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 $\mathrm{GSp}_{2g}$ 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 $\mathrm{GSp}_{2g}$-eigenvariety. A sufficiently small subdomain of the eigenvariety can be described as the rigid analytic space associated with a profinite algebra $\mathbb{T}$. An irreducible component of $\mathbb{T}$ is defined by a profinite ring $\mathbb{I}$ and a morphism $\theta\colon\mathbb{T}\to\mathbb{I}$. In the residually irreducible case we can attach to $\theta$ a representation $\rho_\theta\colon\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GSp}_{2g}(\mathbb{I})$. We study the image of $\rho_\theta$ when $\theta$ describes a positive slope component of $\mathbb{T}$. 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 $\theta$ is residually of symmetric cube type. We prove that $\mathrm{Im}\,\rho_\theta$ is ``big'' and that its size is related to the ``accidental congruences'' of $\theta$ with the subfamilies that are obtained as lifts of families for groups of smaller rank. More precisely, we enlarge a subring $\mathbb{I}_0$ of $\mathbb{I}[1/p]$ to a ring $\mathbb{B}$ and we define a Lie subalgebra $\mathfrak{G}$ of $\mathfrak{gsp}_{2g}(\mathbb{B})$ associated with $\mathrm{Im}\,\rho_\theta$. We prove that there exists a non-zero ideal $\mathfrak{l}$ of $\mathbb{I}_0$ such that $\mathfrak{l}\cdot\mathfrak{sp}_{2g}(\mathbb{B})\subset\mathfrak{G}$. For $g=1$ the prime factors of $\mathfrak{l}$ correspond to the CM points of the family $\theta$. Such points do not define congruences between $\theta$ and a CM family, so we call them accidental congruence points. For $g=2$ the prime factors of $\mathfrak{l}$ correspond to accidental congruences of $\theta$ with subfamilies of dimension $0$ or $1$ that are symmetric cube lifts of points or families of the $\mathrm{GL}_2$-eigencurve.
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Andrea Conti. Big Galois image for $p$-adic families of positive slope automorphic forms. Number Theory [math.NT]. Université Paris 13 - Sorbonne Paris Cité, 2016. English. ⟨tel-01408061⟩

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