分野別セミナー

Let $p\ge5$ 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 $\Gamma(L)$ of $SL_2(\Zp[[T]])$ for a
principal ideal $(L)\ne0$ of $\Zp[[T]]$
for the canonical ``weight'' variable $T$. If nontrivial (i.e.,
$L\ne1$), 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 of $(1+T)^{p^m}-1)$).