In the Langlands program, theta functions are used as integral kernels to construct automorphic forms through theta lifting, and the Rallis inner product formula relates the inner product of two theta lifts to special values of L-functions. In the Kudla program, arithmetic theta functions with values in Chow groups are used to construct algebraic cycles through arithmetic theta lifting, while the arithmetic inner product formula relates the Beilinson–Bloch height pairing of two arithmetic theta lifts to the central derivative of an L-function and could be used to give evidence for the Beilinson–Bloch conjecture.
In this talk, we propose a cohomological framework that could be used to give a uniform construction of arithmetic theta functions and of motivic theta functions with values in higher Chow groups. These motivic theta functions are used to define motivic theta lifting, which provides a new way to construct motivic classes beyond those arising from modular units. The resulting motivic theta lifts satisfy an identity relating their Beilinson regulators to non-critical L-values. This identity can be viewed as a Hodge–Deligne analogue of the Rallis inner product formula and provides evidence for Beilinson’s conjectures.