Big images of two-dimensional pseudorepresentations

Abstract

For an odd prime p, we study the image of a continuous 2-dimensional (pseudo)representation ρ of a profinite group with coefficients in a local pro-p domain A. Under mild conditions, Bellaïche has proved that the image of ρ 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 ρ, 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 Nekovář, and p-adic Hida or Coleman families of elliptic modular forms due to Hida, Lang, and Conti-Iovita-Tilouine.

Semi-ordinary Iwasawa theory for Rankin--Selberg products

“…7 We would like to express our gratitude to Jackie Lang for carefully reading through this appendix and pointing out several inaccuracies, which helped us improve our exposition drastically. In particular, we thank her for bringing the papers of Klingenberg [Kli60,Kli61] to our attention and explaining relevant portions of the results in her joint work [CLM20] with Conti and Medvedovsky (which lead to a significant simplification of some of our statements and proofs).…”