Integrality of $\mathfrak{p}$-Adic $L$-Functions at Eisenstein Primes
Let $f$ be a normalized, ordinary newform of weight $\ge 2$. For each prime $\mathfrak{p}$ of $F=\mathbb{Q}(a_n)_{n\in \mathbb{N}}$, there is an associated $\mathfrak{p}$-adic $L$-function $\mathcal{L}_\mathfrak{p}(f)\in \Lambda \otimes \mathbb{Q}$ interpolating special values of the classical $L$-function. If $f$ is not congruent modulo $\mathfrak{p}$ to an Eisenstein series, one knows $\mathcal{L}_\mathfrak{p}(f)\in \Lambda$. In this paper, we show, under mild hypotheses on the ramification of $f$, that this integrality result holds when $f$ is congruent to an Eisenstein series. Moreover, we also obtain a divisibility in the main conjecture for $\mathcal{L}_\mathfrak{p}(f)$. As an application, we show that the integrality result and the divisibility hold in particular when $f$ is of weight $2$.