Serre's integrality question for the quotient zeta function $ζ_k(s)/ζ_Q(s)$ at negative odd integers

For a number field $k$ containing no subfield abelian over the rationals other than $Q$ itself, set $L(s)=ζ_k(s)/ζ_Q(s)$, the quotient of the Dedekind zeta function of $k$ by the Riemann zeta function. Serre asked (Séminaire Bourbaki exposé 399, 1970/71) whether the rational numbers $L(-1)$, $L(-3)$, …, $L(1-2m)$, … are all integers. The case $m=1$ is settled: $L(-1)=-12ζ_k(-1)$ is an integer divisible by $2^{r-1}$, where $r=[k:Q]$, proved via the $m=1$ case of Harder's Gauss–Bonnet formula for the Euler–Poincaré characteristic of $Sp_{2m}(O_k)$. The hypothesis is necessary, since abelian fields fail (e.g. $k=Q(√5)$ has $L(-1)=-2/5$). No proof or counterexample is recorded for $m≥2$, and the question appears to remain open as posed.

Background

For a number field $k$ (a finite-degree extension of the rational numbers $Q$), the Dedekind zeta function $ζ_k(s)$ is the Dirichlet series $∑1/N(a)^s$ taken over the non-zero integral ideals $a$ of the ring of integers $O_k$, where $N(a)$ is the absolute norm; it converges for $Re(s)>1$ and extends meromorphically to the whole plane. For $k=Q$ it is the Riemann zeta function $ζ_Q(s)$. A classical theorem of Siegel states that the values $ζ_k(1-2m)$ at negative odd integers ($m≥1$) are rational numbers, and the functional equation of $ζ_k$ forces $ζ_k(1-2m)=0$ unless $k$ is totally real (every embedding of $k$ into the complex numbers has real image); hence any integrality question about these values is non-trivial only for totally real $k$. When $k$ is abelian over $Q$, the zeta function factors as a finite product of Dirichlet L-functions $L(s,χ)$, and $L(1-n,χ)=-B_{n,χ}/n$ where $B_{n,χ}$ are generalized Bernoulli numbers attached to the Dirichlet character $χ$ (Leopoldt's formula); these typically carry denominators, so integrality fails in general — for instance $k=Q(√5)$ has $ζ_k(-1)/ζ_Q(-1)=-2/5$. In the relative-abelian setting a deep integrality theorem is known — Coates and Sinnott proved integrality properties of partial zeta values and Deligne and Ribet established the corresponding $p$-integrality for abelian L-functions over totally real base fields — but this machinery concerns abelian extensions, the setting excluded by the hypothesis of the question. For general (non-abelian) $k$ no such factorization exists, though Siegel's explicit formulas still permit exact computation and give bounds on the denominators; such computations have been carried out for totally real cubic fields (Halbritter–Pohst, and later Cheon–Kim–Lee for non-normal cubic fields). Alternatively $ζ_k$ factors into Artin L-series (the L-series attached to the irreducible complex representations of the Galois group of a normal closure of $k$), which gives a second exact-evaluation route. The cohomological input is Harder's Gauss–Bonnet theorem: the Euler–Poincaré characteristic $χ(Γ)$ of the symplectic group $Γ=Sp_{2m}(O_k)$ (the alternating sum of the dimensions of its rational cohomology) equals the product $ζ_k(-1)ζ_k(-3)⋯ζ_k(1-2m)$. Dividing the formula for $k$ by the formula for $Q$ isolates the products $∏_{j=1}^{m}L(1-2j)$ with $L(s)=ζ_k(s)/ζ_Q(s)$. The $m=1$ case underlies the known statement that $L(-1)=-12ζ_k(-1)$ is an integer divisible by $2^{r-1}$, $r=[k:Q]$; the same integrality appears in Bass's exposé on $K_2$ of global fields, where $K_2(O_F)$ denotes the second Quillen K-group of the ring of integers of $F$, and the integrality verifies cases of a Birch–Tate-type conjecture on the order of $K_2(O_F)$ for totally real $F$ (as recorded in that exposé). Since the hypothesis of the question excludes abelian $k$ (such a $k$ is itself an abelian extension of $Q$), the question asks whether removing the Riemann factor from $ζ_k$ also removes the Bernoulli-type denominators whenever $k$ has no abelian part.

Problem Statement

Let $k$ be a number field whose only subfield abelian over $Q$ is $Q$ itself, and let $L(s)=ζ_k(s)/ζ_Q(s)$ be the quotient of the Dedekind zeta function of $k$ by the Riemann zeta function. By Siegel's rationality theorem and the functional equation, each $L(1-2m)$, $m≥1$, is a rational number, and it equals 0 (hence is an integer) unless $k$ is totally real (every embedding of $k$ into the complex numbers has real image). The question, posed by Serre in Bourbaki exposé 399 (1970/71): is $L(1-2m)$ an integer for every $m≥1$ and every such $k$? The case $m=1$ is settled affirmatively: $L(-1)=-12ζ_k(-1)$ is an integer divisible by $2^{r-1}$, where $r=[k:Q]$, via the $m=1$ case of Harder's Gauss–Bonnet formula for the Euler–Poincaré characteristic of $Sp_{2m}(O_k)$. The open core is the case $m≥2$ for totally real $k$ with no abelian subfield other than $Q$: no proof of integrality, and no counterexample, appears in the literature.

References

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