Asai-Flach classes, p-adic L-functions and the Bloch-Kato conjecture for GO(4)

TL;DR

This paper proves the Bloch-Kato conjecture for critical Asai L-values of p-ordinary Hilbert modular forms over quadratic fields and establishes an inclusion in the Iwasawa main conjecture, advancing understanding of p-adic L-functions.

Contribution

It provides the first proof of the Bloch-Kato conjecture for these specific Asai L-functions and extends the p-adic Eichler-Shimura isomorphism to Hida families.

Findings

Proved Bloch-Kato conjecture for critical Asai L-values.

Established an inclusion in the Iwasawa main conjecture.

Extended p-adic Eichler-Shimura comparison to Hida families.

Abstract

We prove the Bloch-Kato conjecture for critical values of Asai L-functions of p-ordinary Hilbert modular forms over quadratic fields (with p split); and one inclusion in the Iwasawa main conjecture for these L-functions (up to a power of p). Along the way, we also prove a version of the p-adic Eichler-Shimura comparison isomorphism for Hida families of Hilbert modular forms.