Higher regulators of Siegel sixfolds and non-critical values of Spin $L$-functions

Abstract

We construct classes in the middle degree plus one motivic cohomology of Siegel sixfolds and we compute their image by Beilinson higher regulator in terms of Rankin-Selberg type automorphic integrals. Using results of Pollack and Shah, we relate the integrals to non-critical special values of the degree 8 Spin L-functions. Along the way, by defining and studying complexes of tempered currents on smooth projective complex varieties endowed with a normal crossings divisor, we provide a new description of Deligne-Beilinson cohomology for any Shimura variety. This is particularly useful for computations of higher regulators and fills a gap in the literature on the subject.

Motivic Action on Coherent Cohomology of Hilbert Modular Varieties

“…The formulation of Beilinson's conjecture via a Poincaré duality pairing on Deligne cohomology is standard; another example for non-critical values of spin L-functions can be found in [CLJ19,CLJ22].…”