von Staudt-Clausen for other special values

Question

The von Staudt-Clausen theorem expresses that the Bernoulli numbers' denominators have a very special form (see the wikipedia page on the theorem for more details).

What interests me is that those Bernoulli numbers have a link with the Riemann $\zeta$ function (also discussed in this other question) : $$\forall n>1, \zeta(1-n)=\frac{-B_n}n$$ (and through the functional equation, likewise to the right -- with an explicit factor with $\pi$ powers)

I interpret this as an example of an arithmetical function with an arithmeticity result for some special values, where a theorem gives results on denominators.

Here comes the question now the big picture is in place : there are also arithmeticity results for special values of $L$ functions of Dirichlet characters. Are there results similar to the von Staudt-Clausen theorem in this case? And for other $L$-functions for which arithmeticity results do exist (say, elliptic modular forms)?

Notice that I checked the proof of the theorem for the Bernoulli numbers, and the one I found (in Hardy&Wright's "Introduction to the theory of numbers") uses the definition of the sequence as coefficients of a series expansion... something which isn't applicable as far as I know in the setting(s) of my question.

Answer 1

The von Staudt-Clausen theorem indeed asserts that the denominators of Bernoulli numbers have special behaviour, but from a modern perspective what is even deeper are the so-called Kummer's congruences , whose proof normally relies upon von Staudt-Clausen.

What these congruences say is that evaluating the Riemann $\zeta$ function at integers that are $p$-adically close to each other, gives you integers which are again $p$-adically closed. In other terms, this asserts that if you want to define a function $\zeta_p:\mathbb{Z}_p\to\mathbb{Z}_p$ simply by insisting that it takes the same value of the usual Riemann $\zeta$ function on the negative integers, you can do this because it will be continuous (by Kummer) and then it will be unique, since $\mathbb{Z}_{<0}$ is dense in $\mathbb{Z}_p$.

The generalization you are looking for can then be restated as asking whether for each Dirichlet character $\chi$ we can find a continuous function $$ L_p(\chi,s):\mathbb{Z}_p\to\mathbb{Z}_p $$ which takes the same values as the complex function $L(\chi,s)$ when evaluated in a dense subset of $\mathbb{Z}_p$ consisting of integers (so that it make sense to compare the two functions). The answer is indeed affirmative, and this function is the so-called Kubota-Leopoldt $p$-adic $L$-function (eventually re-constructed by Iwasawa and Coleman). An extremely detailed, and still accessible, description of this is contained in Chapter 5 of Washington's Introduction to Cyclotomic Fields.

Indeed, much more is true. As BY points out in her/his comment, Deligne-Ribet (their unique joint work, I guess) and Cassou-Noguès (Inventiones Math., 1979) and Barsky proved the existence of a $p$-adic gadget $\zeta_{p,F}:\mathbb{Z}_p\to\mathbb{Z}_p$ which "interpolates" $p$-adically the special values of the Dedekind $\zeta_F$ function attached to any totally real number field $F$, and a similar statement was proved by Katz when replacing $F$ by a CM field. Both Deligne-Ribet's and Cassou-Noguès' or Barsky's proof of the existence of such a continous function heavily realies on proving some congruences satisfied by the special values of the complex functions under consideration, precisely in the same spirit as Kummer's congruences were crucial (or equivalent) to prove the continuity of the Kubota-Leopoldt $p$-adic $L$-function. These congruences are usually referred to as ``Coates' congruences'' and were stated by John Coates in the paper $p$-adic $L$-functions and Iwasawa theory appeared in the proceedings Algebraic Number Fields edited by A. Fröhlich in 1977 (and published by Academic Press).

Then, the whole theory saw a striking explosion, providing for $p$-adic analytic function interpolating the complex $L$-functions attached to many arithmetic objects, and you can look for instance at the complex $L$-function attached to an elliptic curve or, more generally, to a modular form. Such a $p$-adic analytic object indeed exists (it is the so-called Amice-Vélu,Visik,Mazur-Tate-Teitelbaum $p$-adic $L$-function) and you can regard its continuity as a generalization of Kummer's congruences and, hence, of von Staudt-Clausen. The literature is huge, but you can start by reading the comments at the end of Chapter 5 in Washington's book for some hints.

Answer 2

For an elliptic curve over $\mathbb{Q}$, the analogous question is what is the denominator of the algebraic part of $L(E,\chi,1)$ as $\chi$ varies in the Dirichlet characters.

First if $\chi$ is trivial, then the precise question is the denominator of $L(E,1)/\Omega^+$ where $\Omega^+$ is the least positive period of the Néron differential. If it is non-zero, then the only odd primes $p$ that can divide this denominator are those for which the Galois module $E[p]$ is reducible. But even those need not arise depending on which elliptic curve in the isogeny class is taken. BSD would tell you what the denominator is; some divisor of the torsion order.

Now for other twists the same is true, only those odd prime can divide the denominator (and very often the values are integers in fact). Since these values are integral sums of modular symbols it all boils down to study the denominator of modular symbols.

For more general modular forms one has to be careful when making the question precise. It depends a lot on what period is chosen and there may not always be a canonical best choice.

Answer 3

Leopold's article "Eine Verallgemeinerung der Bernoullischen Zahlen", in Abhandlungen aus dem mathematischen Seminar der Universität Hamburg, volume 22, year 1958 and pages 131--140, discusses Bernoulli numbers associated to Dirichlet characters, and discusses congruences and von Staudt-Closen theorems about them. It's short, clear and precise.

(I'm still looking for the equivalent for modular forms... so still not a full answer)