The Beilinson–Soulé vanishing conjecture and Soulé's conjecture on the order of vanishing of zeta functions of arithmetic schemes, via cyclic higher Chow groups

Audit of the open questions recorded in J. Park's preprint 'Algebraic cycles and Connes periodicity' (arXiv:math/0607272, 2006), which constructs cyclic higher Chow groups $CHC^p(X,n)$ through Connes periodicity and uses them to reformulate two famous conjectures: the Beilinson–Soulé vanishing conjecture ($H^i_{\mathcal{M}}(X,\mathbb{Z}(p))=0$ for $i<0$ on smooth quasi-projective varieties, equivalently $CH^p(X,n)=0$ for $n>2p$) and Soulé's conjecture predicting the order of vanishing of $\zeta_X(s)$ at $s=\dim X-r$ by alternating sums of ranks of higher Chow groups.

Background

Algebraic $K$-theory and the theory of motives assign to a scheme $X$ a family of groups meant to organize its algebraic cycles and cohomological invariants. For a smooth variety, Bloch's higher Chow groups $CH^p(X,n)$ (defined algebraically from families of codimension-$p$ algebraic cycles on $X\times\Delta^n$, where $\Delta^n$ is the algebraic $n$-simplex, modulo rational equivalence-type relations) agree with motivic cohomology $H^{2p-n}_{\mathcal{M}}(X,\mathbb{Z}(p))$, the conjectural 'universal cohomology' of algebraic varieties with integral coefficients twisted by weight $p$ (the identification used throughout the source paper is stated there as equation (4.174), following work of Suslin and Voevodsky). Two of the most influential predictions about these groups go back to the early 1980s. The Beilinson–Soulé vanishing conjecture asserts that for a smooth quasi-projective variety $X$ over a field there is no motivic cohomology in negative degrees: $H^i_{\mathcal{M}}(X,\mathbb{Z}(p))=0$ for all $i<0$, which is equivalent to $CH^p(X,n)=0$ for all $n>2p$. Soulé's zeta conjecture, in the reformulation by Bloch (Advances in Mathematics, 1986) used by the source paper, predicts that for a scheme $X$ of finite type over $\mathbb{Z}$ of dimension $d$ whose higher Chow groups are defined, finite-rank, and rationally vanish in high degree, the order of vanishing of the zeta function $\zeta_X(s)$ (the Dirichlet-type generating series counting points of $X$ modulo varying primes) at the integer point $s=d-r$ is given by the alternating sum of ranks $-\operatorname{ord}_{s=d-r}\zeta_X(s)=\sum_i(-1)^i\,\operatorname{rk}(CH^r(X,i))$. Both conjectures are wide open in their stated generality and are cornerstones of the expected structure of motivic cohomology, its relation to Quillen's algebraic $K$-theory (obtained from $K$-theory of vector bundles through a spectrum built by Quillen), and the special values of zeta functions.

The source paper (Park, arXiv:math/0607272, a preprint based on the author's Chicago thesis) imports Connes' technique of cyclic objects: just as the Hochschild complex of an algebra carries a cyclic structure producing cyclic homology and the Connes periodicity long exact sequence linking Hochschild and cyclic homology through the periodicity operator $S$, Bloch's higher Chow complex carries such a structure, yielding cyclic higher Chow groups $CHC^p(X,n)$ and a Connes-type long exact sequence $\cdots\to CHC^p(X,n-1)\to CH^p(X,n)\to CHC^p(X,n)\to CHC^p(X,n-2)\to\cdots$. Park proves (Theorem 4.10) that the Beilinson–Soulé vanishing conjecture is equivalent to concrete periodicity properties of the groups $CHC^p(X,\cdot)$: the shift map $S$ must be an isomorphism $CHC^p(X,n+1)\to CHC^p(X,n-1)$ for all $n>2p$ and injective on $CHC^p(X,2p+1)$. Under a rank-boundedness hypothesis (a weak form of Beilinson–Soulé), Soulé's conjecture is likewise equivalent (Theorem 4.13) to a two-term rank formula for cyclic higher Chow groups. The paper also records two structural open problems: there is no known cyclic analogue of Quillen's higher $K$-theory (needed to transplant Chern classes, the Gersten conjecture — the conjectured description of $K$-theory of a variety through its points of various codimensions — and the projective bundle theorem (the computation of the $K$-theory or cohomology of $X\times\mathbb{P}^m$ in terms of $X$) into the cyclic setting), and the definability plus finite-rank behavior of integral higher Chow groups $CH^r(X,i)$ for schemes of finite type over $\mathbb{Z}$ was, at the time, a hypothesis rather than a theorem.

Problem Statement

Decide, in the generality stated by the source, the two jointly reformulated conjectures on motivic cohomology, where 'decide' means prove or refute each one:

(1) Beilinson–Soulé vanishing (Park, Conjecture 4.9; equivalently his Theorem 4.10). Let $X$ be a smooth quasi-projective variety over a field. Is it true that for all weights $p\ge 0$ and all $i<0$, the motivic cohomology $H^i_{\mathcal{M}}(X,\mathbb{Z}(p))$ vanishes — equivalently, that the higher Chow group $CH^p(X,n)$ vanishes for all $n>2p$ — equivalently, that the cyclic higher Chow groups satisfy $S:CHC^p(X,n+1)\xrightarrow{\sim}CHC^p(X,n-1)$ for all $n>2p$ together with injectivity of $S:CHC^p(X,2p+1)\to CHC^p(X,2p-1)$ for every $p$?

(2) Soulé's zeta conjecture (Park, Conjecture 4.12, in Bloch's reformulation; equivalently his Theorem 4.13). Let $X$ be a scheme of finite type over $\mathbb{Z}$ with $d=\dim X$, under the hypotheses with which the source states the conjecture: the integral higher Chow groups $CH^r(X,i)$ are defined, of finite rank, and rationally zero for $i\gg 0$. Is it true that for every $r$, $-\operatorname{ord}_{s=d-r}\zeta_X(s)=\sum_i(-1)^i\,\operatorname{rk}(CH^r(X,i))$, equivalently (Theorem 4.13) — under the hypothesis that $\operatorname{rk}(CH^r(X,n))=0$ for all $n\ge N$ for some integer $N$, a weak form of (1) — that $\operatorname{ord}_{s=d-r}\zeta_X(s)=(-1)^N\,\operatorname{rk}(CHC^r(X,N-1))+(-1)^{N+1}\,\operatorname{rk}(CHC^r(X,N))$ (for $N=2r+1$, as noted in the source's Remark 4.14, eq. (4.178), this reads $\operatorname{ord}_{s=d-r}\zeta_X(s)=-\operatorname{rk}(CHC^r(X,2r))+\operatorname{rk}(CHC^r(X,2r+1))$)?

The common objective is to determine whether motivic cohomology of varieties possesses exactly the vanishing and rank structure predicted by the two conjectures, with the cyclic higher Chow formulations of the source as equivalent forms; the two parts are logically intertwined through Theorem 4.13, whose hypothesis is a weak form of (1). The source's further structural question — existence of a cyclic analogue of Quillen's higher $K$-theory suitable for the cyclic higher Chow setting — is an adjacent open direction and is not required for completion of (1)–(2).

References

1. Jinhyun Park, "Algebraic cycles and Connes periodicity", arXiv:math/0607272 (2006), https://arxiv.org/abs/math/0607272
2. Spencer Bloch, "Algebraic cycles and higher K-theory", Advances in Mathematics 61 (1986) 267–304, DOI 10.1016/0001-8708(86)90081-2, https://doi.org/10.1016/0001-8708(86)90081-2
3. Baptiste Morin, "Zeta functions of regular arithmetic schemes at s=0", Duke Mathematical Journal 163 (2014), no. 7, DOI 10.1215/00127094-2681387, arXiv:1103.6061, https://arxiv.org/abs/1103.6061
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