Big Galois representations and -adic -functions

Abstract

Let$p\geqslant 5$be a prime. If an irreducible component of the spectrum of the ‘big’ ordinary Hecke algebra does not have complex multiplication, under mild assumptions, we prove that the image of its Galois representation contains, up to finite error, a principal congruence subgroup${\rm\Gamma}(L)$of$\text{SL}_{2}(\mathbb{Z}_{p}[[T]])$for a principal ideal$(L)\neq 0$of$\mathbb{Z}_{p}[[T]]$for the canonical ‘weight’ variable$t=1+T$. If$L\notin {\rm\Lambda}^{\times }$, the power series$L$is proven to be a factor of the Kubota–Leopoldt$p$-adic$L$-function or of the square of the anticyclotomic Katz$p$-adic$L$-function or a power of$(t^{p^{m}}-1)$.