Soulé's conjectured logarithmic bound for torsion in the K-theory of rings of integers

For a number field $F$ of degree $d$ with ring of integers $\mathcal{O}_F$ and absolute discriminant $D_F$, Quillen proved that each algebraic K-theory group $K_n(\mathcal{O}_F)$ is finitely generated, and Borel computed its rank; but the size of the torsion subgroup as $F$ varies is poorly understood. Soulé (2003) proved huge explicit upper bounds and conjectured the optimal shape: there should exist a constant $C(n,d)$ such that $\log\,\mathrm{card}\,K_n(\mathcal{O}_F)_{\mathrm{tors}} \le C(n,d)\,\log|D_F|$ for all number fields $F$ of degree $d\ge 2$ — equivalently, the torsion order should grow at most polynomially in the discriminant, with exponent depending only on $n$ and $d$. The best proven bounds (Bayer-Fluckiger–Emery–Houriet 2015, refining Soulé's method) still give only $\log\,\mathrm{card}\,K_n(\mathcal{O}_F)_{\mathrm{tors}} \le (nd)^{\alpha}|D_F|^{\beta}$ up to small torsion, a polynomial-in-$|D_F|$ bound for the logarithm itself. The problem asks for a proof or refutation of Soulé's conjectured logarithmic rate, in its primary-source form $\mathrm{card}\,K_m(\mathcal{O}_F)_{\mathrm{tors}} \le \alpha\exp(\beta m^{\gamma}\log D_F)$ with $\gamma$ expected independent of the degree. The case $F=\mathbb{Q}$ ($|D_F|=1$) is degenerate and excluded; the conjecture governs fields of each fixed degree $d\ge 2$.

Background

A number field $F$ is a finite extension of $\mathbb{Q}$; its ring of integers $\mathcal{O}_F$ is the subring of elements satisfying monic integer polynomial equations. The absolute discriminant $D_F$ is a positive integer measuring the arithmetic complexity of $F$ (it grows, for example, with the ramification of primes). For $n\ge 0$, Quillen's K-group $K_n(\mathcal{O}_F)$ is a finitely generated abelian group attached to $\mathcal{O}_F$ via an infinite loop space construction; its rank (the number of free generators) was computed by Borel, so the quantitative mystery is the torsion subgroup $K_n(\mathcal{O}_F)_{\mathrm{tors}}$. For $n=0$ this torsion is the ideal class group of $F$, whose order satisfies the classical bound $h(F)\le\alpha(r)\sqrt{D_F}\log^{r-1}(D_F)$ for fields of degree $r$ — that is, $\log h(F) \le C(r)\log D_F$, exactly the type of behavior conjectured in higher degrees. For $F=\mathbb{Q}$ and $n=2k-2$, the Lichtenbaum conjecture (a theorem in this setting, relating $\mathrm{card}\,K_{2k-2}(\mathbb{Z})_{\mathrm{tors}}$ to numerators of Bernoulli numbers) predicts $\mathrm{card}\,K_{2k-2}(\mathbb{Z})_{\mathrm{tors}}$ exponential in $k$ but independent of any discriminant. These two anchors led Soulé to conjecture (Sect. 5.1 of his 2003 paper) that in general $\mathrm{card}\,K_m(\mathcal{O}_F)_{\mathrm{tors}} \le \alpha\exp(\beta m^{\gamma}\log D_F)$, i.e. $\log$ of the order at most a constant times $\log D_F$ for fixed degree, with $\gamma$ expected not to depend on the degree. Proving upper bounds is hard because $K_n(\mathcal{O}_F)$ is related through the Hurewicz map to the integral homology of $GL_N(\mathcal{O}_F)$, which Soulé studied via Ash's well-rounded retract — a contractible cell complex for the symmetric space of $GL_N$ on which $GL_N(\mathcal{O}_F)$ acts with compact quotient — together with a lemma of Gabber bounding torsion in homology of a group action by the number of cells; the cell count is governed by the geometry of numbers of hermitian lattices over $\mathcal{O}_F$ (lattices equipped with a $\mathcal{O}_F$-valued hermitian form). This method controls $K_n$ only up to small torsion: writing $\mathrm{card}_{\ell}(A)$ for the order of the quotient of a finite abelian group $A$ by its subgroup generated by elements of order $\le\ell$, all current bounds bound $\mathrm{card}_{\ell} K_n(\mathcal{O}_F)_{\mathrm{tors}}$, not the full torsion.

Problem Statement

Prove or refute Soulé's conjecture on the torsion in the K-theory of rings of integers: for every integer $n\ge2$ and every degree $d\ge2$, does there exist a constant $C(n,d)$ such that

$\log\,\mathrm{card}\,K_n(\mathcal{O}_F)_{\mathrm{tors}} \le C(n,d)\,\log|D_F|$

for every number field $F$ of degree $d$ over $\mathbb{Q}$, with ring of integers $\mathcal{O}_F$ and absolute discriminant $D_F$? Equivalently, is $\mathrm{card}\,K_n(\mathcal{O}_F)_{\mathrm{tors}}\le|D_F|^{C(n,d)}$ uniformly over all such $F$? The primary-source form of the conjecture (Soulé, 2003, Sect. 5.1) is stronger: for each fixed degree there should exist positive constants $\alpha,\beta,\gamma$ with

$\mathrm{card}\,K_m(\mathcal{O}_F)_{\mathrm{tors}} \le \alpha\,\exp\bigl(\beta\,m^{\gamma}\,\log D_F\bigr)$

for all $m\ge2$ and all number fields $F$ of that degree, with $\gamma$ expected to be independent of the degree. The uniform-in-$m$ form implies the fixed-$(n,d)$ bound, and a refutation of the fixed-$(n,d)$ form — fields of degree $d$ whose torsion orders grow faster than $|D_F|^{C}$ for every constant $C$ — refutes the uniform-in-$m$ form as well. The two forms are not equivalent: the fixed-$(n,d)$ bound does not by itself imply the uniform-in-$m$ form, which for some fixed degree $d$ may fail even when no such fields exist. The degree-$1$ case $F=\mathbb{Q}$ is excluded as degenerate: $|D_{\mathbb{Q}}|=1$ makes $\log|D_F|=0$, while $K_2(\mathbb{Z})\cong\mathbb{Z}/2\mathbb{Z}$ is nonzero, so the displayed inequality is meaningful only for $d\ge2$, where $|D_F|\ge3$ and grows without bound as $F$ varies; the sources themselves frame the conjecture in this range (Bayer-Fluckiger–Emery–Houriet state their results for $F\ne\mathbb{Q}$ and their Corollary 1.2 for $d\ge2$), and for $F=\mathbb{Q}$ the torsion orders are expected to grow exponentially in $n$ via Bernoulli numerators (Soulé's own heuristic in Sect. 5.1), a dependence on $n$ unconstrained by any discriminant. The bound is conjectured for the full torsion subgroup, while all currently proven bounds hold only up to small torsion ($\mathrm{card}_{\ell}$, i.e. after quotienting out the subgroup generated by elements of order $\le\ell$ with $\ell=\max(d+1,2n+2)$). For $n=0$ the analogous statement is the classical class-number bound; the open content is the general case $n\ge2$ with $F$ varying over all number fields of fixed degree $d\ge2$.

References

1. Christophe Soulé, 'A bound for the torsion in the K-theory of algebraic integers', in: A Collection of Manuscripts Written in Honour of Kazuya Kato on the Occasion of His Fiftieth Birthday, Documenta Mathematica Extra Volume Kato (2003), pp. 761–788. DOI: 10.4171/dms/3/21, https://doi.org/10.4171/dms/3/21
2. Eva Bayer-Fluckiger, Vincent Emery, Julien Houriet, 'Hermitian lattices and bounds in K-theory of algebraic integers', in: A Collection of Manuscripts Written in Honour of Alexander S. Merkurjev on the Occasion of His Sixtieth Birthday, Documenta Mathematica Extra Volume Merkurjev (2015), pp. 71–83. DOI: 10.4171/dms/7/4 (arXiv:1406.5953), https://doi.org/10.4171/dms/7/4
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