The Diophantine problem in isotropic reductive groups

We begin to study model-theoretic properties of non-split isotropic reductive group schemes. In this paper we show that the base ring $K$ is e-interpretable in the point group $G(K)$ of every sufficiently isotropic reductive group scheme $G$. In particular, the Diophantine problems in $K$ and $G(K)$ are equivalent. We also compute the centralizer of the elementary subgroup of $G(K)$ and the common normalizer of all its root subgroups.

💡 Research Summary

The paper investigates model‑theoretic aspects of isotropic (i.e., non‑split) reductive group schemes over a commutative base ring K, extending earlier work that dealt only with split Chevalley–Demazure groups. The central goal is to relate the Diophantine problem for the ring K to the analogous problem for the group of K‑points G(K) of a sufficiently isotropic reductive group scheme G.

After recalling the notions of Diophantine sets, elementary (positive‑primitive) formulas, and e‑interpretability, the author explains that for a finitely presented affine group scheme G over K the group G(K) is always e‑interpretable in K. The reverse direction—interpreting K inside G(K)—is non‑trivial and was previously known only when G is split and of rank at least 2 (Bunina‑Myasnikov‑Plotkin).

The paper defines an “isotropic reductive group scheme” via a Tits index: there must exist a surjective map u from a reduced root system eΦ (together with 0) onto the (possibly non‑reduced) root system Φ of G, together with a split Levi subgroup L and root subgroups Uα satisfying certain compatibility conditions. Moreover, the author assumes the existence of local Weyl elements wα that conjugate root subgroups as in the split case. This framework includes all simple reductive groups over a local ring, and the author plans to treat the locally isotropic case in a sequel.

A comprehensive list of all isotropic Tits indices is provided, covering the infinite families (A, B, C, D) and the finite exceptional types (E, F, G). For each index the author specifies the underlying reduced root system, the kernel of u, the size of pre‑images of roots, and the outer automorphism group. Equivalences and isomorphisms among indices are also recorded.

Technical lemmas introduce closed subsets Σ of Φ and classify them as unipotent, closed root subsystems, parabolic, or saturated. For each Σ the associated subgroup schemes GΣ and G⁰Σ are defined; their structural properties (reductive, unipotent, parabolic) are established. Lemma 1 proves a Gauss decomposition G(K)=U⁺(K) T(K) U⁻(K) U⁺(K) for semilocal K, a key tool for later constructions.

The main results are:

1. Theorem 4 – Every root subgroup Uα of G(K) is a Diophantine subset of G(K). The proof uses the explicit description of root subgroups via the chosen Tits index and the existence of Weyl elements.

2. Theorem 5 – The base ring K is e‑interpretable in the group G(K). By encoding ring elements as parameters that describe the interaction of two root subgroups (or a root subgroup with the torus), one obtains a Diophantine set X⊆G(K)ⁿ together with a surjection X→K satisfying the required lifting conditions.

3. Theorem 6 – Consequently, the Diophantine problems D(K) and D(G(K)) are algorithmically equivalent. An algorithm solving one can be transformed into an algorithm solving the other by the explicit reductions given in Theorems 4 and 5.

4. Theorem 7 – The centralizer of the elementary subgroup E(K) (the subgroup generated by all root subgroups) is precisely the torus T(K) together with the subgroup of elements commuting with all Uα; in particular it is a relatively small, explicitly describable abelian group.

5. Theorem 8 – The common normalizer of all root subgroups (and of any finite collection thereof) is the semidirect product L(K)⋊⟨wα⟩, where L is the split Levi subgroup and the wα are the Weyl elements. This generalises earlier results for split groups.

As an illustration, the author notes that for the group SO(E₈⊥H⊥H)² (where E₈ is the root lattice of type E₈ and H a hyperbolic plane over ℤ) the Diophantine problem is undecidable, because ℤ is e‑interpretable in this group.

In summary, the paper succeeds in extending the e‑interpretability and Diophantine equivalence results from split reductive groups to a broad class of isotropic, possibly non‑split, reductive group schemes. It also provides a detailed structural analysis of centralizers and normalizers of elementary subgroups, thereby enriching the algebraic understanding of these groups and opening the way for further applications in model theory, algebraic K‑theory, and computational group theory.