Big images of two-dimensional pseudo representations

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 ▲]

Lifting trianguline Galois representations

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 ▲]

Big image of Galois representations associated with finite slope p-adic families of modular forms

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 ▲]