Open Access Publications from the University of California

## SYMMETRIC POWER CONGRUENCE IDEALS AND SELMER GROUPS

• Author(s): Hida, Haruzo
• Tilouine, Jacques
• et al.

## Published Web Location

https://doi.org/10.1017/S1474748018000476
Abstract

We prove, under some assumptions, a Greenberg type equality relating the characteristic power series of the Selmer groups over $\mathbb{Q}$ of higher symmetric powers of the Galois representation associated to a Hida family and congruence ideals associated to (different) higher symmetric powers of that Hida family. We use $R=T$ theorems and a sort of induction based on branching laws for adjoint representations. This method also applies to other Langlands transfers, like the transfer from $\text{GSp}(4)$ to $U(4)$. In that case we obtain a corollary for abelian surfaces.

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