Arithmetic Theta Kernel and Liftings

Let $K=\Q(\sqrt d)$ be an imaginary quadratic field. Let $L$ be a unimodular $O_K$-lattice of signature $(n-1, 1)$, and let $\mathcal X$ be the compactified Shimura variety over $O_K$ associated at $L$.  In this talk, we describe roughly how to construct an arithmetic theta series (arithmetic theta  kernel), and prove that it is a modular form of level $d$, weight $n$, character $\chi_d^n$ with values in  $\widehat{CH}^1(\mathcal X^*)$.  Using this arithmetic theta kernel, we can produce classical modular forms from arithmetic $1$-cycles (Faltings’ height pairing) and produce arithmetic divisors from classical cusp forms.  This is joint work with Bruinier, Howard, Kudla, and Rapoport for $n \ge 3$. The case $2$ is a joint work in progress with Qiao He and Yousheng Shi.

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