Abstract

We discuss the construction of an Euler system for the ``triple product'': the $8$-dimensional $p$-adic Galois representation $V=V_1\otimes V_2\otimes V_3$, where $V_i$ for $i=1,2,3$ is the representation attached to an ordinary cuspidal eigenform $f_i$ of weight $k_i$ (varying in a Hida family). Our construction is via the pullback method recently introduced by Sangiovanni Vincentelli and Skinner. The image of the Euler system under a Perrin-Riou regulator map is the $p$-adic $L$-function of Hsieh and Yamana that interpolates critical values of $L(f_1\times f_2\times f_3\otimes \chi ,s)$, where $\chi$ is a Dirichlet character and the weights $(k_1,k_2,k_3)$ are in the balanced range. The cyclotomic variable $\chi$ is not seen by previous constructions, such as the diagonal cycles of Darmon and Rotger.

Organizer

Ming-Lun Hsieh (NTU)