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Linear algebraic groups with good reduction

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Abstract

This article is a survey of conjectures and results on reductive algebraic groups having good reduction at a suitable set of discrete valuations of the base field. Until recently, this subject has received relatively little attention, but now it appears to be developing into one of the central topics in the emerging arithmetic theory of (linear) algebraic groups over higher-dimensional fields. The focus of this article is on the Main Conjecture (Conjecture 5.7) asserting the finiteness of the number of isomorphism classes of forms of a given reductive group over a finitely generated field that have good reduction at a divisorial set of places of the field. Various connections between this conjecture and other problems in the theory of algebraic groups (such as the analysis of the global-to-local map in Galois cohomology and the genus problem) are discussed in detail. The article also includes a brief review of the required facts about discrete valuations, forms of algebraic groups, and Galois cohomology.

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Notes

  1. More formally, let \(v_p\) denote the (normalized) p-adic valuation on \({{\mathbb {Q}}}\) and let \({{\mathbb {Z}}}_{(p)}\) be the corresponding valuation ring. The elliptic curve E is said to have good reduction at p if there exists an abelian scheme \(E_{(p)}\) over the valuation ring \({{\mathbb {Z}}}_{(p)}\) with generic fiber E (the scheme \(E_{(p)}\) is then unique, which leads to a well-defined notion of reduction modulo p).

  2. We recall that one defines the unipotent radical of a connected algebraic group G to be the largest connected unipotent normal subgroup, and one says that G is reductive if its unipotent radical is trivial. For example, all tori (i.e., connected diagonalizable algebraic groups) are reductive. An algebraic group is (absolutely almost) simple if it does not contain any proper connected normal subgroups, and semi-simple if it admits a surjective morphism from a direct product of simple groups. We refer the reader to [10] and [32] for the details.

  3. In more technical terms, this system defines a scheme over \({{\mathbb {Z}}}_{(p)}\) with generic fiber G.

  4. For comparison, we would like to point out the following finiteness theorem for the forms of abelian varieties (cf. [89, §3, Finiteness theorem for forms]): Let X be an abelian variety over a field K, and let F/K be a finite separable extension. Then the set of K-isomorphism classes of abelian K-varieties \(X'\)such that there exists an F-isomorphism \(X \times _K F \simeq X' \times _K F\)is finite. On the contrary, for a semi-simple linear algebraic K-group G, and a finite separable extension F/K, the set of K-isomorphism classes of F/K-forms \(G'\) of G is infinite in many cases, even when K is a number field (see, however, the discussion of fields of type (F) in Sect. 5.2). So, the problem of classifying forms of (semi-simple) linear algebraic groups with special properties, which is central to the current article, comprises some challenges that do not arise in the context of abelian varieties.

  5. As defined in [32, §5].

  6. Using twisting, one shows that the properness of \(\theta _{\mathrm {PSL}_n , V}\) in fact implies the properness of \(\theta _{\mathrm {PSL}_{1 , A} , V}\) for any central simple K-algebra A of degree n.

  7. We observe that if the genus \(\mathbf{gen }(D)\) is infinite for a central division K-algebra D, then the genus \(\mathbf{gen }_K(G)\) is also infinite for the corresponding algebraic group \(G = \mathrm {SL}_{1 , D}\).

  8. We will not go into the details of this analysis here and would only like to point out that one of the important factors is the existence of so-called generic elements in every Zariski-dense subgroup—see [99] for a detailed discussion. The reader interested in the technical ingredients can also review the Isogeny Theorem (Theorem 4.2) in [96], which provides a far-reaching generalization of the following fact used in section Sect. 8.3: if \(\gamma _1 , \gamma _2 \in \mathrm {SL}_2(F)\)are semi-simple elements of infinite order that are weakly commensurable, then for any subfield K that contains the traces of \(\gamma _1\)and \(\gamma _2\), the subalgebras \(K[\gamma _1]\)and \(K[\gamma _2]\)are K-isomorphic.

  9. This means that that there exists an F-isomorphism \(\varphi :{\overline{G}}_1 \rightarrow {\overline{G}}_2\) between the corresponding adjoint groups such that \(\varphi ({\overline{\Gamma }}_1)\) is commensurable with \({\overline{\Gamma }}_2\), where \({\overline{\Gamma }}_i\) denotes the image of \(\Gamma _i\) in \({\overline{G}}_i(F)\).

  10. See Sect. 10.2 for some rigidity results over rings more general than rings of algebraic S-integers

  11. Of course, the traces of elements in the adjoint representation that generate the field of definition can be easily expressed in terms of the eigenvalues, but in our set-up, all we can work with are relations like (WC) in Definition 9.1 for \(\gamma _1 \in \Gamma _1\) and \(\gamma _2 \in \Gamma _2\), which do not immediately yield any relation between \(\mathrm {tr}(\mathrm {Ad}\, \gamma _1)\) and \(\mathrm {tr}(\mathrm {Ad}\, \gamma _2)\).

  12. Let us point out that here we deviate from the standard terminology. Namely, recall that in the classical setting where K is a number field, \(V^K\) is the set of all places of K, and \(S \subset V^K\) is a finite subset with complement \(V = V^K \setminus S\), we say that G has strong approximation with respect to S if the diagonal embedding \(G(K) \hookrightarrow G({\mathbb {A}}(K , V))\) has dense image (cf. [94, Ch. 7]).

  13. We will say that \((\Phi , R)\) is a nice pair if 2 is a unit in R whenever \(\Phi \) contains a subsystem of type \({\mathsf {B}}_2\), and 2 and 3 are units in R whenever \(\Phi \) is of type \({\mathsf {G}}_2\).

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Acknowledgements

Special thanks are due to Brian Conrad, who carefully read the article and made a number of suggestions that helped to improve the exposition. We are also grateful to Uriya First, Ariyan Javanpeykar, Boris Kunyavskiī, Daniel Loughran, Alexander Merkurjev, Dipendra Prasad, C. Rajan, Zinovy Reichstein, Charlotte Ure, Uzi Vishne, and the anonymous referee for helpful comments.

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Rapinchuk, A.S., Rapinchuk, I.A. Linear algebraic groups with good reduction. Res Math Sci 7, 28 (2020). https://doi.org/10.1007/s40687-020-00226-3

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