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Galois level and congruence ideal for $p$-adic families of finite slope Siegel modular forms

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  27 March 2019

Andrea Conti
Andrea Conti*
Affiliation:
Department of Mathematics and Statistics, Concordia University, 1400 De Maisonneuve Boulevard West, Montreal, Quebec, Canada H3G 1M8 email [email protected]
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Abstract

We consider families of Siegel eigenforms of genus $2$ and finite slope, defined as local pieces of an eigenvariety and equipped with a suitable integral structure. Under some assumptions on the residual image, we show that the image of the Galois representation associated with a family is big, in the sense that a Lie algebra attached to it contains a congruence subalgebra of non-zero level. We call the Galois level of the family the largest such level. We show that it is trivial when the residual representation has full image. When the residual representation is a symmetric cube, the zero locus defined by the Galois level of the family admits an automorphic description: it is the locus of points that arise from overconvergent eigenforms for $\operatorname{GL}_{2}$, via a $p$-adic Langlands lift attached to the symmetric cube representation. Our proof goes via the comparison of the Galois level with a ‘fortuitous’ congruence ideal. Some of the $p$-adic lifts are interpolated by a morphism of rigid analytic spaces from an eigencurve for $\operatorname{GL}_{2}$ to an eigenvariety for $\operatorname{GSp}_{4}$, while the remainder appear as isolated points on the eigenvariety.

Keywords

Siegel modular formseigenvarietiesGalois representationsbig imagecongruence idealtrianguline representations

MSC classification

Primary: 11F80: Galois representations 11F33: Congruences for modular and $p$-adic modular forms 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 4 , April 2019 , pp. 776 - 831
Copyright
© The Author 2019 

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Footnotes

1

Current address: Computational Arithmetic Geometry – IWR – Universität Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany

The author was supported by the Programs ArShiFo ANR-10-BLAN-0114 and PerCoLaTor ANR-14-CE25-0002-01.

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