Eichler integrals, period relations and Jacobi forms. (English) Zbl 1251.11032
The authors write themselves: “This paper contains three main results: The first one is to derive two ‘period relations’ and the second one is a complete characterization of ‘period functions’ of Jacobi forms in terms of period relations. These are done by introducing the concept of ‘Jacobi integrals’ on the Jacobi group. The last one is to show, for a given holomorphic function \(P(\tau,z)\) having two period relations, there exists a unique Jacobi integral, up to Jacobi forms, with the given \(P(\tau,z)\) as its period function.”
Let \(\mathcal{H}\) be the usual upper complex half plane and \(\tau \in \mathcal{H}\) a variable living in it. Let \(z \in \mathbb{C}^j\) for some \(j \in \mathbb{N}\). Recall the Jacobi group \(\Gamma^J\) for a given subgroup \(\Gamma \subset \mathrm{SL}(2,\mathbb{Z})\) of finite index: \[ \Gamma^J:= \Gamma \propto \mathbb{Z}^{2j} = \big\{ \big[M, (\lambda,\mu)\big] \;\; M \in \Gamma, \, \lambda, \mu \in \mathbb{Z}^j \big\}. \] The Jacobi group satisfies the group law \[ \big[M_1, (\lambda_1,\mu_1)\big] \, \big[M_2, (\lambda_2,\mu_2)\big] = \big[M_1M_2, (\lambda_1,\mu_1)M_2 + (\lambda_2,\mu_2)\big] \] and acts on \(\mathcal{H} \times \mathbb{C}^j\) by \[ \gamma(\tau,z) = \left( \frac{a\tau + b}{c\tau+d} , \frac{z+\lambda \tau + \mu}{c\tau + d} \right) \] for all \(\gamma = \big[ {a \; b \choose c \; d}, (\lambda,\mu)\big] \in \Gamma^J\).
Define furthermore \[ j_{k,\mathcal{M}^{(j)}} \big( \gamma,(\tau,z) \big) := (c\tau+d)^{-k} \, e^{2\pi i \, \mathrm{Tr}\big(\mathcal{M}^{(j)}(-z^t \frac{c}{c\tau+d} z + \lambda^t \frac{a\tau + b}{c\tau + d} \lambda + 2 \lambda^t \frac{z}{c\tau d} + \lambda^t \mu)\big)} \] for \(\gamma = \bigg[ {a \; b \choose c \; d}, (\lambda,\mu)\bigg] \in \Gamma^J\), \(k \in \frac{1}{2} \mathbb{Z}\) and \(\mathcal{M}^{(j)} \in M_{j \times j}\big(\frac{1}{2} \mathbb{Z}\big)\). The slash operator on a function \(f: \mathcal{H} \times \mathbb{C}^j \to \mathbb{C}\) is defined as \[ \big(f \big|_{\omega,k,\mathcal{M}^{(j)}} \gamma \big) (\tau,z) := \omega(\gamma) \, j_{k,\mathcal{M}^{(j)}} \big(\gamma,(\tau,z)\big) \, f\big(\gamma(\tau,z)\big) \] for all \(\gamma \in \Gamma^J\) where \(\omega(\gamma)\) is the multiplier system of weight \(k\) on \(\Gamma^J\) (satisfying the usual comparability relation).
A real analytic function \(f:\mathcal{H}\times \mathbb{C}^j \to \mathbb{C}\) is called a (real analytic) Jacobi function of weight \(k \in \frac{1}{2}\mathbb{Z}\) and index \(\mathcal{M}^{(j)}\) with multiplier system \(\omega\) if \[ f \big|_{\omega,k,\mathcal{M}^{(j)}} \gamma = f \qquad \text{for all } \gamma \in \Gamma^J \] holds and if \(f\) satisfies a certain growth condition.
The space of Jacobi forms is denoted by \(J_{\omega,k,\mathcal{M}^{(j)}}\big(\Gamma^J\big)\).
Let \(\mathcal{M}^{(j)}\) be a fixed element in \(M_{j\times j}(\mathbb{Z})\) and \(\mathcal{P}_{\mathcal{M}^{(j)}}\) be the space of functions \(g\) which are holomorphic in \(\mathcal{H} \times \mathbb{C}^j\) and satisfy the growth condition \[ \big| g(\tau,z)\big| \leq K \big( |\tau|^\rho + \text{im}(\tau)^{-\sigma} \big) \, e^{2 \pi \text{Tr}\left( \frac{\mathcal{M}^{(j)} \, y^ty}{\text{im}(\tau)}\right)} \] for some positive constants \(K\), \(\rho\) and \(\sigma\), and \(y= \text{im}(z)= \big(\text{im}(z_1),\ldots, \text{im}(z_j)\big)\). The set \(\mathcal{P}_{\mathcal{M}^{(j)}}\) is a complex vector space and is preserved under the slash operator for any real \(k\).
Definition. A real analytic function \(f: \mathcal{H}\times \mathbb{C}^j \to \mathbb{C}\), which is periodic with respect to both variables \(\tau\) and \(z\), is called a Jacobi integral of weight \(k \in \frac{1}{2} \mathbb{Z}\) and index \(\mathcal{M}^{(j)}\) with a multiplier system \(\omega\) and holomorphic period functions \(P_\gamma \in \mathcal{P}_{\mathcal{M}^{(j)}}\) on \(\Gamma^J\) if it satisfies the following conditions:
(1) \(\big(f \big|_{\omega,k,\mathcal{M}^{(j)}} \gamma \big) (\tau,z) = f(\tau,z) + P_\gamma(\tau,z)\) for all \(\gamma \in \Gamma^J\),
(2) \(f(\tau,z) = \mathcal{O} \big( e^{\alpha \text{im}(\tau)}\, e^{2 \pi \text{Tr}\left( \frac{\mathcal{M}^{(j)} \, y^ty}{\text{im}(\tau)}\right)} \big)\) as \(\text{im}(\tau) \to \infty\) and for some \(\alpha > 0\).
Then, the first two main results are the following
Theorem. Let \(r\) be any real number, \(m >0\) and \(\omega\) a multiplier system of weight \(r\) and index \(m\). Suppose that \(\big\{ \varphi_\gamma ;\; \gamma \in \mathrm{SL}(2,\mathbb{Z})^J\big\}\) is a parabolic cocycle of weight \(r\) and index \(m\) in \(\mathcal{P}_m\). then there is a unique meromorphic function, up to Jacobi forms, \(f: \mathcal{H} \times \mathbb{C}^1 \to \mathbb{C}\) such that \[ \big(f \big|_{\omega,k,\mathcal{M}^{(j)}} \gamma \big) (\tau,z) - f(\tau,z) = \varphi_\gamma(\tau,z) \] for all \(\gamma \in \mathrm{SL}(2,\mathbb{Z})^J\). This \(f\) is called a Jacobi integral of weight \(r\) and index \(m\) with period functions \(\big\{ \varphi_\gamma ;\; \gamma \in \mathrm{SL}(2,\mathbb{Z})^J\big\}\).
Put \(I={1 \; 0 \choose 0 \; 1}\), \(S={1 \; 1 \choose 0 \; 1}\) and \(T={0 \; -1 \choose 1 \; 0}\).
Theorem. Take a multiplier system \(\omega\) with \(\omega(-I)=1\). If a Jacobi integral \(f\) is even and periodic with respect to \(z\), i.e., \(P_{\big[-I,(0,0)\big]}(\tau,z) = P_{\big[I,(0,0)\big]}(\tau,z) = 0\), then the period functions \(P_{\big[T,(0,0)\big]}(\tau,z)\) and \(P_{\big[I,(1,0)\big]}(\tau,z)\) satisfy the following properties:
(1) \(P_{\big[T,(1,0)\big]} = P_{\big[T,(0,0)\big]}\),
(2) \(P_{\big[T,(0,0)\big]} + P_{\big[T,(0,0)\big]}\big|_{\omega,k,m}\big[T,(0,0)\big] = 0\),
(3) \(P_{\big[T,(0,0)\big]} + P_{\big[T,(0,0)\big]}\big|_{\omega,k,m}\big[ST,(0,0)\big] + P_{\big[T,(0,0)\big]}\big|_{\omega,k,m}\big[(ST)^2,(0,0)\big] = 0\),
(4) \(P_{\big[I,(1,0)\big]}\big|_{\omega,k,m}\big[T,(0,0)\big] = -P_{\big[T,(0,0)\big]} + P_{\big[T,(0,0)\big]}\big|_{\omega,k,m}\big[I,(0,-1)\big]\),
(5) \(P_{\big[I,(1,0)\big]} = P_{\big[T,(0,0)\big]} - P_{\big[T,(0,0)\big]}\big|_{\omega,k,m}\big[-I,(1,0)\big]\) and
(6) \(P_{\big[I,(1,0)\big]} + P_{\big[I,(1,0)\big]}\big|_{\omega,k,m}\big[-I,(1,0)\big] = 0\).
The third main result says basically: the set of all \(P \in \mathcal{P}_m\) satisfying the properties (2) and (3) above, i.e., \[ P + P\big|_{\omega,k,m}\big[T,(0,0)\big] = P + P\big|_{\omega,k,m}\big[ST,(0,0)\big] + P\big|_{\omega,k,m}\big[(ST)^2,(0,0)\big] = 0, \] is a generating set of all parabolic cocycles of \(\mathrm{SL}(2,\mathbb{Z})^J\).
Let \(\mathcal{H}\) be the usual upper complex half plane and \(\tau \in \mathcal{H}\) a variable living in it. Let \(z \in \mathbb{C}^j\) for some \(j \in \mathbb{N}\). Recall the Jacobi group \(\Gamma^J\) for a given subgroup \(\Gamma \subset \mathrm{SL}(2,\mathbb{Z})\) of finite index: \[ \Gamma^J:= \Gamma \propto \mathbb{Z}^{2j} = \big\{ \big[M, (\lambda,\mu)\big] \;\; M \in \Gamma, \, \lambda, \mu \in \mathbb{Z}^j \big\}. \] The Jacobi group satisfies the group law \[ \big[M_1, (\lambda_1,\mu_1)\big] \, \big[M_2, (\lambda_2,\mu_2)\big] = \big[M_1M_2, (\lambda_1,\mu_1)M_2 + (\lambda_2,\mu_2)\big] \] and acts on \(\mathcal{H} \times \mathbb{C}^j\) by \[ \gamma(\tau,z) = \left( \frac{a\tau + b}{c\tau+d} , \frac{z+\lambda \tau + \mu}{c\tau + d} \right) \] for all \(\gamma = \big[ {a \; b \choose c \; d}, (\lambda,\mu)\big] \in \Gamma^J\).
Define furthermore \[ j_{k,\mathcal{M}^{(j)}} \big( \gamma,(\tau,z) \big) := (c\tau+d)^{-k} \, e^{2\pi i \, \mathrm{Tr}\big(\mathcal{M}^{(j)}(-z^t \frac{c}{c\tau+d} z + \lambda^t \frac{a\tau + b}{c\tau + d} \lambda + 2 \lambda^t \frac{z}{c\tau d} + \lambda^t \mu)\big)} \] for \(\gamma = \bigg[ {a \; b \choose c \; d}, (\lambda,\mu)\bigg] \in \Gamma^J\), \(k \in \frac{1}{2} \mathbb{Z}\) and \(\mathcal{M}^{(j)} \in M_{j \times j}\big(\frac{1}{2} \mathbb{Z}\big)\). The slash operator on a function \(f: \mathcal{H} \times \mathbb{C}^j \to \mathbb{C}\) is defined as \[ \big(f \big|_{\omega,k,\mathcal{M}^{(j)}} \gamma \big) (\tau,z) := \omega(\gamma) \, j_{k,\mathcal{M}^{(j)}} \big(\gamma,(\tau,z)\big) \, f\big(\gamma(\tau,z)\big) \] for all \(\gamma \in \Gamma^J\) where \(\omega(\gamma)\) is the multiplier system of weight \(k\) on \(\Gamma^J\) (satisfying the usual comparability relation).
A real analytic function \(f:\mathcal{H}\times \mathbb{C}^j \to \mathbb{C}\) is called a (real analytic) Jacobi function of weight \(k \in \frac{1}{2}\mathbb{Z}\) and index \(\mathcal{M}^{(j)}\) with multiplier system \(\omega\) if \[ f \big|_{\omega,k,\mathcal{M}^{(j)}} \gamma = f \qquad \text{for all } \gamma \in \Gamma^J \] holds and if \(f\) satisfies a certain growth condition.
The space of Jacobi forms is denoted by \(J_{\omega,k,\mathcal{M}^{(j)}}\big(\Gamma^J\big)\).
Let \(\mathcal{M}^{(j)}\) be a fixed element in \(M_{j\times j}(\mathbb{Z})\) and \(\mathcal{P}_{\mathcal{M}^{(j)}}\) be the space of functions \(g\) which are holomorphic in \(\mathcal{H} \times \mathbb{C}^j\) and satisfy the growth condition \[ \big| g(\tau,z)\big| \leq K \big( |\tau|^\rho + \text{im}(\tau)^{-\sigma} \big) \, e^{2 \pi \text{Tr}\left( \frac{\mathcal{M}^{(j)} \, y^ty}{\text{im}(\tau)}\right)} \] for some positive constants \(K\), \(\rho\) and \(\sigma\), and \(y= \text{im}(z)= \big(\text{im}(z_1),\ldots, \text{im}(z_j)\big)\). The set \(\mathcal{P}_{\mathcal{M}^{(j)}}\) is a complex vector space and is preserved under the slash operator for any real \(k\).
Definition. A real analytic function \(f: \mathcal{H}\times \mathbb{C}^j \to \mathbb{C}\), which is periodic with respect to both variables \(\tau\) and \(z\), is called a Jacobi integral of weight \(k \in \frac{1}{2} \mathbb{Z}\) and index \(\mathcal{M}^{(j)}\) with a multiplier system \(\omega\) and holomorphic period functions \(P_\gamma \in \mathcal{P}_{\mathcal{M}^{(j)}}\) on \(\Gamma^J\) if it satisfies the following conditions:
(1) \(\big(f \big|_{\omega,k,\mathcal{M}^{(j)}} \gamma \big) (\tau,z) = f(\tau,z) + P_\gamma(\tau,z)\) for all \(\gamma \in \Gamma^J\),
(2) \(f(\tau,z) = \mathcal{O} \big( e^{\alpha \text{im}(\tau)}\, e^{2 \pi \text{Tr}\left( \frac{\mathcal{M}^{(j)} \, y^ty}{\text{im}(\tau)}\right)} \big)\) as \(\text{im}(\tau) \to \infty\) and for some \(\alpha > 0\).
Then, the first two main results are the following
Theorem. Let \(r\) be any real number, \(m >0\) and \(\omega\) a multiplier system of weight \(r\) and index \(m\). Suppose that \(\big\{ \varphi_\gamma ;\; \gamma \in \mathrm{SL}(2,\mathbb{Z})^J\big\}\) is a parabolic cocycle of weight \(r\) and index \(m\) in \(\mathcal{P}_m\). then there is a unique meromorphic function, up to Jacobi forms, \(f: \mathcal{H} \times \mathbb{C}^1 \to \mathbb{C}\) such that \[ \big(f \big|_{\omega,k,\mathcal{M}^{(j)}} \gamma \big) (\tau,z) - f(\tau,z) = \varphi_\gamma(\tau,z) \] for all \(\gamma \in \mathrm{SL}(2,\mathbb{Z})^J\). This \(f\) is called a Jacobi integral of weight \(r\) and index \(m\) with period functions \(\big\{ \varphi_\gamma ;\; \gamma \in \mathrm{SL}(2,\mathbb{Z})^J\big\}\).
Put \(I={1 \; 0 \choose 0 \; 1}\), \(S={1 \; 1 \choose 0 \; 1}\) and \(T={0 \; -1 \choose 1 \; 0}\).
Theorem. Take a multiplier system \(\omega\) with \(\omega(-I)=1\). If a Jacobi integral \(f\) is even and periodic with respect to \(z\), i.e., \(P_{\big[-I,(0,0)\big]}(\tau,z) = P_{\big[I,(0,0)\big]}(\tau,z) = 0\), then the period functions \(P_{\big[T,(0,0)\big]}(\tau,z)\) and \(P_{\big[I,(1,0)\big]}(\tau,z)\) satisfy the following properties:
(1) \(P_{\big[T,(1,0)\big]} = P_{\big[T,(0,0)\big]}\),
(2) \(P_{\big[T,(0,0)\big]} + P_{\big[T,(0,0)\big]}\big|_{\omega,k,m}\big[T,(0,0)\big] = 0\),
(3) \(P_{\big[T,(0,0)\big]} + P_{\big[T,(0,0)\big]}\big|_{\omega,k,m}\big[ST,(0,0)\big] + P_{\big[T,(0,0)\big]}\big|_{\omega,k,m}\big[(ST)^2,(0,0)\big] = 0\),
(4) \(P_{\big[I,(1,0)\big]}\big|_{\omega,k,m}\big[T,(0,0)\big] = -P_{\big[T,(0,0)\big]} + P_{\big[T,(0,0)\big]}\big|_{\omega,k,m}\big[I,(0,-1)\big]\),
(5) \(P_{\big[I,(1,0)\big]} = P_{\big[T,(0,0)\big]} - P_{\big[T,(0,0)\big]}\big|_{\omega,k,m}\big[-I,(1,0)\big]\) and
(6) \(P_{\big[I,(1,0)\big]} + P_{\big[I,(1,0)\big]}\big|_{\omega,k,m}\big[-I,(1,0)\big] = 0\).
The third main result says basically: the set of all \(P \in \mathcal{P}_m\) satisfying the properties (2) and (3) above, i.e., \[ P + P\big|_{\omega,k,m}\big[T,(0,0)\big] = P + P\big|_{\omega,k,m}\big[ST,(0,0)\big] + P\big|_{\omega,k,m}\big[(ST)^2,(0,0)\big] = 0, \] is a generating set of all parabolic cocycles of \(\mathrm{SL}(2,\mathbb{Z})^J\).
Reviewer: Tobias Mühlenbruch (Hagen)
MSC:
| 11F50 | Jacobi forms |
| 11F37 | Forms of half-integer weight; nonholomorphic modular forms |
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 11F75 | Cohomology of arithmetic groups |
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