Two-primary comparison of even algebraic K-groups of arithmetically equivalent number fields with real primes

For arithmetically equivalent number fields (fields sharing the same Dedekind zeta function), the odd-degree Quillen K-groups of the rings of integers are isomorphic, and the p-primary parts of the even-degree K-groups are isomorphic for every odd prime p outside an explicit finite exceptional set; at the prime 2 this is proved only when the fields are totally imaginary. The open problem, stated explicitly as a natural conjecture in Komiya (arXiv:2607.26685), is the remaining case: whether the 2-primary parts of the even K-groups of two number fields with real embeddings must be isomorphic whenever the fields have isomorphic 2-adic permutation modules (as holds for locally integrally equivalent fields, and for arithmetically equivalent fields whose arithmetic-equivalence defect integer $\nu$ is odd). The obstruction is that at the prime 2 the Quillen–Lichtenbaum identification of K-groups with étale cohomology acquires correction terms coming from the real primes. The problem is to prove or refute this comparison without any total-imaginary hypothesis.

Background

A number field $F$ has a ring of integers $\mathcal{O}_F$, and its Dedekind zeta function $\zeta_F(s)$ is the Euler product over the nonzero prime ideals of $\mathcal{O}_F$. Two number fields $k_1,k_2$ are called arithmetically equivalent if $\zeta_{k_1}(s)=\zeta_{k_2}(s)$ as complex functions; they need not be isomorphic. Perlis showed that $\zeta_{k_1}=\zeta_{k_2}$ if and only if, writing $L$ for a common finite Galois closure over $\mathbb{Q}$ with $G=\mathrm{Gal}(L/\mathbb{Q})$ and $H_i=\mathrm{Gal}(L/k_i)$, the rational permutation representations are isomorphic: $\mathbb{Q}[G/H_1]\cong\mathbb{Q}[G/H_2]$ (here $G/H_i$ is the set of left cosets and $\mathbb{Q}[G/H_i]$ the permutation module, i.e., the vector space with basis $G/H_i$ on which $G$ acts by left translation). Arithmetic equivalence forces equality of the degree, the discriminant, and the numbers $r_1$ of real embeddings and $r_2$ of complex pairs of embeddings. It also forces the $p$-primary parts of the ideal class groups $\mathrm{Cl}(k_i)$ to be isomorphic for every prime $p$ not dividing an explicit integer $\nu_{k_1k_2}$: by definition, $\nu_{k_1k_2}$ is the smallest positive integer such that $\mathbb{Z}_p[G/H_1]\cong\mathbb{Z}_p[G/H_2]$ holds as $\mathbb{Z}_p[G]$-modules exactly for the primes $p\nmid\nu_{k_1k_2}$, and one always has $\nu_{k_1k_2}\mid[L:k_1]$; the full class groups need not be isomorphic (de Smit–Perlis).

Quillen's algebraic K-groups $K_n(\mathcal{O}_F)$ generalize these classical invariants: $K_0(\mathcal{O}_F)\cong\mathbb{Z}\oplus\mathrm{Cl}(F)$ and $K_1(\mathcal{O}_F)\cong\mathcal{O}_F^\times$, while for $n>0$ the even group $K_{2n}(\mathcal{O}_F)$ is a finite abelian group that behaves like a higher analogue of the class group, and $K_{2n+1}(\mathcal{O}_F)$ like a higher analogue of the unit group. For an abelian group $A$, $A[p^\infty]$ denotes its $p$-primary torsion part. The Quillen–Lichtenbaum conjecture, now a theorem (for odd $p$ by the Rost–Voevodsky theorem on the norm residue isomorphism, and at the prime 2 by Rognes–Weibel using Voevodsky's results), identifies $K_{2n}(\mathcal{O}_F)[p^\infty]$ with the second étale cohomology group $H^2(\mathrm{Spec}\,\mathcal{O}_F[1/p],\mathbb{Z}_p(n+1))$; étale cohomology is the cohomology theory for schemes that plays the role of singular cohomology, and $\mathbb{Z}_p(n+1)$ is the $n+1$-st twisted-constant $p$-adic coefficient system built from roots of unity. At the prime 2 this identification holds as stated when $F$ is totally imaginary (has no real embedding, $r_1=0$); when $F$ has real primes, the real places contribute additional $2$-torsion (for example, a field with exactly one prime ideal above $2$ and trivial narrow class group — the narrow class group being the class group modulo principal ideals generated by elements positive at every real embedding — is called 2-regular, and for 2-regular fields one has $K_{2n}(\mathcal{O}_F)[2^\infty]\cong(\mathbb{Z}/2)^{r_1}$ for $n\equiv 1\pmod 4$ and $0$ otherwise, for $n>0$), a correction encoded in the general structure theorem of Weibel's Handbook of K-Theory chapter.

Komatsu proved in 1990 that arithmetically equivalent fields have isomorphic $p$-primary parts $K_n(\mathcal{O}_{k_1})[p^\infty]\cong K_n(\mathcal{O}_{k_2})[p^\infty]$ for all $n$ and all primes $p\nmid[L:k_1]$. Komiya (2026) introduced the equivalence relations called K-equivalence ($K_n(\mathcal{O}_{k_1})\cong K_n(\mathcal{O}_{k_2})$ for all $n$) and almost K-equivalence (isomorphism of $p$-primary parts for almost all primes $p$ independent of $n$), and, working with continuous étale cohomology (Jannsen's variant of étale cohomology adapted to inverse systems of coefficient sheaves) and coinduced Galois modules, proved: if $\mathbb{Z}_p[G/H_1]\cong\mathbb{Z}_p[G/H_2]$ as $\mathbb{Z}_p[G]$-modules for an odd prime $p$ (a condition implied by local integral equivalence, i.e., by such isomorphisms for every $p$, and by arithmetic equivalence whenever $p\nmid\nu_{k_1k_2}$), then $K_{2n}(\mathcal{O}_{k_1})[p^\infty]\cong K_{2n}(\mathcal{O}_{k_2})[p^\infty]$ for all $n\ge 0$; the same conclusion at $p=2$ is proved only under the additional hypothesis that at least one of $k_1,k_2$ is totally imaginary. Consequently arithmetic equivalence implies almost K-equivalence, local integral equivalence plus total-imaginary implies K-equivalence, and for totally real fields arithmetic equivalence, almost K-equivalence, and the condition that $|K_n(\mathcal{O}_{k_1})[p^\infty]|=|K_n(\mathcal{O}_{k_2})[p^\infty]|$ for almost all $p$ are all equivalent (using the Lichtenbaum special-value formulae relating $\zeta_F$ at negative integers to orders of even K-groups, and Oh's theorem that these special values reconstruct $\zeta_F$). The gap between the odd-prime theory and the prime 2 for fields with real embeddings is the subject of this problem.

Problem Statement

Let $k_1,k_2$ be number fields contained in a finite Galois extension $L/\mathbb{Q}$, and set $G=\mathrm{Gal}(L/\mathbb{Q})$ and $H_i=\mathrm{Gal}(L/k_i)$. Suppose the permutation modules are isomorphic over the $2$-adic integers,

$$\mathbb{Z}_2[G/H_1]\;\cong\;\mathbb{Z}_2[G/H_2]\quad\text{as }\mathbb{Z}_2[G]\text{-modules},$$

with no assumption that $k_1,k_2$ are totally imaginary. This hypothesis holds for locally integrally equivalent fields, and for arithmetically equivalent fields — those with $\zeta_{k_1}=\zeta_{k_2}$ — whenever $2\nmid\nu_{k_1k_2}$; it already implies arithmetic equivalence, since an integral $\mathbb{Z}_2[G]$-isomorphism of permutation modules forces equal permutation characters and hence equal zeta functions by Perlis's criterion, and arithmetic equivalence forces $r_1(k_1)=r_1(k_2)$, so both fields have real primes simultaneously. Prove or disprove: for every integer $n\ge 1$ there is an isomorphism of the $2$-primary parts of the even Quillen K-groups of the rings of integers,

$$K_{2n}(\mathcal{O}_{k_1})[2^\infty]\;\cong\;K_{2n}(\mathcal{O}_{k_2})[2^\infty].$$

The case $n=0$ (i.e., the $2$-primary parts of the class groups) is already Perlis's theorem under exactly this hypothesis, the analogue for every odd prime $p$ with $\mathbb{Z}_p[G/H_1]\cong\mathbb{Z}_p[G/H_2]$ is Komiya's Theorem 4.2, and the case $p=2$ when at least one of $k_1,k_2$ is totally imaginary is settled in the same theorem; the open content is precisely the prime 2 for fields with real embeddings. A disproof may consist of a single explicit pair of number fields satisfying the hypothesis together with one $n\ge 1$ for which the two finite abelian groups $K_{2n}(\mathcal{O}_{k_i})[2^\infty]$ are not isomorphic.

References

1. Ryo Komiya, "On the higher algebraic $K$-groups of arithmetically equivalent number fields", arXiv:2607.26685 (29 July 2026), https://arxiv.org/abs/2607.26685
2. Robert Perlis, "On the equation $\zeta_k(s)=\zeta_{k'}(s)$", Journal of Number Theory 9 (1977), 342–360, DOI 10.1016/0022-314X(77)90070-1, https://doi.org/10.1016/0022-314X(77)90070-1
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