Let \(G\) be a reductive group over a field \(k\). Given a discrete valuation \(v\) of \(k\), one says that \(G\) has good reduction at \(v\) if there is a reductive group scheme \(\mathcal{G}\) over \(\mathcal{O}_v\) whose generic fiber is isomorphic to \(G\times_kk_v\). (Here \(k_v\) denotes the completion of \(k\) at \(v\) and \(\mathcal{O}_v\) denotes the valuation ring of \(k_v\).) When \(G\) is an absolutely almost simple group, the genus \(\mathbf{gen}_k(G)\) is the set of \(k\)-isomorphism classes of inner \(k\)-forms of \(G\) that have the same isomorphism classes of maximal tori as \(G\).
The first main result of this paper is the following theorem relating the genus problem to good reduction.
Theorem 1.1. Suppose \(G\) is absolutely almost simple. Let \(v\) be a discrete valuation of \(k\) whose residue field \(k^{(v)}\) is a finitely generated field. Assume further that \(\mathrm{char}(k^{(v)})\neq 2\) if \(G\) is of type \(B_l\) (\(l\ge 2\)). If \(G\) has good reduction at \(v\), then any \(G'\in\mathbf{gen}_k(G)\) also has good reduction at \(v\), and the reduction \(\underline{G}'{}^{(v)}\) of \(G'\) at \(v\) lies in the genus \(\mathbf{gen}_{k^{(v)}}(\underline{G}^{(v)})\) of the reduction \(\underline{G}^{(v)}\).
In the proof of Theorem 1.1, a new approach to good reduction is developed, and it shows that for absolutely almost simple groups the existence of good reduction can be characterized in terms of the existence of maximal tori with certain specific properties.
The paper exhibits a number of applications of Theorem 1.1 and the new characterization of good reduction in terms of maximal tori. Some of these applications are concerned with the case where \(k\) is a finitely generated field of good characteristic (and \(G\) is absolutely almost simple). In this case we have:
Corollary 1.2. Assume \(k\) is infinite. Let \(V\) be a divisorial set of valuations of \(k\). (That is, \(V\) is the set of discrete valuations of \(k\) defined by prime divisors of a normal model of \(k\) over \(\mathbb{Z}\).) Then there exists a finite subset \(S\subseteq V\) such that every \(G'\in\mathbf{gen}_k(G)\) has good reduction at all \(v\in V\setminus S\).
Let us mention some results that reveal some effects of a purely transcendental extension on the genus.
Theorem 1.3. Let \(K=k(x)\) be a one-variable rational function field. Then every \(H\in \mathbf{gen}_K(G\times_kK)\) is of the form \(H_0\times_kK\) for some \(H_0\in\mathbf{gen}_k(G)\).
When \(k\) is a number field and \(G\) is simply connected, Theorem 1.3 implies that \(\mathbf{gen}_K(G\times_kK)\) is finite for any purely transcendental extension \(K=k(x_1,\cdots, x_m)\) of transcendence degree \(m\ge 1\).
The following theorem is an example of a new phenomenon which the authors have termed “killing the genus by a purely transcendental extension”.
Theorem 1.6. Suppose \(G\) is of type \(G_2\) and \(K=k(x_1,\cdots, x_6)\). Then \(\mathbf{gen}_K(G\times_kK)\) is a singleton.
The authors conjecture that for any Cartan-Killing type there is an integer \(m\ge 1\) such that for any absolutely almost simple group \(G\) of that type, every \(H\in\mathbf{gen}_K(G\times_kK)\) is of the form \(H_0\times_kK\), where \(H_0\) has the property that \(H_0\times_kF\in\mathbf{gen}_F(G\times_kF)\) for any field extension \(F/k\).
As another application of the new characterization of good reduction, the authors prove some results (e.g., Theorems 1.8 and 9.5) that relate the presence of finitely generated Zariski dense subgroups in \(G(k)\) weakly commensurable to a given one with good reduction. They also obtain a first example of applying arithmetic geometrical techniques to non-arithmetic Riemann surfaces (Theorem 1.9).
In the last part of the paper, the authors prove several results about the genus problem and good reduction form simple groups of type \(F_4\). See, e.g., Theorems 1.10–1.13.