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

We consider families of Siegel eigenforms of genus 2 and nite slope, de ned 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 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 de ned by the Galois level of the family admits an automorphic description: it is the locus of points that arise from overconvergent eigenforms for GL2, 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, that describes the zero- and one-dimensional subvarieties of symmetric cube type appearing in the family. We show that some of the p-adic lifts are interpolated by a morphism of rigid analytic spaces from an eigencurve for GL2 to an eigenvariety for GSp4. The remaining lifts appear as isolated points on the eigenvariety.