Transcendence of Hecke operators in the big Hecke algebra

Abstract

We prove a transcendence result of Hecke operator $T\left(\mathfrak{l}\right)$ over the ring generated by diamond operators over $\mathbb{Q}$ in a non-CM component of the cyclotomic ordinary big $p$-adic Hecke algebra, and we discuss its conjectural implication of a modulo-$p$ version to the vanishing of the cyclotomic $\mu$-invariant.