Arithmetic of the Asai L-function for Hilbert modular forms.

摘要

We prove two results on rationality of special values of the Asai L-function attached to Hilbert modular forms at critical points. Such L-functions only admit critical values when the Hilbert modular form has non-parallel weight. Our rationality results generalize previous work of Shimura on algebraicity. The first result uses a period defined by transferring the Hilbert modular form to a Shimura curve. The second result uses a period defined using rational structures on the coherent cohomology of Hilbert modular surfaces. We also give some partial results towards integrality of such L-values. Our results are motivated by the study of a p-adic analog of the Beilinson conjecture, which is a deep conjecture relating algebraic cycles (and motivic cohomology) to values of L-functions.