Maass-Jacobi Poincaré series and Mathieu moonshine. (English) Zbl 1327.11035
The main result here is a technical one: given an element \(g\in M_{24}\) there is an associated semi-holomorphic Maass-Jacobi form \(\phi_g\) of weight one and index two, which is characterised by its level, multiplier and principal parts at the cusps. The important part, though, is the construction of \(\phi_g\) and its relation to representations of \(M_{24}\).
The starting point is the observation of T. Eguchi et al. [Exp. Math. 20, No. 1, 91–96 (2011; Zbl 1266.58008)] that the coefficients of the unique Jacobi form of weight zero and index one, after some correction terms and factors, are simple linear combinations of the dimensions of irreducible representations of the Mathieu group \(M_{24}\). Attempts to explain this moonshine phenomenon have been partially successful and yield an \(M_{24}\) module \(K\) whose properties are encoded by weak Jacobi forms \(Z_{g}(\tau;z) \). However, no actual construction of this module is known. The authors propose an alternative in which the weak Jacobi forms are replaced by semi-holomorphic Maass-Jacobi forms, which are the analogues for Jacobi forms of Maass forms for modular forms.
This is a reformulation rather than a different version, because the \(\phi_g\) are modifications of mock modular forms \(H_g\) which in turn are mock modular forms associated with \(Z_g\) and thus already encode the information about the module \(K\). The advantage is that \(\phi_g\) can be identified with a Poincaré series that the authors construct: the characterisation theorem is used to show that the Poincaré series agrees with \(\phi_g\). The main technical difficulty in this part is showing that the Poincaré series converge and have a suitable analytic continuation. For this the authors use some formulae for certain Gauss sums, proved by them here, to reduce to a known case and thus avoid having to extend a substantial amount of theory (spectral theory, Kloosterman sums) from modular forms to Jacobi forms.
The starting point is the observation of T. Eguchi et al. [Exp. Math. 20, No. 1, 91–96 (2011; Zbl 1266.58008)] that the coefficients of the unique Jacobi form of weight zero and index one, after some correction terms and factors, are simple linear combinations of the dimensions of irreducible representations of the Mathieu group \(M_{24}\). Attempts to explain this moonshine phenomenon have been partially successful and yield an \(M_{24}\) module \(K\) whose properties are encoded by weak Jacobi forms \(Z_{g}(\tau;z) \). However, no actual construction of this module is known. The authors propose an alternative in which the weak Jacobi forms are replaced by semi-holomorphic Maass-Jacobi forms, which are the analogues for Jacobi forms of Maass forms for modular forms.
This is a reformulation rather than a different version, because the \(\phi_g\) are modifications of mock modular forms \(H_g\) which in turn are mock modular forms associated with \(Z_g\) and thus already encode the information about the module \(K\). The advantage is that \(\phi_g\) can be identified with a Poincaré series that the authors construct: the characterisation theorem is used to show that the Poincaré series agrees with \(\phi_g\). The main technical difficulty in this part is showing that the Poincaré series converge and have a suitable analytic continuation. For this the authors use some formulae for certain Gauss sums, proved by them here, to reduce to a known case and thus avoid having to extend a substantial amount of theory (spectral theory, Kloosterman sums) from modular forms to Jacobi forms.
Reviewer: G. K. Sankaran (Bath)
MSC:
| 11F50 | Jacobi forms |
| 11F20 | Dedekind eta function, Dedekind sums |
| 11F22 | Relationship to Lie algebras and finite simple groups |
| 11F37 | Forms of half-integer weight; nonholomorphic modular forms |
| 20C34 | Representations of sporadic groups |
| 20C35 | Applications of group representations to physics and other areas of science |
Citations:
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