Existence of Burns–de Jeu–Gangl special elements in odd-degree algebraic K-groups for general Artin characters

For a finite Galois extension F/k of number fields with group G, a strictly negative integer r, and an irreducible character χ of G whose contragredient Artin L-function vanishes to order exactly one at s=r, Burns, de Jeu and Gangl conjectured (Conjecture 1.2, Advances in Mathematics 2012) that for every embedding σ of F into C there is an element ε_σ(χ,S) in the coefficients-tensored K-group O⊗Z K_{1−2r}(F) whose Borel regulator values are explicit multiples of the first derivatives L'_S(r, χ̌^γ) of the S-truncated Artin L-functions. Apart from abelian characters over Q (cyclotomic elements) and a conditional derivation from the equivariant Tamagawa number conjecture, no construction or existence proof is known; in the rank-one imaginary-quadratic case Deninger proved Gross's ratio-level conjecture and Levin's dilogarithm constructions produce candidate elements, but without the denominator control that Conjecture 1.2 requires, so this case remains open as well. The conjecture is open for general characters, in particular irreducible non-abelian ones.

Background

Algebraic K-groups K_n(F) of a number field F, introduced by Quillen, generalize the unit group (K_1) and the class group-related groups; for strictly negative integers r the group K_{1−2r}(F) is a finitely generated abelian group in odd degree (K_3 for r=−1, K_5 for r=−2, ...). Its free part is measured by the Borel (Beilinson) regulator, a canonical homomorphism reg_{1−r} from K_{1−2r}(F) to a real vector space spanned by the complex embeddings σ: F→C; in weight two (K_3) the regulator is computed by the Bloch–Wigner dilogarithm on the Bloch group, the universal receiver for the five-term dilogarithm relation. Borel's theorem expresses the leading term of the Dedekind zeta function ζ_F(s) at s=r through this regulator, and the Lichtenbaum conjecture relates |ζ_F(r)| to the orders of K-groups; the torsion of K_3(O_F) is the classical tame kernel.

Stark's conjecture predicts, for an Artin L-function L(s,χ) (the L-function attached to a finite-dimensional complex representation of G=Gal(F/k) with character χ) vanishing to order one at s=0, that the first derivative divided by a regulator of S-units is rational in a suitable sense; Tate's integral refinement predicts a specific unit. Gross (2005) formulated the analogue in which the unit group is replaced by K_{1−2r}(F): the ratio of L'(r,χ̌) (χ̌ the contragredient, i.e. dual-representation, character) to a K_{1−2r}(F)-regulator determinant should be algebraic and Galois-equivariant under the action of Γ=Gal(E/Q), where E⊇Q(χ) is a finite Galois coefficient field with ring of integers O. Chinburg, Kolster, Pappas and Snaith (1998) stated a finer 'modified Lichtenbaum–Gross conjecture' predicting that this ratio is the inverse of an explicit fractional ideal built from Fitting ideals (the determinantal ideals attached to a presentation matrix of a module); Burns, de Jeu and Gangl (2012) showed it is equivalent to the relevant case of the equivariant Tamagawa number conjecture (ETNC) of Burns–Flach, which predicts the vanishing of an equivariant Tamagawa number TΩ(h^0(Spec F)(r), Z[G]) in a relative K-group.

Burns, de Jeu and Gangl went beyond these ratio-conjectures and predicted the actual arithmetic objects: their Conjecture 1.2 asserts the existence, for each σ in the set Σ_F of embeddings F→C and each finite set S of places of k containing the archimedean ones, of an element ε_σ(χ,S) in O⊗Z K_{1−2r}(F) whose regulator values under all σ'∈Σ_F and all γ∈Γ are prescribed explicitly in terms of L'S(r,χ̌^γ). Here reg{1−r,σ'} applies the regulator after embedding K_{1−2r}(F) into K_{1−2r}(C) via σ', w_r(χ) is the standard archimedean period factor (a factor given by the number of cyclotomic-twisted roots of unity in an invariant subfield, raised to the power of the character degree), and c^χ_{σ',σ} is an explicit coefficient depending only on archimedean data (complex conjugation τ_σ and the parity (−1)^r). An analysis of Gamma factors and the functional equation shows the conjecture has content exactly when either k has exactly one complex place and χ(1)=1, or k is totally real and χ(1)+(−1)^rχ(τ_σ)=2 for one G-orbit of embeddings while χ(1)+(−1)^rχ(τ_{σ'})=0 on the complementary orbit; in the totally real case χ may be irreducible and non-abelian. The elements ε_σ(χ,S), unique up to torsion, are higher-K-theoretic analogues of Stark units; via Theorem 3.1(ii) of the same paper they conjecturally yield annihilators of the p≠2 parts of étale cohomology groups H²_ét(Spec(O_{F,S}[1/p]), Z_p(1−r)) (which the Quillen–Lichtenbaum conjecture identifies with K_{−2r}(O_{F,S})), paralleling the annihilation of class groups by cyclotomic units.

Known cases: for k=Q and χ(1)=1 the conjecture holds via the cyclotomic elements in K-theory and the Burns–Greither proof of ETNC for abelian extensions of Q (2-primary part corrected by Flach); for k imaginary quadratic and χ(1)=1 Gross's conjecture was proved by Deninger and dilogarithm constructions exist (Levin), but without control of denominators. Theorem 3.1 of the paper proves Conjecture 1.2 conditionally on the modified Lichtenbaum–Gross conjecture for χ. Numerical evidence (k=Q, r=−1) covers two dihedral cases, a tetrahedral case with the standard representation of S_4, and a Tate–Buhler–Chinburg representation, verified to over one hundred decimal places. For general characters, no theoretical construction is known; the authors state explicitly that they are 'not aware of any theoretical constructions that can account for the existence of the elements ... for other classes of characters χ'.

Problem Statement

Prove, or disprove, Conjecture 1.2 of Burns–de Jeu–Gangl in full generality. Let $F/k$ be a finite Galois extension of number fields with group $G$, let $\chi$ be an irreducible character of $G$ that is realizable over a finite Galois extension $E\supseteq\mathbb{Q}(\chi)$ of $\mathbb{Q}$, with ring of integers $\mathcal{O}$ and Galois group $\Gamma=\mathrm{Gal}(E/\mathbb{Q})$, let $S$ be a finite set of places of $k$ containing the archimedean ones, and let $r$ be a strictly negative integer such that $L_S(r,\check{\chi})=0\neq L'_S(r,\check{\chi})$, where $\check{\chi}$ denotes the contragredient (dual-representation) character and $L_S(s,\psi)$ the $S$-truncated Artin $L$-function of $\psi$. Write $\Sigma_F$ for the set of embeddings $\sigma:F\to\mathbb{C}$. For every $\sigma\in\Sigma_F$, construct (or prove the existence of) an element

$\epsilon_\sigma(\chi,S)\;\in\;\mathcal{O}\otimes_{\mathbb{Z}} K_{1-2r}(F)$

such that for every $\sigma'\in\Sigma_F$ and every $\gamma\in\Gamma$,

$(2\pi i)^{r}\,\mathrm{reg}_{1-r,\sigma'}\big(\epsilon_\sigma(\chi,S)^{\gamma}\big)\;=\;w_r(\chi)\,\gamma\big(c^{\chi}_{\sigma',\sigma}\big)\,L'_S\big(r,\check{\chi}^{\,\gamma}\big),$

where the ingredients are defined exactly as in the source: $\Gamma$ acts semi-linearly on $\mathcal{O}\otimes_{\mathbb{Z}}K_{1-2r}(F)$ via $(e\otimes x)^{\gamma}=\gamma(e)\otimes x$ for $e\in\mathcal{O}$ and $x\in K_{1-2r}(F)$ (this is the superscript action $\epsilon_\sigma(\chi,S)^{\gamma}$ above), and it acts on the $E$-valued quantities $c^{\chi}_{\sigma',\sigma}$ and $\check{\chi}^{\,\gamma}$ through its action on $E$; $\mathrm{reg}_{1-r}$ is the Borel regulator on $K_{1-2r}(\mathbb{C})$ and $\mathrm{reg}_{1-r,\sigma'}(e\otimes x)=e\cdot\mathrm{reg}_{1-r}(\sigma'(x))$ for $e\in\mathcal{O}$, $x\in K_{1-2r}(F)$, with $\sigma'$ also denoting the induced map $K_{1-2r}(F)\to K_{1-2r}(\mathbb{C})$; $\tau_\sigma$ is the generator of the decomposition subgroup in $G$ of the place of $F$ corresponding to $\sigma$ (the complex conjugation relative to $\sigma$), and $\tau$ denotes complex conjugation acting on $\Sigma_F$ by post-composition; $w_r(\chi)=w_{1-r}(F^{\ker(\chi^{\gamma})})^{\chi^{\gamma}(1)}$, a value independent of the choice of $\gamma\in\Gamma$ (here $w_a(F)=|\mu_a(F)|$ and $\mu_a(F)=H^{0}(G_{\mathbb{Q}^{c}/F},\mathbb{Q}/\mathbb{Z}(a))$ is the finite module of the $a$-th cyclotomic twist of $\mathbb{Q}/\mathbb{Z}$, with $G_{\mathbb{Q}^{c}/F}$ acting through the $a$-th power of the cyclotomic character); and

$c^{\chi}_{\sigma',\sigma}\;=\;\begin{cases}\check{\chi}(g)+(-1)^{r}\,\check{\chi}(g\tau_\sigma) & \text{if } \sigma(k)\subset\mathbb{R} \text{ and } \sigma'=g(\sigma) \text{ for some } g\in G,\\ \check{\chi}(g) & \text{if } \sigma(k)\not\subset\mathbb{R} \text{ and } \sigma'=g(\sigma) \text{ for some } g\in G,\\ (-1)^{r}\,\check{\chi}(g) & \text{if } \sigma(k)\not\subset\mathbb{R} \text{ and } \sigma'=\tau\circ g(\sigma) \text{ for some } g\in G,\\ 0 & \text{otherwise.}\end{cases}$

By Remark 1.5 of the source the conjecture has content exactly in the situations (i) $k$ has exactly one complex place, $\chi(1)=1$ and $\chi(1)+(-1)^r\chi(\tau_{\sigma'})=0$ for all $\sigma'\in\Sigma_F$ with $\sigma'(k)\subset\mathbb{R}$, or (ii) $k$ is totally real and $\chi(1)+(-1)^r\chi(\tau_\sigma)=2$ for one $\sigma\in\Sigma_F$ while $\chi(1)+(-1)^r\chi(\tau_{\sigma'})=0$ for all $\sigma'$ outside its $G$-orbit — in particular irreducible non-abelian characters over totally real base fields $k$ (e.g. the dihedral and tetrahedral-type characters for which numerical evidence exists) and rank-one characters over base fields that are neither $\mathbb{Q}$ nor imaginary quadratic. The elements must genuinely lie in $\mathcal{O}\otimes_{\mathbb{Z}} K_{1-2r}(F)$, so that integrality and denominator control are part of the claim. The cases already settled in the source are $k=\mathbb{Q}$ with $\chi(1)=1$ (cyclotomic elements, Corollary 3.2 of the source) and the case covered by the conditional Theorem 3.1 of the source (validity under the modified Lichtenbaum–Gross conjecture). For $k$ imaginary quadratic with $\chi(1)=1$, Deninger's theorem settles Gross's ratio-level conjecture, but the existence of elements $\epsilon_\sigma(\chi,S)$ satisfying the displayed identity — with the denominator control that Levin's dilogarithm constructions lack — remains open. A disproof would consist of a specific $F/k$, $r$, $\chi$, $S$ satisfying the order-one vanishing hypothesis above for which no element of $\mathcal{O}\otimes_{\mathbb{Z}} K_{1-2r}(F)$ satisfies the displayed regulator identity for all $\sigma'\in\Sigma_F$ and $\gamma\in\Gamma$.

References

1. David Burns, Rob de Jeu, Herbert Gangl, On special elements in higher algebraic K-theory and the Lichtenbaum–Gross Conjecture, Advances in Mathematics 230 (2012), 1502–1529, DOI 10.1016/j.aim.2012.03.014, https://doi.org/10.1016/j.aim.2012.03.014 (arXiv:1101.5477, https://arxiv.org/abs/1101.5477)
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