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  • Haruzo Hida (a1) and Jacques Tilouine (a2)


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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The first author is partially supported by the NSF grant: DMS 1464106. The second author is partially supported by the ANR grant: PerCoLaTor ANR-14-CE25.



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