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Images of Galois representations Conti, Andrea Speeches/Talks (2019) The absolute Galois group of a local or global field can be better understood by studying its representations, important classes of which are constructed from geometric objects such as elliptic or modular ... [more ▼] The absolute Galois group of a local or global field can be better understood by studying its representations, important classes of which are constructed from geometric objects such as elliptic or modular curves. The results of a line of work initiated by Serre, Ribet and Momose suggest that certain interesting symmetries of a Galois representation constructed this way are in bijection with the symmetries of the underlying geometric object. We present a recent result in this direction, obtained in a joint work with J. Lang and A. Medvedovsky. [less ▲] Detailed reference viewed: 46 (3 UL)Big images of two-dimensional pseudo representations Conti, Andrea ; ; E-print/Working paper (2019) For an odd prime p, we study the image of a continuous 2-dimensional (pseudo)representation rho of a profi nite group with coe cients in a local pro-p domain A. Under mild conditions, Bella che has proved ... [more ▼] For an odd prime p, we study the image of a continuous 2-dimensional (pseudo)representation rho of a profi nite group with coe cients in a local pro-p domain A. Under mild conditions, Bella che has proved that the image of rho contains a nontrivial congruence subgroup of SL2(B) for a certain subring B of A. We prove that the ring B can be slightly enlarged and then described in terms of the conjugate self-twists of rho, symmetries that naturally constrain its image; hence this new B is optimal. We use this result to recover, and in some cases improve, the known large-image results for Galois representations arising from elliptic and Hilbert modular forms due to Serre, Ribet and Momose, and Nekov a r, and p-adic Hida or Coleman families of elliptic modular forms due to Hida, Lang, and Conti-Iovita-Tilouine. [less ▲] Detailed reference viewed: 77 (2 UL)Galois level and congruence ideal for p-adic families of finite slope Siegel modular forms Conti, Andrea in Compositio Mathematica (2019), 155(4), 776-831 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 ... [more ▼] 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. [less ▲] Detailed reference viewed: 45 (1 UL)Lifting trianguline Galois representations Conti, Andrea E-print/Working paper (2017) For a central isogeny $H^\prime\to H$ of group schemes over a $p$-adic field $E$ and a continuous representation $\rho\colon\mathr{Gal}(\overline F/F)\to H(E)$ of the absolute Galois group of a number ... [more ▼] For a central isogeny $H^\prime\to H$ of group schemes over a $p$-adic field $E$ and a continuous representation $\rho\colon\mathr{Gal}(\overline F/F)\to H(E)$ of the absolute Galois group of a number field $F$, trianguline at the $p$-adic places, we give sufficient conditions for the existence of a lift of $\rho$ to a continuous representation $\mathrm{Gal}(\overline F/F)\to H^\prime(E)$ that is also trianguline at the $p$-adic places. This is an analogue in the world of non-de Rham representations of results of Wintenberger, Conrad, Patrikis, and Hoang Duc for $p$-adic Hodge-theoretic properties of $\rho$. Our main tool for studying $p$-adic Galois representation locally at $p$ is an abstract Tannakian construction combined with the theory of $B$-pairs; our result and concrete manipulation of the $B$-pairs are inspired by recent work of Berger and Di Matteo. [less ▲] Detailed reference viewed: 57 (5 UL)Big Galois image for p-adic families of positive slope automorphic forms Conti, Andrea Doctoral thesis (2016) Detailed reference viewed: 48 (8 UL)Big image of Galois representations associated with finite slope p-adic families of modular forms Conti, Andrea ; ; in Loeffler, David; Zerbes, Sarah Livia (Eds.) Elliptic Curves, Modular Forms and Iwasawa Theory: In Honour of John H. Coates' 70th Birthday (2016) We consider the Galois representation associated with a finite slope, non-CM p-adic family of Hecke eigenforms, and prove that the Lie algebra of its image contains a congruence Lie subalgebra of non ... [more ▼] We consider the Galois representation associated with a finite slope, non-CM p-adic family of Hecke eigenforms, and prove that the Lie algebra of its image contains a congruence Lie subalgebra of non-trivial level. We describe the largest such level in terms of the congruences of the family with p-adic CM eigenforms. [less ▲] Detailed reference viewed: 49 (0 UL) |
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