elliptic and modular transformation laws for indeﬁnite theta (1) g g functions. In all of these results, we assume that z ∈ C , ∈ H , c ∈ C satisfying g j c Im()c ,and f is a function of one variable satisfying the conditions speciﬁed in j j Deﬁnition 3.10. Proposition 3.12 The indeﬁnite theta function attached to f satisﬁes the following trans- g g formation law with respect to the z variable, for a + b ∈ Z + Z : c ,c c ,c 1 2 1 2 [f ](z + a + b; ) = e − b b − b z [f ](z; ). (3.42) Proof By deﬁnition, c ,c 1 2 [f ](z + a + b; ) c Im(n + (z + a + b)) = E e Q (n) + n (z + a + b) . − c Im()c n∈Z c=c (3.43) Because a ∈ Z ,Im(a)iszeroand e(n a) = 1, so c ,c 1 2 [f ](z + a + b; ) c Im((n + b) + z) = E e Q (n) + n (z + b) (3.44) − c Im()c n∈Z c=c 1 c Im((n + b) + z) = e − b b E e Q (n + b) + n z − c Im()c n∈Z c=c (3.45) 1 c Im(n + z) = e − b b E e Q (n) + (n − b) z (3.46) − c Im()c n∈Z 2 c=c c ,c 1 2 = e − b b − b z [f ] (z; ). (3.47) The identity is proved. Proposition 3.13 The indeﬁnite theta function satisﬁes the following condition with respect to the c variable: c ,c c ,c c ,c 1 3 1 2 2 3 [f ](z; ) = [f ](z; ) + [f ](z; ). (3.48) Proof Add the series termwise. Theorem 3.14 The indeﬁnite theta function satisﬁes the following transformation laws with respect to the variable, where A ∈ GL (Z),B ∈ M (Z),B = B : g g c ,c Ac ,Ac − 1 2 1 2 (1) [f ](z; A A) = [f ](A z; ). G. S. Kopp Res Math Sci (2021) 8:17 Page 17 of 34 17 c ,c c ,c 1 2 1 2 (2) [f ](z; + 2B) = [f ](z; ). (3) In the case where f (u) = 1(u) = 1,wehave πiz z −1 −1 c ,c −1 − c ,− c 1 2 1 2 (z; − ) = (z; ). (3.49) −1 det(i ) Proof The proof of (1) is a direct calculation. c ,c 1 2 [f ](z; A A) ⎛ ⎞ c Im(A An + z) 1 ⎝ ⎠ = E e n A An + n z (3.50) n∈Z − c Im()c c=c ⎛ ⎞ c Im(A m + z) 1 −1 ⎝ ⎠ = E e m m + A m z (3.51) m∈Z − c Im()c c=c by the change of basis m = An,so c ,c 1 2 [f ](z; A A) ⎛ ⎞ (Ac) Im(m + A z) 1 ⎝ ⎠ = E e m m + m A z (3.52) 1 2 − c Im()c m∈Z c=c Ac ,Ac − 1 2 = [f ](A z; ). (3.53) The proof of (2) is also a direct calculation. c ,c 1 2 [f ](z; + 2B) ⎛ ⎞ c Im(( + 2B)n + z) 1 ⎝ ⎠ = E e n ( + 2B)n + n z (3.54) n∈Z − c Im()c c=c ⎛ ⎞ c (Im(()n + z)) + 2Im(B)n ⎝ ⎠ = E e Q (n) + n Bn + n z n∈Z − c Im()c c=c (3.55) ⎛ ⎞ c Im(()n + z) ⎝ ⎠ = E e Q (n) + n z (3.56) − c Im()c n∈Z c=c c ,c 1 2 = [f ](z; ); (3.57) where e n Bn = 1 because the n Bn are integers, and Im(B) = 0 because B isareal matrix. The proof of (3) is more complicated, and, like the proof of the analogous property for deﬁnite (Jacobi and Riemann) theta functions, uses Poisson summation. The argument that follows is a modiﬁcation of the argument that appears in the proof of Lemma 2.8 of Zwegers’s thesis [27]. We will ﬁnd a formula for the Fourier transform of the terms of our theta series. Most of the work is done in the calculation of the integral that follows. In this calculation, 17 Page 18 of 34 G. S. Kopp Res Math Sci (2021) 8:17 M = Im ,and z = x + iy for x, y ∈ C . The diﬀerential operator is a row vector with entries , and similarly for . ∂x ⎛ ⎞ ⎛ ⎞ c Mn + c y ⎜ ⎟ −1 ⎝ ⎠ E e Q n + z dn ⎝ ⎠ n∈R − c Mc c=c ⎛ ⎞ c Mn + c y −1 ⎝ ⎠ = E e Q n + z dn (3.58) n∈R − c Mc c=c ⎛ ⎞ ⎛ ⎞ c Mn + c y ⎜ ⎟ −1 −1 ⎝ ⎠ = E e Q n + z dn (3.59) ⎝ ⎠ n∈R − c Mc c=c ⎛ ⎞ ⎛ ⎛ ⎞ ⎞ c Mn + c y ⎜ ⎟ −1 −1 ⎝ ⎝ ⎠ ⎠ = − E e Q n + z dn (3.60) ⎝ n ⎠ n∈R − c Mc c=c −1 = k e c Im()n e (Q (n + a )) dn c M , −c Mc n∈R c=c (3.61) −2 −1 −1 g where k := ∈ C, a := z − M y ∈ C , and integration by parts was used − c Mc in Eq. (3.60). Continuing the calculation, ⎛ ⎞ ⎛ ⎞ c Mn + c y ⎜ ⎟ −1 ⎝ ⎠ E e Q n + z dn x ⎝ ⎠ n∈R − c Mc c=c −1 = k e Q 2i (n) + a n + a a dn c M (3.62) − Mcc M g 2 c Mc n∈R c=c −1 1 2i 1 (c) −1 = ke − a − Mcc M a + a a I c M , 2 c Mc 2 c=c (3.63) where −1 2i (c) I := e Q 2i n + − Mcc M a dn (3.64) − Mcc M c Mc c Mc n∈R = (3.65) 2i det −i − Mcc M c Mc by Lemma 2.5. We can check (by multiplication) that −1 2i 2i −1 −1 −1 − Mcc M = − Mcc M . (3.66) −1 c Mc c Mc − 2ic M Mc Thus, −1 2i 2i − − Mcc M = Mcc M. (3.67) −1 c Mc c Mc − 2ic M Mc G. S. Kopp Res Math Sci (2021) 8:17 Page 19 of 34 17 Now compute, using Lemma 3.7, −1 −1 −1 −1 Ma = M z − y = Im − z − y = Im − z − y (3.68) 1 1 −1 −1 −1 −1 −1 = − + z − (z − z) = − z + z (3.69) 2i 2i −1 = Im − z . (3.70) −1 −1 Also by Lemma 3.7, M − 2iM M = Im − ,and −1 −c Im − c 2i det −i − Mcc M = det(−i) √ . (3.71) c Mc −c Mc We have now shown that ⎛ ⎞ ⎛ ⎞ c Im n + z ( ) ⎜ ⎟ −1 ⎝ ⎠ E e Q n + z dn ⎝ ⎠ n∈R − c Mc c=c −2e (c Ma) −1 (c)Im(− )(c) −1 = c M (3.72) −1 det(−i) − (c)Im(− )(c) c=c −2e (c Ma) −1 (c)Im(− )(c) −1 = (c) Im( ) (3.73) −1 det(−i) − (c)Im(− )(c) c=c ⎛ ⎞ −1 −1 (c) Im(− )n + Im(− z) ⎝ ⎠ = E . (3.74) det(−i) −1 − (c)Im(− )(c) c=c Deﬁne the following function on C , ⎛ ⎞ c Im (n + z) −1 ⎝ ⎠ C(z): = E e Q n + z dn n∈R − c c c=c ⎛ ⎞ −1 1 (c) Im(− z) ⎝ ⎠ − E , (3.75) det(i) −1 − (c)Im(− )(c) c=c suppressing the dependence of C(z)on and c.Wehavejustshown that C(z) = 0, g −1 g so C(z + a) = C(z) for any a ∈ R . By inspection, C(z + b) = C(z) for any b ∈ R . It follow from both of these properties that C(z) is constant. Moreover, by inspection, C(−z) =−C(z); therefore, C(z) = 0. In other words, ⎛ ⎞ c Im (n + z) −1 ⎝ ⎠ E e Q n + z dn n∈R − c c c=c ⎛ ⎞ −1 1 (c) Im(− z) ⎝ ⎠ = E . (3.76) det(−i) −1 − (c)Im(− )(c) c=c 1 17 Page 20 of 34 G. S. Kopp Res Math Sci (2021) 8:17 c ,c 1 2 Now set g(z):= (z; ), which has Fourier coeﬃcients ⎛ ⎞ c Im (n + z) 1 ⎝ ⎠ c (g)(z) = E e n n + n z . (3.77) − c c c=c −1 By plugging in z − ν for z in Eq. (3.76) and multiplying both sides by e − (z − ν) (z − ν) , we obtain the following expression for the Fourier coeﬃcients of g: ⎛ ⎞ c Im (n + z) 1 ⎝ ⎠ c ( g) (z) = E e n n + n z e(−n ν)dn (3.78) 1 2 n∈R − c c c=c ⎛ ⎞ 1 −1 −1 −1 e − (z − ν) (z − ν) (c) Im(− ν − z) ⎝ ⎠ = E (3.79) det(−i) −1 − (c)Im(− )(c) c=c ⎛ ⎞ 1 −1 −1 −1 e − z z (c) Im(− (−ν) − z) ⎝ ⎠ = E (3.80) det(−i) −1 − (c)Im(− )(c) c=c −1 −1 · e ν (− )ν + (−ν) (− z) . (3.81) It follows by Poisson summation that c ,c 1 2 (z; ) = c (g) (z) (3.82) ν∈Z −1 e − z z 2 c ,c −1 −1 1 2 = − z; − . (3.83) det(−i) −1 We obtain (3) by replacing with − . 3.5 Indeﬁnite theta functions with characteristics Now we restate the transformation laws using “characteristics” notation, which will be used when we deﬁne indeﬁnite zeta functions in Sect. 5. Deﬁnition 3.15 Deﬁne the indeﬁnite theta null with characteristics p, q ∈ R : 2πi q q+p q c ,c c ,c 1 2 2 1 2 [f ]():= e [f ] (p + q; ) ; (3.84) p,q 2πi q q+p q c ,c c ,c 1 2 2 1 2 ():= e (p + q; ) . (3.85) p,q c ,c 1 2 The transformation laws for [f ]() follow from the transformation laws for p,q c ,c 1 2 [f ](z; ). Proposition 3.16 The elliptic transformation law for the indeﬁnite theta null with char- acteristics is: c ,c 1 2 c ,c 1 2 [f ]() = e(a (q + b)) [f ](). (3.86) p,q p+a,q+b Proposition 3.17 The modular transformation laws for the indeﬁnite theta null with characteristics are as follows. G. S. Kopp Res Math Sci (2021) 8:17 Page 21 of 34 17 c ,c Ac ,Ac 1 2 1 2 (1) [f ](A A) = [f ](). p,q − A p,Aq c ,c c ,c 1 2 1 2 (2) [f ]( + 2B) = e(−q Bq) [f ](). p,q p+2Bq,q −1 −1 c ,c e(p q) − c ,− c 1 2 −1 1 2 (3) (− ) = (). p,q −q,p −1 det(i ) 3.6 P-stable indeﬁnite theta functions We now introduce a special symmetry that may be enjoyed by the parameters (c ,c ,z, ), 1 2 which we call P-stability. In this section, c ,c will always be real vectors. 1 2 (1) g g Deﬁnition 3.18 Let P ∈ GL (Z)beﬁxed. Let z ∈ C , ∈ H , c ,c ∈ R satisfying g g 1 2 c Im()c < 0. The quadruple (c ,c ,z, ) is called P-stable if P P = , Pc = c ,and j 1 2 1 2 P z ≡ z mod Z . Remarkably, P-stable indeﬁnite theta functions attached to f (u) =|u| turn out to be independent of r (up to a constant factor). r+1 c ,c 2 1 2 π c ,c 1 2 Theorem 3.19 (P-Stability Theorem) Set (z; ):= [f ](z; ) when r+1 c ,c r 1 2 f (u) = |u| for Re(r) > −1.If (c ,c ,z, ) is P-stable, then (z; ) is independent 1 2 r of r. Proof Let M = Im()and y = Im(z). If α ∈ R and Re(r) > 1, then r −πu E (α) = |u| e du (3.87) |α| r −πu = sgn(α) u e du (3.88) |α| sgn(α) 2 r−1 −πu =− u d e (3.89) 2π |α| |α| sgn(α) 2 2 r−1 −πu −πu r−1 =− u e − e d u (3.90) u=0 2π |α| sgn(α) 2 2 r−1 −πα r−2 −πu =− |α| e − (r − 1) u e du (3.91) 2π 1 2 r−1 −πα = − sgn(α) |α| e + (r − 1)E (α) . (3.92) r−2 2π c Im(n+z) −1 c c c c 1 2 Let α = √ .Set A := M + M Re −Q (c) cc M, so that A and A are ( ) −Q (c) positive deﬁnite, as in the proof of Proposition 3.11.Thus, r/2 c ,c c ,c 1 2 1 2 (z; ) =− S + (z; ), (3.93) r−2 r+1 where r−1 2 c c c S = sgn α α exp −π α e n n + n z . (3.94) n n n c=c 2 n∈Z The c and c terms in this sum decay exponentially, because 1 2 c −1 exp −π α e n n + n z = exp −2πQ n + M y) . (3.95) 2 17 Page 22 of 34 G. S. Kopp Res Math Sci (2021) 8:17 Thus, the series may be split as a sum of two series: r−1 2 c c c 2 2 2 S = sgn α α exp −π α e n n + n z n n n n∈Z r−1 2 c c c 1 1 1 − sgn α α exp −π α e n n + n z . (3.96) n n n n∈Z Now we use the P-symmetry to show that these two series are, in fact, equal. Note that Im(P z) = Im(z) because P z ≡ z mod Z ,so (Pc ) Im(Pn + z) α (c ) = (3.97) Pn 2 −Q (Pc ) M 1 c Im(P Pn + P z) = (3.98) −Q (c ) P MP c Im(n + z) = (3.99) −Q (c ) M 1 = α (c ). (3.100) n 1 Moreover, 1 1 (Pn) (Pn) + (Pn) z = n (P P)n + n (P z) (3.101) 2 2 ≡ n n + n z mod Z . (3.102) Thus, we may substitute Pn for n in the ﬁrst series (involving c ) to obtain the second (involving c ). We’ve now shown the periodicity relation c ,c c ,c 1 2 1 2 (z; ) = (z; ). (3.103) r r−2 c ,c 1 2 Note that this identity provides an analytic continuation of (z, ) to the entire r- plane. To show that it is constant in r, we will show that it is bounded on vertical strips in the r-plane. As in the proof of Proposition 3.11,bound (x, λ) → x A(λ)x, considered as a positive real-valued continuous function on the product of the unit ball {x x = 1} and the interval [0, 1], from below by its global minimum ε> 0. Thus, ⎛ ⎞ c (Mn + y) 1 ⎝ ⎠ E e n n + n z − c Mc c=c c Im(n+z) − c Im()c −1 −1 2 2 Re(r) πy M y −πε n+M y ≤ |u| du e e (3.104) c Im(n+z) − c Im()c 2 1 −1 −πε n+M y ≤ p (n)e , (3.105) Re(r) c ,c 1 2 where p (n) is a polynomial independent of Im(r). Hence, (z, ) is bounded on Re(r) −1 −πε n+M y the line Re(r) = σ by p (n)e . It follows that it is bounded on any n∈Z c ,c 1 2 vertical strip. Along with periodicity, this implies that (z, ) as a function of r is bounded and entire, thus constant. G. S. Kopp Res Math Sci (2021) 8:17 Page 23 of 34 17 4 Deﬁnite zeta functions and real analytic Eisenstein series We will now consider deﬁnite zeta functions—the Mellin transforms of deﬁnite theta functions—in preparation for studying the Mellin transforms of indeﬁnite theta functions in the next section. In dimension 2, deﬁnite zeta functions specialise to real analytic Eisenstein series for the congruence subgroup (N) (which specialise further to ray class zeta functions of imaginary quadratic ideal classes). 4.1 Deﬁnition and Dirichlet series expansion We deﬁne the deﬁnite zeta function as a Mellin transform of the indeﬁnite theta null with real characteristics. (0) Deﬁnition 4.1 Let ∈ H and p, q ∈ R .The deﬁnite zeta function is dt s g (t)t if q ∈ / Z , p,q 0 t ζ (,s):= (4.1) p,q dt s g (t) − 1 t if q ∈ Z . p,q 0 t By direct calculation, ζ (,s) has a Dirichlet series expansion. p,q −s −s ζ (,s) = (2π) (s) e(p (n + q))Q (n + q) , (4.2) p,q −i n∈Z n=−q −s where Q (n + q) is deﬁned using the standard branch of the logarithm (with a branch −i cut on the negative real axis). 4.2 Specialisation to real analytic Eisenstein series Now, suppose g = 2, = iM for some real symmetric, positive deﬁnite matrix M, p = ,and q ∈ / Z . Then the deﬁnite zeta function may be written as follows. −s −s ζ (,s) = (2π) (s) Q (n + q) (4.3) 0,q M n∈Z −s −s = (2π) (s) Q (n) . (4.4) n∈Z +q 1Re(τ) Up to scaling, M is of the form M = for some τ ∈ H;scaling M by Im(τ) Re(τ) τ τ −s λ ∈ R simply scales ζ (,s)by λ ,soweassume M is of this form. Write p,q n 1 2 2 Q = n + 2Re τn n + τ τn (4.5) M 1 2 1 2 n 2Im(τ) | | = n + n τ . (4.6) 1 2 2Im(τ) Thus, −s s −2s ζ (,s) = π (s)Im(τ) |n τ + n | . (4.7) 0,q 1 2 ∈Z +q 2 17 Page 24 of 34 G. S. Kopp Res Math Sci (2021) 8:17 If q ∈ Q and the gcd of the denominators of the entries of q is N, this is essen- tially an Eisenstein series of associated with (N). Choose k, ∈ Z such that q ≡ k/N (mod 1) and gcd(k, ) = 1. Then, we have /N −2s −s s ζ (,s) = (πN) (s)Im(τ) cτ + d . (4.8) 0,q c≡k (mod N) d≡ (mod N) The Eisenstein series associated with the cusp ∞ of (N)is ∞ s E (τ,s) = Im(γ · τ) (4.9) (N) γ ∈ (N)\ (N) −2s = Im(τ) cτ + d (4.10) c≡0 (mod N) d≡1 (mod N) gcd(c,d)=1 −s (1 − p ) p|N −2s = Im(τ) cτ + d . (4.11) ζ (s) c≡0 (mod N) d≡1 (mod N) Here, (N) is the stabiliser of ∞ under the fractional linear transformation action; that 1 n is, (N) = ± : n ∈ Z . uv Choose u, v ∈ Z such that det = 1. We have s −2s ζ (s) uτ + v uτ + v uτ + v E ,s = Im c + d (4.12) (N) −s 1 (1 − p ) kτ + kτ + kτ + p|N c≡0 mod N ( ) d≡1 mod N ( ) −2s = Im (τ) (cu + dk)τ + (cv + d) (4.13) c≡0 (mod N) d≡1 mod N ( ) −2s = Im (τ) c τ + d . (4.14) c ≡k (mod N) d ≡ (mod N) Combining Eqs. (4.8)and (4.12), we see that −s (πN) (s)ζ (s) uτ + v ζ (,s) = E ,s . (4.15) 0,q (N) −s 1 (1 − p ) kτ + p|N 5 Indeﬁnite zeta functions: deﬁnition, analytic continuation, and functional equation We now turn our attention to the primary objects of interest, (completed) indeﬁnite zeta functions—the Mellin transforms of indeﬁnite theta functions. We will generally omit the word “completed” when discussing these functions. (1) g g As usual, let ∈ H , p, q ∈ R , c ,c ∈ C , c Mc < 0, c Mc < 0. g 1 2 1 1 2 2 We deﬁne the indeﬁnite zeta function using a Mellin transform of the indeﬁnite theta function with characteristics. G. S. Kopp Res Math Sci (2021) 8:17 Page 25 of 34 17 Deﬁnition 5.1 The (completed) indeﬁnite zeta function is dt c ,c c ,c s 1 2 1 2 ζ (,s):= (t)t . (5.1) p,q p,q c ,c 1 2 The terminology “zeta function” here should not be taken to mean that ζ (,s)has a p,q Dirichlet series—it (usually) doesn’t (although it does have an analogous series expansion involving hypergeometric functions, as we’ll see in Sect. 6). Rather, we think of it as a zeta function by analogy with the deﬁnite case, and (as we’ll see) because is sometimes specialises to certain classical zeta functions. By deﬁning the zeta function as a Mellin transform, we’ve set things up so that a proof of the functional equation Theorem 1.1 is a natural ﬁrst step. Analytic continuation and a functional equation will follow from Theorem 3.14 by standard techniques. Our analytic continuation also gives an expression that converges quickly everywhere and is therefore useful for numerical computation, unlike Eq. (5.1) or the series expansion in Sect. 6. c ,c 1 2 Theorem 1.1 The function ζ (,s) may be analytically continued to an entire function p,q on C. It satisﬁes the functional equation g e(p q) c ,c c ,c −1 1 2 1 2 ζ , − s = ζ − ,s . (5.2) p,q −q,p det(−i) Proof Fix r > 0, and split up the Mellin transform integral into two pieces, dt c ,c c ,c s 1 2 1 2 ζ ,s = (t)t (5.3) ( ) p,q p,q ∞ r dt dt c ,c s c ,c s 1 2 1 2 = (t)t + (t)t . (5.4) p,q p,q t t r 0 −1 Replacing t by t , and then using part (3) of Theorem 3.14, the second integral is r ∞ dt dt c ,c s c ,c −1 −s 1 2 1 2 (t)t = (t )t (5.5) p,q p,q −1 t t 0 r e(p q) dt tc ,tc 1 2 −1 −1 −s = (−(t ) )t (5.6) −q,p −1 t det(−it) e(p q) dt c ,c −1 −s 1 2 = (t(− ))t . (5.7) −q,p −1 det(−i) r c ,c 1 2 (Recall that scaling the c does not aﬀect the value of ().) Putting it all together, we j p,q have dt c ,c c ,c s 1 2 1 2 ζ (,s) = (t)t p,q p,q e(p q) dt c ,c 1 2 −1 −s + (t(− ))t . (5.8) −q,p −1 t det(−i) r As we showed in the proof of Proposition 3.11,the -functions in both integrals decay exponentially as t →∞, so the right-hand side converges for all s ∈ C. The right-hand c ,c 1 2 side is obviously analytic for all s ∈ C, so we’ve analytically continued ζ (,s)toan p,q entire function of s. Finally, we must prove the functional equation. If we plug − s for s 2 17 Page 26 of 34 G. S. Kopp Res Math Sci (2021) 8:17 in Eq. (5.8), factor out the coeﬃcient of the second term, and switch the order of the two terms, we obtain g e(p q) dt c ,c c ,c 1 2 −1 s 1 2 ζ , − s = (t(− ))t p,q −q,p −1 2 t det(−i) r e(−p q) dt c ,c −s 1 2 − (t)t . (5.9) p,q −1 det(i ) r c ,c c ,c 1 2 −1 1 2 Reusing Eq. (5.8)on ζ − ,s , and appealing to the fact that () = p,q −q,p c ,c 1 2 − (), we have −p,−q dt c ,c c ,c 1 2 −1 1 2 −1 s ζ − ,s = (t(− ))t −q,p −q,p −1 t e(−p q) dt c ,c −s 1 2 − (t)t . (5.10) p,q −1 det(i ) r The functional equation now follows from Eqs. (5.9)and (5.10). The formula for the analytic continuation is useful in itself. In particular, we have used this formula for computer calculations, as it may be used to compute the indeﬁnite zeta function to arbitrary precision in polynomial time. Corollary 5.2 The following expression is valid on the entire s-plane. dt c ,c c ,c s 1 2 1 2 ζ ,s = (t)t ( ) p,q p,q e(p q) dt c ,c 1 2 −1 −s + (t(− ))t . (5.11) −q,p −1 det(−i) r Proof This is Eq. (5.8). 6 Series expansion of indeﬁnite zeta function In this section, we give a series expansion for indeﬁnite zeta functions, under the assump- c ,c 1 2 tion that c and c are real. Speciﬁcally, we write ζ (,s) as a sum of three series, the ﬁrst 1 2 p,q of which is a Dirichlet series and the others of which involve hypergeometric functions. This expansion is related to the decomposition of a weak harmonic Maass form into its holomorphic “mock” piece and a non-holomorphic piece obtained from a “shadow” form in another weight. To proceed, we will need to introduce some special functions and review some of their properties. 6.1 Hypergeometric functions and modiﬁed beta functions Let a, b, c be complex numbers, c not a negative integer or zero. If z ∈ C with |z| < 1, the power series (a) (b) z n n F (a, b; c; z):= · (6.1) 2 1 (c) n! n=0 converges. Here we are using the Pochhammer symbol (w) := w(w + 1) ··· (w + n − 1). n G. S. Kopp Res Math Sci (2021) 8:17 Page 27 of 34 17 Proposition 6.1 There is an identity −b F (a, b; c; z) = (1 − z) F b, c − a; c; , (6.2) 2 1 2 1 z − 1 −b valid about z = 0 and using the principal branch for (1 − z) . Proof This is part of Theorem 2.2.5 of [2]. Using this identity, we extend the domain of deﬁnition of F (a, b; c; x) from the unit disc 2 1 {|z| < 1} to the union of the unit disc and a half-plane {|z| < 1}∪{Re(z) < }.We −b interpret (1 − z) = exp(−b log(1 − z)) with the logarithm having a branch cut along the negative real axis. At the boundary point z = 1, the hypergeometric series converges when Re(c) > Re(a + b), and its evaluation is a classical theorem of Gauss. Proposition 6.2 If Re(c) > Re(a + b),then (c)(c − a − b) F (a, b; c;1) = . (6.3) 2 1 (c − a)(c − b) Proof This is Theorem 2.2.2 of [2]. Of particular interest to us will be a special hypergeometric function which is a modiﬁed version of the beta function. Deﬁnition 6.3 Let x > 0and a, b ∈ C.The beta function is a−1 b−1 B(x; a, b):= t (1 − t) dt, (6.4) and the modiﬁed beta function is a−1 b−1 B(x; a, b):= t (1 + t) dt. (6.5) The following proposition enumerates some properties of the modiﬁed beta function. Proposition 6.4 Let x > 0, and let a, b be complex numbers with Re(a), Re(b) > 0 and Re(a + b) < 1.Then, (1) B(x; a, b) = B ; a, 1 − a − b , x + 1 (2) B(x; a, b) = x F (a, 1 − b; a + 1; −x), 2 1 1 (a)(1 − a − b) (3) B ; a, b = − B(x;1 − a − b, b),and x (1 − b) (a)(1 − a − b) (4) B(+∞; a, b) = B(1; a, 1 − a − b) = . (1 − b) Proof To prove (1), we use the substitution t = . 1−u a−1 b−1 B(x; a, b) = t (1 + t) dt (6.6) a−1 b−1 x+1 u u du = 1 + (6.7) 1 − u 1 − u (1 − u) x+1 a−1 −a−b = u (1 − u) du (6.8) 0 17 Page 28 of 34 G. S. Kopp Res Math Sci (2021) 8:17 = B ; a, 1 − a − b . (6.9) x + 1 To prove (2), expand G(x; a, b) as a power series in x (up to a non-integral power). a−1 b−1 B(x; a, b) = t (1 + t) dt (6.10) b − 1 n+a−1 = t dt (6.11) n=0 b − 1 1 n+a = x (6.12) n n + a n=0 (b − n) · (b − n + 1) ··· (b − 1) 1 n+a = · x (6.13) n! n + a n=0 n n (−1) (1 − b) · (2 − b) ··· (n − b) x = x · (6.14) n + a n! n=0 (a) (1 − b) (−x) n n = x · (6.15) a(a + 1) n! n=0 = x F (a, 1 − b; a + 1; −x). (6.16) 2 1 To prove(3),use thesubstitution t = . 1/x a−1 b−1 B ; a, b = t (1 + t) dt (6.17) b−1 1 du −a+1 = u 1 + − (6.18) u u −a−b b−1 = u (1 + u) du (6.19) = G(+∞, 1 − a − b, b) − G(x, 1 − a − b, b). (6.20) To complete the proof of (3), we need to prove (4). Note that it follows from (4) that (a)(1−a−b) B(+∞, 1−a−b, b) = . The ﬁrst equality of (4) follows from (1) with x →+∞; (1−b) we will now derive the second. By (2), B(x; a, b) = x F (a, 1 − b; a + 1; −x) (6.21) 2 1 = x F (1 − b, a; a + 1; −x) (6.22) 2 1 1 −x a −a = x · (1 − (−x)) F a, (a + 1) − (1 − b); a + 1; (6.23) 2 1 a (−x) − 1 1 x x = F a, a + b; a + 1; . (6.24) 2 1 a 1 + x x + 1 Proposition 6.1 wasusedinEq. (6.23). Sending x →+∞ and applying Proposition 6.2 yields the second equality of (4). G. S. Kopp Res Math Sci (2021) 8:17 Page 29 of 34 17 Lemma 6.5 Let λ, μ> 0,and Re(s) > 0.Then dt 1 1 πλ 1 1 s −1/2 −s E( λt)exp(−μt)t = π μ s + B ; , − s . (6.25) t 2 2 μ 2 2 Proof First of all, note that the left-hand side of Equation (6.25) converges: The integrand λt Re s− 1 −1/2 −πu is exp(−O(t)) as t →∞ and O t as t → 0. Write E( λt) = u e du. 2 0 μtv The left-hand side of Equation (6.25) may be rewritten, using the substitution u = in the inner integral, as ∞ ∞ λt dt 1 dt s −1/2 −(πu+μt) s E( λt)exp(−μt)t = u e t du (6.26) t 2 t 0 0 0 πλ −1/2 1 μtv μt dt −(μtv+μt) s = e t dv . (6.27) 2 π π t 0 0 The double integral is absolutely convergent (indeed, the integrand is nonnegative, and we already showed convergence), so we may swap the integrals. We compute πλ ∞ ∞ 1/2 μ dt 1 μ 1 dt s −1/2 −μt(v+1) s+ E( λt)exp(−μt)t = v e t dv t 2 π t 0 0 0 (6.28) πλ 1/2 μ 1 1 μ 1 −(s+ ) −1/2 = v s + (μ(v + 1)) dv 2 π 2 (6.29) πλ μ 1 1 1 −1/2 −s −1/2 −(s+ ) = π μ s + v (v + 1) dv 2 2 (6.30) 1 1 πλ 1 1 −1/2 −s = π μ s + B ; , − s . (6.31) 2 2 μ 2 2 This proves Equation (6.25). 2 2 Lemma 6.6 Let ν , ν ∈ R and μ ∈ C satisfying Re(μ) > −π max{ν , ν } if sgn(ν ) = 1 2 1 1 2 sgn(ν ) and Re(μ) > 0 otherwise. Then, dt 1/2 s E νt exp(−μt)t ν=ν −s = sgn(ν ) − sgn(ν ) (s)μ ( ) 2 1 sgn(ν ) 1 1 μ − s+ −2s − π s + |ν | F s, s + ,s + 1; − 2 2 1 2s 2 2 πν sgn(ν ) 1 1 μ 1 − s+ 2 −2s + π s + |ν | F s, s + ,s + 1; − . (6.32) 1 2 1 2s 2 2 πν Proof Initially, consider λ, μ> 0, as in Lemma 6.5.Wehave dt E λt exp(−μt)t 0 17 Page 30 of 34 G. S. Kopp Res Math Sci (2021) 8:17 1 1 1 πλ 1 1 − −s = π μ s + B ; , − s (6.33) 2 2 μ 2 2 1 1 1 ( )(s) μ − −s 2 = π μ s + − B ; s, 1 − s (6.34) 2 2 πλ (s + ) 1 1 1 μ −s − −s = (s)μ − π μ s + B ; s, 1 − s (6.35) 2 2 2 πλ 1 1 1 1 1 μ −s −(s+ ) −s = (s)μ − π s + λ F s, s + ,s + 1; − , (6.36) 2 1 2 2s 2 2 πλ using parts (2) and (3) of Proposition 6.4. Equation (6.32) follows for positive real μ.But 2 2 the integral on the left-hand side of Eq. (6.32) converges for Re(μ) > −π max{ν , ν } if 1 2 sgn(ν ) = sgn(ν )and Re(μ) > 0 otherwise, and both sides are analytic functions in μ on 1 2 this domain. Thus, Eq. (6.32) holds in general by analytic continuation. 6.2 The series expansion We are now ready to prove Theorem 1.2, which we ﬁrst restate here for convenience. Theorem 1.2 If c ,c ∈ R ,and Re(s) > 1, then the indeﬁnite zeta function may be 1 2 written as 1 1 c ,c −s c ,c −(s+ ) c c 1 2 1 2 2 1 ζ (,s) = π (s)ζ (,s) − π s + ξ (,s) − ξ (,s) ,(6.37) p,q p,q p,q p,q where M = Im(), c ,c −s 1 2 ζ (,s) = sgn(c Mn) − sgn(c Mn) e p n Q (n) , (6.38) −i p,q 1 2 n∈Z +q and −s 1 (c Mn) ξ (,s) = sgn(c Mn)e p n p,q 2 Q (c) ν∈Z +q 1 2Q (c)Q (n) M −i × F s, s + ,s + 1; . (6.39) 2 1 2 (c Mn) Proof Take the Mellin transform of the theta series term-by-term, and apply Lemma 6.6. Note that the series for ξ (,s) converges absolutely, so the series may be split up like p,q this. c ,c 1 2 The function ζ (,s) here is a Dirichlet series summed over a double cone, with any p,q lattice points on the boundary of the cone weighted by . The coeﬃcients of the terms are ±e p n , where the sign is determined by whether one is in the positive or negative part of the double cone. c c 1 2 Theorem 6.7 Suppose (c ,c ,p + q, ) is P-stable. Then, ξ (,s) = ξ (,s) and 1 2 p,q p,q c ,c c ,c 1 2 −s 1 2 ζ (,s) = π (s)ζ (,s). p,q p,q Proof The equality of the ξ (,s) follows by the substitution n → Pn and the deﬁnition p,q of P-stability. The equation c ,c −s c ,c 1 2 1 2 ζ (,s) = π (s)ζ (,s) (6.40) p,q p,q then follows from Theorem 1.2. G. S. Kopp Res Math Sci (2021) 8:17 Page 31 of 34 17 7 Zeta functions of ray ideal classes in real quadratic ﬁelds In this section, we will specialise indeﬁnite zeta functions to obtain certain zeta functions to obtain certain zeta functions attached to real quadratic ﬁelds. We deﬁne two Dirichlet series, ζ (s, A)and Z (s), attached to a ray ideal class A of the ring of integers of a number ﬁeld. Deﬁnition 7.1 (Ray class zeta function) Let K be any number ﬁeld and c an ideal of the maximal order O .Let S be a subset of the real places of K (i.e. the embeddings K → R). Let A be a ray ideal class modulo cS, that is, an element of the group {nonzero fractional ideals of O coprime to c} Cl := Cl (O ):= . cS cS K {aO : a ≡ 1 (mod c) and a is positive at each place in S} (7.1) Deﬁne the zeta function of A to be −s ζ (s, A):= N(a) . (7.2) a∈A This function has a simple pole at s = 1 with residue independent of A. The pole may be eliminated by considering the function Z (s), deﬁned as follows. Deﬁnition 7.2 (Diﬀerenced ray class zeta function) Let R be the element of Cl deﬁned cS by R :={aO : a ≡−1 (mod c) and a is positive at each place in S}. (7.3) Deﬁne the diﬀerenced zeta function of A to be Z (s):= ζ (s, A) − ζ (s, RA). (7.4) The function Z (s) is holomorphic at s = 1. Now, specialise to the case where K = Q( D) be a real quadratic ﬁeld of discriminant D.Let O be the maximal order of K,and let c be an ideal of O .Let A be a narrow K K ray ideal class modulo c, that is, an element of the group Cl (O ). We show, as c∞ ∞ K 1 2 promised in the introduction, that the indeﬁnite zeta function specialises to the L-series Z (s) attached to a ray class of an order in a real quadratic ﬁeld. −1 Theorem 1.3 For each A ∈ Cl and integral ideal b ∈ A , there exists a real c∞ ∞ 1 2 2 2 symmetric 2 × 2 matrix M, vectors c ,c ∈ R ,and q ∈ Q such that 1 2 −s c ,c 1 2 (2πN(b)) (s)Z (s) = ζ (iM, s). (7.5) 0,q Proof The diﬀerenced zeta function Z (s)is −s −s Z (s) = N(a) − N(a) . (7.6) a∈A a∈RA We have −s −s −s N(b) Z (s) = N(ba) − N(ba) (7.7) a∈A a∈RA −s −s = N(b) − N(b) . (7.8) b∈b b∈b (b)∈I (b)∈R up to units up to units 17 Page 32 of 34 G. S. Kopp Res Math Sci (2021) 8:17 Write bc = γ Z + γ Z. The norm form N(n γ + n γ ) = Q for some real 1 2 1 1 2 2 M symmetric matrix M with integer coeﬃcients. The signature of M is (1, 1), just like the norm form for K.Since b and c are relatively prime (meaning b + c = O ), there exists by the Chinese remainder theorem some b ∈ O such that b ≡ b (mod bc) if and only 0 K 0 if b ≡ 0 (mod b) and b ≡ 1 (mod c).Express b = p γ + p γ for rational numbers 0 1 1 2 2 p ,p ,and set p = . 1 2 Let ε be the fundamental unit of O ,and let ε (= ε for some k) be the smallest totally 0 K positive unit of O greater than 1 such that ε ≡ 1 (mod c). Choose any c ∈ R such that Q (c ) < 0. Let P be the matrix describing the linear 1 M 1 action of ε on b by multiplication, i.e. ε(β n) = β (Pn). Set c = Pc . 2 1 Thus, we have −s N(β) Z (s) = sgn(c Mn) − sgn(c Mn) Q (n). (7.9) A M 2 1 n∈Z +q Moreover, (c ,c ,p, )is P-stable. So, by Theorem 6.7,Eq. (7.9) may be rewritten as 1 2 c ,c −s 1 2 (2πN(b)) (s)Z (s) = ζ (iM, s), (7.10) 0,q completing the proof. 7.1 Example √ √ Let K = Q( 3), so O = Z[ 3], and let c = 5O . The ray class group Cl = Z/8Z. K K c∞ The fundamental unit ε = 2 + 3 is totally positive: εε = 1. It has order 3 modulo 5: ε = 26 + 15 3 ≡ 1 (mod 5). In this section, we use the analytic continuation Eq. (5.11) for indeﬁnite zeta functions to compute Z (0), where I is the principal ray class of Cl . c∞ By deﬁnition, Z = ζ (s, I) − ζ (s, R) where R ={aO : a ≡−1 mod c and ais positive at ∞ } (7.11) ( ) K 2 ={aO : a ≡ 1 mod c and ais negative at ∞ }. (7.12) ( ) K 2 Write I = I I and R = R R , where I and R are the following ray ideal classes + − + − ± ± in Cl : c∞ ∞ 1 2 I :={aO : a ≡ 1 mod c and ahas sign ± at ∞ and + at ∞ }, (7.13) ( ) ± K 1 2 R :={aO : a ≡ 1 mod c and ahas sign ± at∞ and − at ∞ }. (7.14) ( ) ± K 1 2 Thus, Z (s) = ζ (s, I ) + ζ (s, I ) − ζ (s, R ) − ζ (s, R ). The Galois automorphism (a + I + − + − 1 √ √ a 3) = (a − a 3) deﬁnes a norm-preserving bijection between I and R ,sothe 2 1 2 − + middle terms cancel and Z (s) = ζ (s, I ) − ζ (s, R ) = Z (s). (7.15) I + − I To the principal ray class I of Cl , we associate = iM where M = and + c∞ ∞ 1 2 0 −6 1/5 0 q = .Wemay choose c ∈ R arbitrarily so long as c Mc < 0; take c = . 1 1 1 0 1 G. S. Kopp Res Math Sci (2021) 8:17 Page 33 of 34 17 The left action of ε on Z + 3Z is given by the matrix P = .ByTheorem 1.3, −s c ,P c 1 1 (2π) (s)Z (s) = ζ (iM, s). (7.16) + 0,q Taking a limit as s → 0, and using Eqs. (7.15), (7.16) becomes c ,P c 1 1 Z (0) = Z (s) = ζ (iM, 0). (7.17) I I 0,q For the purpose of making the numerical computation more eﬃcient, we split up the right-hand side as 2 2 3 c ,Pc Pc ,P c P c ,P c 1 1 1 1 1 1 Z (0) = ζ (iM, 0) + ζ (iM, 0) + ζ (iM, 0) (7.18) I 0,q 0,q 0,q c ,Pc c ,Pc c ,Pc 1 1 1 1 1 1 = ζ (iM, 0) + ζ (iM, 0) + ζ (iM, 0), (7.19) 0,q 0,q 0,q 0 1 2 1 2 2 1 1 1 where q = q = , q = q = ,and q = q = are obtained from the 0 1 2 5 5 5 0 1 4 0 1 2 residues of ε , ε , ε modulo 5. Using Eq. (5.11), we computed Z (0) to 100 decimal digits. The decimal begins Z (0) = 1.35863065339220816259511308230 ... . (7.20) The conjectural Stark unit is exp(Z (0)) = 3.89086171394307925533764395962 ....We used the RootApproximant[] function in Mathematica, which uses lattice basis reduc- tion internally, to ﬁnd a degree 16 integer polynomial having this number as a root, and we factored that polynomial over Q( 3). To 100 digits, exp(Z (0)) is equal to a root of the polynomial √ √ √ √ 8 7 6 5 4 x − (8 + 5 3)x + (53 + 30 3)x − (156 + 90 3)x + (225 + 130 3)x √ √ √ 3 2 − (156 + 90 3)x + (53 + 30 3)x − (8 + 5 3)x + 1. (7.21) We have veriﬁed that this root generates the expected class ﬁeld H . We have also computed Z (0) a diﬀerent way in PARI/GP, using its internal algorithms for computing Hecke L-values. We obtained the same numerical answer this way. Acknowledgements This research was partially supported by National Science Foundation (USA) Grants DMS-1401224, DMS-1701576, and DMS-1045119, and by the Heilbronn Institute for Mathematical Research (UK). This paper incorporates material from the author’s PhD thesis [12]. Thank you to Jeﬀrey C. Lagarias for advising my PhD and for many helpful conversations about the content of this paper. Thank you to Marcus Appleby, Jeﬀrey C. Lagarias, Kartik Prasanna, and two anonymous referees for helpful comments and corrections. Author details 1 2 School of Mathematics, University of Bristol, Bristol, UK, Heilbronn Institute for Mathematical Research, Bristol, UK. Received: 2 July 2020 Accepted: 27 January 2021 References 1. Alexandrov, S., Banerjee, S., Manschot, J., Pioline, B.: Indeﬁnite theta series and generalized error functions. Sel. Math. 24(5), 3927–3972 (2018) 2. Andrews, G.E., Askey, R., Roy, R.: Special Functions. Encyclopedia of Mathematics and its Applications, vol. 71. Cam- bridge University Press, Cambridge (1999) 3. Bringmann, K., Folsom, A., Ono, K., Larry, R.: Theory and Applications. American Mathematical Society, Harmonic Maass Forms and Mock Modular Forms (2017) 17 Page 34 of 34 G. S. Kopp Res Math Sci (2021) 8:17 4. Bringmann, K., Ono, K.: The f (q) mock theta function conjecture and partition ranks. Invent. Math. 165(2), 243–266 (2006) 5. Dabholkar, A., Murthy, S., Zagier, D.: Quantum black holes, wall crossing, and mock modular forms. Preprint arXiv:1208.4074, (2012) 6. 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PhD thesis, Universiteit Utrecht, Utrecht, The Netherlands, October (2002) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

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