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We show that every Fricke-invariant meromorphic modular form for
whose divisor on
is defined over
and supported on Heegner divisors and the cusps is a generalized Borcherds product associated to a harmonic Maass form of weight
. Further, we derive a criterion for the finiteness of the multiplier systems of generalized Borcherds products in terms of the vanishing of the central derivatives of
-functions of certain weight
newforms. We also prove similar results for twisted Borcherds products.
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.
Jan Hendrik Bruinier, Mathematisches Institut, Universität zu Köln, Weyertal 86–90, D-50931 Köln, Germany,
Jens Funke, Department of Mathematical Sciences, New Mexico State University, P.O.Box 30001, 3MB, Las Cruces, NM 88003, USA
We consider the Kudla-Millson lift from elliptic modular forms of weight (p + q)/2 to closed q-forms on locally symmetric spaces corresponding to the orthogonal group O(p, q). We study the L2-norm of the lift following the Rallis inner product formula. We compute the contribution at the Archimedian place. For locally symmetric spaces associated to even unimodular lattices, we obtain an explicit formula for the L2-norm of the lift, which often implies that the lift is injective. For O(p, 2) we discuss how such injectivity results imply the surjectivity of the Borcherds lift.
In previous work, we studied the Kudla-Millson theta lift (see e.g.) and Borcherds' singular theta lift (e.g.) and established a duality statement between these two lifts. Both of these lifts have played a significant role in the study of certain cycles in locally symmetric spaces and Shimura varieties of orthogonal type. In this paper, we study the injectivity of the Kudla-Millson theta lift, and revisit part of the material of from the viewpoint of, to obtain surjectivity results for the Borcherds lift. Moreover, we provide evidence for the following principle: The vanishing of the standard L-function of a cusp form of weight 1 + p/2 at s0 = p/2 corresponds to the existence of a certain “exceptional automorphic product” on O(p, 2) (see Theorem 1.8).
We derive lower bounds for the rank of Picard groups of modular varieties associated with natural congruence subgroups of the orthogonal group of an even lattice of signature (2, l). As an example we consider the Siegel modular group of genus 2. The analytic part of this paper also leads to certain class number identities.
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