Formal Fourier Jacobi expansions and special cycles of codimension two (Q2786459)

This rather interesting paper is about Fourier-Jacobi expansions of Siegel modular forms. Suppose that \(F\) is a vector-valued Siegel modular form with respect to a (finite dimensional unitary) representation \(\det^k \otimes \rho\) of the Siegel modular group of degree two. Then it has a Fourier Jacobi (F-J) expansion NEWLINE\[NEWLINE F(Z)= \sum_{0 \leq m \in \mathbb{Q}} \phi_m(\tau,z) e(m \tau'), \tag{1} NEWLINE\]NEWLINE where \(Z = \begin{pmatrix} \tau & z \\ z & \tau' \end{pmatrix}\) is in the Siegel upper half-space of degree \(2\) and the \(\phi_m\) are certain Jacobi forms. The author proves the remarkable converse result: if a formal Fourier Jacobi expansion (1) is given, then under rather general conditions (for example with half-integral weights allowed and additional automorphy with respect to symmetric powers), (1) actually is the F-J expansion of a Siegel modular form.NEWLINENEWLINEThe proof is inspired by the results of \textit{T. Ibukiyama} et al. [Abh. Math. Semin. Univ. Hamb. 83, No. 1, 111--128 (2013; Zbl 1297.11037)] who treated the case \(\rho=1\) and integral weights. The author proves that the graded (with respect to the scalar weight) module, say \(G(\rho)\) over \(M=\oplus_k M^2_k\) (algebra of scalar weight Siegel modular forms) consisting of such formal F-J expansions coincides with with the corresponding module \(M(\rho)\) for classical modular forms. For this he proves that \(G(\rho)\) is free over \(M\) and compares the dimensions of the \(k\)-graded parts in a way reminiscent of the work of \textit{H. Aoki} [J. Math. Kyoto Univ. 40, No. 3, 581--588 (2000; Zbl 0972.11034)]. The proof follows from this.NEWLINENEWLINEThe result enables the author to prove one of Kudla's conjectures which says that ``the generating function of special cycles of codimension \(r\) is a Siegel modular form of degree \(r\)'', in the case \(r=2\). Some application towards the computation of Fourier expansions of Siegel modular forms is also discussed.