Big Galois image for p-adic families of positive slope automorphic forms

Abstract : Let g = 1 or 2 and p > 3 be a prime. For the symplectic group GSp2g the Hecke eigensystems appearing in the spaces of classical automorphic forms, of a fixed tame level and varying weight, are p-adically interpolated by a rigid analytic space, the GSp2g-eigenvariety. A sufficiently small subdomain of the eigenvariety can be described as the rigid analytic space associated with a profinite algebra T. An irreducible component of T is defined by a profinite ring I and a morphism θ : T → I. In the residually irreducible case we can attach to θ a representation ρθ : Gal(Q/Q) → GSp2g(I). We study the image of ρθ when θ describes a positive slope component of T. In the case g = 1 this is a joint work with A. Iovita and J. Tilouine. Suppose either that g = 1 or that g = 2 and θ is residually of symmetric cube type. We prove that Im ρθ is “big” and that its size is related to the “accidental congruences” of θ with the subfamilies that are obtained as lifts of families for groups of smaller rank. More precisely, we enlarge a subring I0 of I[1/p] to a ring B and we define a Lie subalgebra G of gsp2g(B) associated with Im ρθ. We prove that there exists a non-zero ideal l of I0 such that l · sp2g(B) ⊂ G. For g = 1 the prime factors of l correspond to the CM points of the family θ. Such points do not define congruences between θ and a CM family, so we call them accidental congruence points. For g = 2 the prime factors of l correspond to accidental congruences of θ with subfamilies of dimension 0 or 1 that are symmetric cube lifts of points or families of the GL2-eigencurve.
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Andrea Conti. Big Galois image for p-adic families of positive slope automorphic forms. Algebraic Geometry [math.AG]. Université Sorbonne Paris Cité, 2016. English. ⟨NNT : 2016USPCD081⟩. ⟨tel-02152381⟩

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