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SYMMETRIC POWER CONGRUENCE IDEALS AND SELMER GROUPS

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  14 November 2018

Haruzo Hida  and
Jacques Tilouine
Haruzo Hida
Affiliation:
Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA (hida@math.ucla.edu)
Jacques Tilouine
Affiliation:
Département de Mathématiques, LAGA, Institut Galilée, U. Paris 13, 99 av. J.-B. Clément, Villetaneuse 93430, France (tilouine@math.univ-paris13.fr)
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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.

Keywords

Congruence moduleHecke algebrauniversal Galois deformation ring

MSC classification

Primary: 11F33: Congruences for modular and $p$-adic modular forms 11F80: Galois representations
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 5 , September 2020 , pp. 1521 - 1572
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Copyright
© Cambridge University Press 2018

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Footnotes

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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