In their appendix to Deligne's paper "Valeurs de fonctions L et périodes d'intégrales" (PSPM 33, 1979), Koblitz and Ogus prove that functions $N^{-1}\mathbf{Z}/\mathbf{Z}-\{0\}\to \mathbf{Q}$ verifying certain relations are spanned by very specific ones; in the remark at the end of this appendix (p. 345 of the article), they say that Kubert has obtained an integral refinement of this result. I looked for it in Kubert's papers but could not find it there: maybe it is put in a different formulation. Does anyone know?
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We need the result you mention in work we're currently doing together with Heidi Goodson and my PhD student Andrea Gallese. We've also been unable to locate this precise result anywhere in Kubert's paper. However, deducing the statement from Kubert's work is not too hard; I try to sketch a proof here (hopefully it will also appear soon-ish in our preprint).
The functions $f : (\frac{1}{N}\mathbb{Z}/\mathbb{Z} \setminus \{0\}) \to \mathbb{Q}$ in Bruno's question are called distributions. Let $\mathbb{U}$ be the universal distribution group, namely the quotient $\mathbb{A}/\mathbb{D}$, where $\mathbb{A}=\mathbb{Z}[(\frac{1}{N}\mathbb{Z}/\mathbb{Z} \setminus \{0\})]$ and $\mathbb{D}$ is the module of distribution relations (these are essentially the `very special' functions in Bruno's question). Let $\mathbb{U}^-$ be the quotient through which all odd distributions factor (a distribution is odd if $f([-a])=-f([a])$ for all $a$). Concretely, $\mathbb{U}^- = \mathbb{A} / \langle \mathbb{D}, [a]+[-a] : a \in \frac{1}{N}\mathbb{Z}/\mathbb{Z} \setminus \{0\} \rangle$.
The group $\mathbb{U}$ is a $(\mathbb{Z}/N\mathbb{Z})^\times$-module, and in particular a $\{\pm 1\}$-module. Using Section 1 of Kubert's paper [1], one shows that $\mathbb{U}^-_{\operatorname{tors}} \cong H^2( \{\pm 1\}, \mathbb{U})$. It follows that all the torsion in $\mathbb{U}^-$ is killed by $2$.
Now suppose that $f$ is a distribution for which $\langle f \rangle$ is constant. Koblitz and Ogus (in the appendix to Deligne's paper) show that some integer multiple of $f$ lies in the submodule $\langle \mathbb{D}, [a]+[-a] : a \in \frac{1}{N}\mathbb{Z}/\mathbb{Z} \setminus \{0\}\rangle$. In particular, the class of $f$ is torsion in $\mathbb{U}^-$. By what we already showed, $2f$ is in $\langle \mathbb{D}, [a]+[-a] : a \in \frac{1}{N}\mathbb{Z}/\mathbb{Z} \setminus \{0\}\rangle$, as desired.
[1] The $\mathbb{Z}/2\mathbb{Z}$ cohomology of the universal ordinary distribution. Bull. Soc. Math. France, 107(2):203–224, 1979
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$\begingroup$ Thanks a lot! In turn, we need this result in a joint work with Noriyuki Otsubo. $\endgroup$ Commented Apr 10 at 16:51
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$\begingroup$ The preprint is now on the ArXiv: arxiv.org/abs/2405.20394. The statements concerning distributions are in §6.3. $\endgroup$ Commented Jun 14 at 14:25
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Probably D. Kubert, The universal ordinary distribution, Bull. Soc. Math. Fr 107 (1979), 179-202.
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$\begingroup$ Thank you; I had consulted it. Can you point out where? $\endgroup$ Commented Mar 28 at 23:14