Abstract
We prove modularity of formal series of Jacobi forms that satisfy a natural symmetry condition. They are formal analogs of Fourier–Jacobi expansions of Siegel modular forms. From our result and a theorem of Wei Zhang, we deduce Kudla’s conjecture on the modularity of generating series of special cycles of arbitrary codimension and for all orthogonal Shimura varieties.
Highlights
Fourier Jacobi expansions are one of the major tools to study Siegel modular forms
We formalize the notion of Fourier–Jacobi expansions by combining two features of Siegel modular forms: Fourier–Jacobi coefficients are Siegel–Jacobi forms, and Fourier expansions of genus-g Siegel modular forms have symmetries with respect to GLg (Z)
We restrict this exposition to classical Siegel modular forms of even weight
Summary
Fourier Jacobi expansions are one of the major tools to study Siegel modular forms. For example, they appeared prominently in the proof of the Saito– Kurokawa conjecture [1, 24,25,26, 38]. We call f a symmetric formal Fourier– Jacobi series (of weight k, genus g, and cogenus l) if its coefficients satisfy (4) for all u ∈ GLg(Z). As a special case of earlier joint work with Millson, Kudla [21] attached classes in CHg(X ) of special cycles Z (t) of codimension g to positive semidefinite matrices t ∈ Symg(Q), and considered their generating series He observed that his results with Millson implied that the analogous (but coarser) generating series for the images clhom Z (t) of the cycle classes in cohomology is a Siegel modular form of weight 1+n/2 and genus g. Zhang’s result states that Ag is a symmetric formal Fourier– Jacobi series of weight 1 + n/2, genus g, and cogenus g − 1 Combining it with our main theorem, we obtain the following.
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