Abstract
We prove that Heegner cycles of codimension m+1 inside Kuga-Sato type varieties of dimension 2m+1 are coefficients of modular forms of weight 3/2+m in the appropriate quotient group. The main technical tool for generating the necessary relations is a Borcherds style theta lift with polynomials. We also show how this lift defines a new singular Shimura-type correspondence from weakly holomorphic modular forms of weight 1/2-m to meromorphic modular forms of weight 2m+2.
Highlights
A celebrated theorem of Gross, Kohnen, and Zagier in [15] states that the Heegner divisors on modular curves correspond, in parts of the Jacobian variety of the modular curve, to coefficients of a modular form of weightThis result was proved using height evaluations
The latter proof generalizes to Shimura curves, and the argument extends to yield the modularity of Hirzebruch–Zagier divisors from [16] on Hilbert modular surfaces, of Humbert surfaces on Siegel threefolds, etc
The subsequent work investigates the results obtained for the case b− = 2. It requires a non-trivial analysis of the theta lift, which is based on the machinery from [2], but has to be examined explicitly since these particular functions do not appear in that reference
Summary
A celebrated theorem of Gross, Kohnen, and Zagier in [15] states that the Heegner divisors on modular curves correspond, in parts of the Jacobian variety of the modular curve, to coefficients of a modular form of weight. In the case b− = 1 of Shimura and modular curves, the images of these theta lifts under the weight raising operators are meromorphic (with known poles). This establishes a new singular Shimura-type correspondence, as stated in the following. The subsequent work investigates the results obtained for the case b− = 2 It requires a non-trivial analysis of the theta lift, which is based on the machinery from [2], but has to be examined explicitly since these particular functions do not appear in that reference. My special thanks are delivered to the anonymous referee, whose suggestions greatly improved the readability of this paper
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