There is a generalization of the Kummer congruences to totally real number fields with characters due to Deligne-Ribet. For example, see the exposition here, more precisely see Theorem 2.1.

What is confusing me is that they don't seem to use the L-functions with the Euler factor at p removed (which I will call the p-adic L function) but I thought that the Kummer congruences only held for the p-adic L functions. For instance, for the case of Dirichlet characters, I have only seen the theorem stated for p-adic L functions, for instance see Theorem 2.2 here. See also the wiki page for Kummer congruences without a character but with a higher order congruence.

Conceptually, I thought of the Kummer congruences as equivalent to the continuity of the p-adic L function but I don't believe the L-function without the p Euler factor removed is continuous.

What am I missing? Is it true that the Kummer congruences (even in the case of just Dirichlet characters) or am I misinterpreting the Deligne-Ribet result?

  • You are right in saying that continuity of $L$-functions corresponds to Kummer congruences, and indeed what Deligne--Ribet do is to remove the factors at $p$, as you imagine. Indeed, Theorem 2.1 in Ribet's paper says that the sum of values $\Delta_c(1-k\varepsilon_k)$ is integral: and, one page before, he defines $\Delta_c$ and observes that $\Delta_c(1-k,\varepsilon)=(1-\varepsilon(c)\mathbf{N}(c)^k)L(1-k,\varepsilon).$ – Filippo Alberto Edoardo May 15 at 21:50
  • @FilippoAlbertoEdoardo why is that equivalent to removing an euler factor? c is different from p and moreover the power of N(c ) in the factor is off by one, no? That factor corresponds to the euler factor of c of L(−k,ε), no? – user404577 May 16 at 2:38
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    you are perfectly right, I was writing nonsense (the ideal $c$ is there only for technical reasons to make the integration against a pseudo-measure meaningful). I have posted in the answer below what, I hope, makes sense in place of my previous comment. – Filippo Alberto Edoardo May 18 at 13:37
up vote 1 down vote accepted

I think that the point lies in the difference between a primitive and imprimitive $L$-function. Before entering the details, let me observe that Washington's definition of $p$-adic $L$-functions (as the one found in Washington's Introduction to Cyclotomic fields, Theorem 5.11) gives the interpolation $$ L_p(1-n,\chi)=(1-\chi\omega^{-n}(p)p^{n-1})L(1-n,\chi)\qquad n\geq 1 $$ and this is shown to be equivalent to the Kummer congruences that you quote, namely $$ (1-\chi\omega^{-n}(p)p^{n-1})\frac{B_{\chi\omega^{-n},n}}{n}\equiv (1-\chi\omega^{-m}(p)p^{m-1})\frac{B_{\chi\omega^{-m},m}}{m}\pmod{p^a} $$ whenever $m\equiv n\pmod{p^{a-1}}$ and $\chi=\omega^r$ is a power of the Teichmüller character. The value $\chi\omega^{-n}(p)=\omega^{r-n}(p)$ is considered, in both statements, as the value of the primitive character attached to $\omega^{r-n}$ at $p$, in the following sense (the discussion is taken from Chapter 3 of Washington's book). When $r\not\equiv n\pmod{p-1}$, the character $\omega^{r-n}$ has conductor $p$ and is primitive; the value $\omega^{r-n}(p)$ is $0$, hence the "correction factor" disappears. The interesting part arises when $r\equiv n\pmod{p-1}$, so that $\omega^{r-n}$ is the trivial character of conductor $p$: this is an imprimitive character, whose primitivisation is the trivial character of conductor $1$, whose value at $p$ is $1$: the "correction factor" is there and takes the value $(1-p^{n-1})$. This, as Washington remarks right after his Theorem 5.11 is the Euler factor at $p$ of the complex $L$-function defined on page 31 of his book ($2^\text{nd}$ edition), where the product runs over all primes.

In Deligne--Ribet the situation is different, and so is in Ribet's report that you quote. Indeed, in equation $(1.1)$ of his report he defines the $L$-function of a character $\varepsilon$ on a ray class group $G_\mathfrak{f}$ as a Dirichlet series over all prime-to-$\mathfrak{f}$ integral ideals. In section 4 of the report, he applies this to the limit $G_{\mathfrak{f}\mathfrak{p}^\infty}$. To make the bridge with Washington's situation, take $K=\mathbb{Q}$ as base field, and $\mathfrak{f}=1$. As character, chose $\varepsilon=\omega^{r-n}$ with $r\equiv n\pmod{p-1}$ (since otherwise, as discussed, Washington's and Ribet's expressions coincide). Then the $L$-function considered by Ribet is given by $$ L(s,\omega^{r-n})=\prod_{\ell\neq p}(1-\omega^{r-n}(\ell)\ell^{-s})^{-1}\qquad\operatorname{Re}(s)>1 $$ because $\omega^{r-n}$ is seen as character on $G=G_{p^\infty}$ of conductor $p$, and the sum defining $L(s,\omega^{r-n})$ is performed only over primes $\ell\neq p$. The corresponding Kummer congruences get modified accordingly.

It might be helpful, to convince yourself of this discussion, to compare the interpolation formula proposed by Washington in Theorem 5.11 and the one proposed by Ribet in his final section on page 13, in the paragraph preceeding equation $(4.6)$: he claims that $L_p(1-k,\varepsilon)=L(1-k,\varepsilon\omega^{-k})$, showing the difference with Washington's.

  • Thanks, I think this is helpful but I need to think through it once more to be sure. – user404577 May 19 at 18:01
  • After thinking it through, I think I understand where I went wrong and also why Deligne-Ribet matches Washington. Thanks a lot! – user404577 Jun 2 at 15:37
  • By the way, would you know of a reference that does Deligne-Ribet in the case of Dirichlet characters? Washington isn't enough because they don't treat the case of a general conductor I think and I don't know of a published reference for the case of Kummer congruences with conductors of arbitrary conductor. – user404577 Jun 2 at 15:39
  • You mean the construction of the $p$-adic $L$-function for characters over $\mathbb{Q}$? What do you have in mind when you say that Washington doesn't treat it fully? – Filippo Alberto Edoardo Jun 2 at 16:17
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    The answer is yes if you restrict a bit the set of $k$'s, for instance to those prime to both conductors, and may be $p$. Just combine Theorem 4.2 with Proposition 4.1 in Washington, together with the interpolation property of $L_p(s,\chi)$. – Filippo Alberto Edoardo Jun 3 at 8:43

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