Anticyclotomic diagonal classes and Beilinson--Flach elements

Abstract

We present a comparison between the anticyclotomic Euler system of diagonal cycles associated with the convolution of two modular forms and the cyclotomic Beilinson--Flach Euler system. This extends the seminal work of Bertolini, Darmon, and Venerucci, who established a link between (anticyclotomic) Heegner points and the Beilinson--Kato system. Our approach hinges on a detailed analysis of $p$-adic $L$-functions and Perrin-Riou maps and exploits the Eisenstein degeneration of diagonal cycles along Hida families, working with a CM family which specializes to an irregular Eisenstein series in weight one. We use these results to derive some arithmetic applications.