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CM points and weight 3/2 modular forms.

Abstract

The theta correspondence has been an important tool in the theory of automorphic forms with plentiful applications to arithmetic questions. In this paper, we consider a specific theta lift for an isotropic quadratic space V over Q of signature (1, 2). The theta kernel we employ associated to the lift has been constructed by Kudla-Millson (e.g., [29, 30]) in much greater generality for O(p, q) (U(p, q)) to realize generating series of cohomo-logical intersection numbers of certain, ’special ’ cycles in locally symmetric spaces of orthogonal (unitary) type as holomorphic Siegel (Hermitian) mod-ular forms. In our case for O(1, 2), the underlying locally symmetric space M is a modular curve, and the special cycles, parametrized by positive in-tegers N, are the classical CM points Z(N); i.e., quadratic irrationalities of discriminant −N in the upper half plane. We survey the results of [16] and of our joint work with Bruinier [12] on using this particular theta kernel to define lifts of various kinds of functions F on the underlying modular curve M. The theta lift is given b

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