In Coates' *$p$-adic L-functions and Iwasawa's theory*, the first of his congruence hypotheses is that $\delta_n(\mathfrak{b},\mathfrak{c},\mathfrak{f})\in \mathbb{Z}_p$, where $\mathfrak{b},\mathfrak{c},\mathfrak{f}$ are ideals of the totally real base field $F$ with $\mathfrak{bc}$ prime to $\mathfrak{f}$ and $\mathfrak{c}$ prime to $p$, and
$$\delta_n(\mathfrak{b},\mathfrak{c},\mathfrak{f}) := (\mathbb{N}\mathfrak{c})^{n+1}\zeta_{\mathfrak{f}}(\mathfrak{b},-n)-\zeta_{\mathfrak{f}}(\mathfrak{bc},-n)$$
is a difference of partial zeta functions. Later, Deligne-Ribet proved this en route to more general congruence results that encompass all of Coates' hypotheses, and in expositions of this (e.g. here) they are phrased in terms of the similar-looking function defined as
$$\Delta_{\mathfrak{c}}(-n,\chi) :=L(-n, \chi) - \chi(\mathfrak{c})(\mathbb{N}\mathfrak{c})^{n+1}L(-n, \chi),$$
and Coates' first congruence is restated as $\Delta_{\mathfrak{c}}(-n,\chi) \in \mathbb{Z}_p$ (or an extension thereof) where $\chi$ is a character of the ray class group mod $\mathfrak{f}$ and $\mathfrak{c}$ is an integral ideal prime to $p\mathfrak{f}$. I can't see how Coates' congruence follows from this. The other direction straightforward enough, since
$$\Delta_{\mathfrak{c}}(-n, \chi) = -\sum_{\mathfrak{b}} \chi(\mathfrak{bc}) \delta_n(\mathfrak{b},\mathfrak{c},\mathfrak{f})$$
where $\mathfrak{b}$ runs over any representatives of ray ideal classes mod $\mathfrak{f}$, but the finite Fourier inverted formula
$$\delta_n(\mathfrak{b},\mathfrak{c},\mathfrak{f}) = -\frac{1}{|C_\mathfrak{f}|}\sum_{\chi} \overline{\chi}(\mathfrak{bc}) \Delta_\mathfrak{c}(n,\chi)$$
where $\chi$ runs over $\widehat{C_{\mathfrak{f}}}$ introduces this denominator, which could have factors of $p$. What am I missing here?

## 1 Answer

I think you are misinterpreting Ribet's claim here.

In Ribet's article he defines $\Delta_c(1 -k, \epsilon)$ for an **arbitrary function** $\epsilon: G_f \to V$ where $V$ is a $\mathbf{Q}_p$-vector space. It needn't be a group homomorphism into the unit group of a field extension of $\mathbf{Q}_p$, even though this is by far the most familiar case.

When Ribet restates Coates' congruences, he doesn't say precisely what he intends $\epsilon$ to be in congruence (A), but he certainly doesn't require that it should be a character; and the fact that it is assumed always to be $\mathbf{Z}_p$-valued is a strong hint that it won't be a character in general (because $\mathbf{Z}_p^\times$ contains too few roots of unity to be the target of interesting characters of most finite abelian groups). So it seems clear that he intends $\epsilon$ to be an arbitrary $\mathbf{Z}_p$-valued function.

So you can recover Coates' formulation of the first congruence from Ribet's formulation just by setting $\epsilon$ to be the indicator function of a specific element of $G_f$.