A finiteness theorem for special unitary groups of quaternionic skew-Hermitian forms with good reduction (Q2205216)

The main result of the paper under review is the following: Let \(K = k(C)\) be the function field of a geometrically integral curve over a field \(k\) of characteristic \(\neq 2\) that satisfying \((F'_2)\) (i.e., for every finite separable extension \(L/k\), the quotient \(L^\times /(L^\times)^2\) is finite) and let \(V\) be the set of discrete valuations on \(K\) corresponding to closed points of \(C\). Then the number of \(K\)-isomorphism classes of the universal coverings of the special unitary groups \(SU_m(h)\) of non-degenerate \(m\)-dimensional skew-hermitian forms \(h\) over some quaternion \(K\)-algebra with the canonical involution, having good reduction at all places in \(V\), is finite. The author is proving this theorem by developing a general theory that relates reduction properties of skew-hermitian forms over a quaternion \(K\)-algebra \(Q\) to quadratic forms over the function field \(K(Q)\). As a result of the main theorem, the author proves Conjecture 7.3 from [\textit{V. I. Chernousov} et al., Compos. Math. 155, No. 3, 484--527 (2019; Zbl 1443.11031)] for groups of this type.