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SYMMETRIC POWER CONGRUENCE IDEALS AND SELMER GROUPS
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 19 / Issue 5 / September 2020
- Published online by Cambridge University Press:
- 14 November 2018, pp. 1521-1572
- Print publication:
- September 2020
-
- Article
- Export citation
-
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.
