The congruence subgroup problem for anisotropic and low S-rank arithmetic groups: the remaining cases of the Bass–Milnor–Serre program

Bass, Milnor and Serre (1967) solved the congruence subgroup problem (CSP) for $SL_n$ ($n \ge 3$) and $Sp_{2n}$ ($n \ge 2$) and posed it for all simply connected, simple, split (Chevalley) groups of rank > 1 over global fields, together with an equivalent 'metaplectic' formulation. Later work closed those conjectures: Matsumoto (1969) settled all Chevalley groups, and Raghunathan, Prasad and Rapinchuk settled the general isotropic case ($k$-rank $\ge 1$) together with the associated cohomological computation of the metaplectic kernel. What survives of the Bass–Milnor–Serre problem, in the formulation the research literature itself uses, is twofold: (1) the determination of the congruence kernel $C(S,G)$ for anisotropic absolutely almost simple simply connected groups over number fields (especially the type $A_n$ forms, such as $SL_1(D)$ for a central division algebra $D$) for general sets $S$ of places of $S$-rank at least 2 — is the kernel central, is it finite, is it trivial, and is the expected dichotomy $C(S,G) \in \{1, \mu_k\}$ correct — together with the corresponding metaplectic-kernel determination, which is still conditional on Conjecture (U) for 'special' outer-type-$A_n$ groups; and (2) the expected infinitude of $C(S,G)$ when the $S$-rank is 1 (the $S$-rank 0 case being settled: the $S$-arithmetic group is then finite and the kernel trivial), proved so far only in many classical cases.

Background

Let $k$ be a number field (a finite extension of the rationals $\mathbb{Q}$) and let $V$ be its set of places, i.e. equivalence classes of absolute values, partitioned into the archimedean ones $\infty$ and the finite (non-archimedean) ones $V_f$. For a subset $S$ with $\infty \subseteq S \subsetneq V$, the ring of $S$-integers is $\mathcal{O}_S = \{x \in k : |x|_v \le 1 \text{ for all } v \notin S\}$. Let $G$ be an algebraic group over $k$, that is, a group defined by polynomial equations with coefficients in $k$ (for example $G = SL_n$, the group of determinant-one matrices). The group $G(\mathcal{O}_S)$ carries two group topologies: in the arithmetic (profinite) topology a basis of neighborhoods of the identity is given by all subgroups of finite index, while in the congruence topology the basic neighborhoods are the principal congruence subgroups

$$G(\mathfrak{q}) = \ker\bigl(G(\mathcal{O}_S) \to G(\mathcal{O}_S/\mathfrak{q})\bigr),$$

where $\mathfrak{q}$ ranges over the nonzero ideals of $\mathcal{O}_S$. Completing $G(\mathcal{O}_S)$ with respect to these topologies yields a profinite arithmetic completion $\hat{G}(S,a)$ and a congruence completion $\hat{G}(S,c)$ (profinite: an inverse limit of finite quotient groups), and there is a natural surjection $\pi : \hat{G}(S,a) \to \hat{G}(S,c)$. The congruence kernel is $C(S,G) = \ker \pi$; it measures the failure of the congruence subgroup problem (CSP). Indeed $C(S,G) = \{1\}$ exactly when every subgroup of finite index in $G(\mathcal{O}_S)$ — and more generally every $S$-arithmetic subgroup of $G(k)$, i.e. a subgroup commensurable with $G(\mathcal{O}_S)$ — contains some principal congruence subgroup, i.e. is a congruence subgroup.

Bass–Milnor–Serre (1967) determined $C(S,G)$ for $G = SL_n$ ($n \ge 3$) and $G = Sp_{2n}$ ($n \ge 2$) over a global field (a number field, or a function field of one variable over a finite field): the kernel is trivial unless every place of $S$ is complex, in which case it is the group $\mu_k$ of roots of unity in $k$. They conjectured the same dichotomy, together with centrality of the kernel, for all simply connected, simple Chevalley groups of rank > 1 — a Chevalley group being a split group, one containing a split maximal torus of dimension equal to its rank, such as the groups $SL_n$, $Sp_{2n}$, $Spin_n$ (the double cover of the special orthogonal group) and the exceptional groups. Their companion 'Metaplectic Conjecture' asserted the existence of a central extension

$$1 \to \mu_k \to \tilde{G} \to G\bigl(\mathbb{A}(S)\bigr) \to 1$$

of the $S$-adèle group $G(\mathbb{A}(S))$ (the restricted direct product of the groups $G(k_v)$ of points over the completions $k_v$, taken over the places $v \notin S$) that splits over the rational points $G(k)$ and is universal among central extensions with profinite kernel; its local components at non-complex places should have order $[\mu_k : 1]$. Moore (1968) had computed the second continuous cohomology $H^2$ of $p$-adic and adèle linear groups, which supplies the local ingredients. Bass–Milnor–Serre also proved (their Theorem 16.2) that any homomorphism of an arithmetic group $\Gamma$ into $GL_m(\mathbb{Q})$ coincides on a finite-index subgroup with the restriction of a morphism of algebraic groups; the 1974 erratum to the paper (by Serre alone) corrects only an auxiliary lemma that was not used in the main arguments.

The subsequent half century resolved the conjectures for split groups and, much more generally, for isotropic groups. Matsumoto (1969) extended the results to arbitrary Chevalley groups. Raghunathan (1976) proved that for $k$-rank at least 2 the kernel is finite and the extension is central (over number fields; in characteristic $p$ for the prime-to-$p$ part, with finiteness if $G$ is quasi-split outside $S$). Here the $k$-rank of $G$ is the dimension of a maximal $k$-split torus, and $G$ is called isotropic if its $k$-rank is positive — equivalently, if $G(k)$ contains nontrivial unipotent elements (matrices whose only eigenvalue is 1) — and anisotropic otherwise. The $S$-rank is the sum $\sum_{v \in S} \mathrm{rank}_{k_v} G$ of the local ranks over the places of $S$. Raghunathan (1986) settled the remaining isotropic case of $k$-rank 1 under the hypothesis of $S$-rank at least 2. The cohomological part of the program was carried out for all $G$: the metaplectic kernel

$$M(S,G) = \ker\bigl(H^2_{\mathrm{cont}}\bigl(G(\mathbb{A}(S)), \mathbb{Q}/\mathbb{Z}\bigr) \to H^2\bigl(G(k), \mathbb{Q}/\mathbb{Z}\bigr)\bigr),$$

where $H^2_{\mathrm{cont}}$ denotes continuous group cohomology with $\mathbb{Q}/\mathbb{Z}$ coefficients, was computed by Moore, Matsumoto, Prasad–Raghunathan (isotropic case) and Prasad–Rapinchuk (anisotropic case, conditionally on Conjecture (U)). Whenever $C(S,G)$ is central and $G(k)$ is perfect (equal to its own commutator subgroup), $C(S,G)$ is the Pontrjagin dual (the group of continuous characters into $\mathbb{Q}/\mathbb{Z}$) of $M(S,G)$, so the two computations are equivalent.

Two regimes remain open, and they are the subject of this problem. First, the anisotropic case: when $k$-rank $G = 0$, the group $G(k)$ has no nontrivial unipotent elements and the entire Bass–Milnor–Serre strategy (generation by elementary unipotent matrices) is unavailable. The prototypes are the inner forms of type $A_{d-1}$, namely $G = SL_1(D)$, the group of reduced-norm-one elements of a central division algebra $D$ over $k$ (a noncommutative analogue of a number field in which every nonzero element is invertible), and the outer forms of type $A_n$ (special unitary groups attached to a division algebra with involution over a quadratic extension of $k$). Raghunathan's 2004 survey states that for type $A_n$ 'the greatest challenge' remains. Triviality of the kernel is known for such groups only when $S$ is restricted: the cofinite sets $S$ with $\infty \subseteq S \subsetneq V$ (those for which $V \setminus S$ is a finite set of places, nonempty because $S \subsetneq V$; here the normal subgroup structure of the norm-one group $D^1$ is known), sets containing a generalized arithmetic progression (Prasad–Rapinchuk 2016), and $d$-amenable sets (Radhika–Raghunathan, published online 2019, issue 2020), a class of sets of places introduced in that work, which includes the cofinite sets $S \subsetneq V$ and sets containing a generalized arithmetic progression. Second, the low-$S$-rank regime: when the $S$-rank is 1 the kernel is expected to be infinite, and this has been proved only in many classical cases; the $S$-rank 0 case is settled ($G(\mathcal{O}_S)$ is then finite and the kernel is trivial). The anisotropic metaplectic-kernel computation of Prasad–Rapinchuk (1996) is conditional on Conjecture (U) for the 'special' groups — outer-type-$A_n$ groups attached to a central division algebra over a quadratic extension of $k$ that remains a noncommutative division algebra over that extension. Conjecture (U) asserts that for such a special group $G$ the kernel $M_W(G) = \ker\bigl(H^2_{\mathrm{cont}}(G(W), \mathbb{Q}/\mathbb{Z}) \to H^2(G(k), \mathbb{Q}/\mathbb{Z})\bigr)$, with $G(W) = \prod_{v \in W} G(k_v)$, depends only on the finite places in $W$: for every finite set $W$ of places of $k$, writing $W_0 = W \cap V_f$ for its subset of finite places, one has $M_W(G) = M_{W_0}(G)$.

Problem Statement

Determine the congruence kernel $C(S,G)$ — equivalently, solve the congruence subgroup problem — in the two regimes left open by the Bass–Milnor–Serre program and its isotropic successors, over number fields $k$:

(1) Anisotropic regime. Let $G$ be an absolutely almost simple, simply connected $k$-group with $k$-rank $G = 0$ — absolutely almost simple means simple as an algebraic group over an algebraic closure of $k$, and simply connected means admitting no nontrivial central isogeny from another connected algebraic group. Examples: $G = SL_1(D)$ for a central division algebra $D$ over $k$ of degree $d$, an inner form of type $A_{d-1}$, or an outer form of type $A_n$. Let $S$ be a set of places of $k$ with $\infty \subseteq S \subsetneq V$; assume the $S$-rank of $G$ is at least 2 and $G(k_v)$ is noncompact for the finite $v \in S$. Determine $C(S,G)$: is the extension $1 \to C(S,G) \to \hat{G}(S,a) \to \hat{G}(S,c) \to 1$ central; is $C(S,G)$ finite; and what is it explicitly? The expected answer, extending the Bass–Milnor–Serre dichotomy, is that $C(S,G)$ is central and is either trivial or $\mu_k$, with triviality when $S$ contains a non-archimedean place at which $G$ is isotropic. The cases that remain open are the inner-type-$A_{d-1}$ case for general $S$ (beyond the cofinite, generalized-arithmetic-progression, and $d$-amenable cases already settled), the outer-type-$A_n$ case, and — on the cohomological side, where the two determinations are equivalent — the unconditional determination of the metaplectic kernel $M(S,G)$ for anisotropic groups, in particular the resolution of Conjecture (U) of Prasad–Rapinchuk for the 'special' outer-type-$A_n$ groups.

(2) Low-$S$-rank regime. Let $G$ be an absolutely almost simple simply connected $k$-group and $S$ a set of places of $k$ with $\infty \subseteq S \subsetneq V$ and $S$-rank at most 1. Determine $C(S,G)$. The $S$-rank 0 case is settled ($G(\mathcal{O}_S)$ is then finite and the kernel trivial), so the open content is the case of $S$-rank exactly 1: prove or refute the expectation that $C(S,G)$ is infinite.

References

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