In recent work with Kartik Prasanna, we propose an explicit relationship between the cohomology of vector bundles on Siegel modular threefolds and higher Chow groups (aka motivic cohomology groups). We prove three results:
(1) Our conjecture is implied by Beilinson’s conjecture on special values of L-functions. This uses recent work of Chen—Ichino and Yang—Radzwill.
(2) Our conjecture holds for Yoshida lifts of Hilbert modular forms. This uses a result of Ramakrishnan on Asai L-functions for Hilbert modular forms.
(3) Our conjecture implies the conjecture of Prasanna—Venkatesh for Yoshida lifts of Bianchi modular forms.