Reference : Big images of two-dimensional pseudo representations
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Physical, chemical, mathematical & earth Sciences : Mathematics
Big images of two-dimensional pseudo representations
Conti, Andrea mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
Lang, Jaclyn [Université Paris 13 > Mathematics > LAGA > Postdoctorante]
Medvedovsky, Anna [Boston University > Mathematics]
[en] 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.

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