Spinor groups with good reduction

Abstract

Let K be a 2-dimensional global field of characteristic = 2, and let V be a divisorial set of places of K. We show that for a given n 5, the set of K-isomorphism classes of spinor groups G = Spin n (q) of nondegenerate n-dimensional quadratic forms over K that have good reduction at all v ∈ V , is finite. This result yields some other finiteness properties, such as the finiteness of the genus gen K (G) and the properness of the global-to-local map in Galois cohomology. The proof relies on the finiteness of the unramified cohomology groups H i (K, µ2)V for i 1 established in the paper. The results for spinor groups are then extended to some unitary groups and to groups of type G2.

The finiteness of the genus of a finite-dimensional division algebra, and some generalizations

“…The finiteness of n Br(K) V implies the truth of this property for inner forms of type A ℓ over a finitely generated field K of characteristic prime to ℓ + 1. Recently, we established this property for the spinor groups of quadratic forms, some unitary groups, and groups of type G 2 over the function field K = k(C) of a smooth geometrically integral curve C over a number field k (see [11]). This relied on the finiteness of the unramified cohomology groups H i (K, µ 2 ) V for all i ≥ 1 with coefficients in µ 2 = {±1}, and the case that required the most effort was i = 3.…”

On residue maps for affine curves

“…). Note that using the usual identification betweenétale and Galois cohomology, we can rewrite the group on the right as (4). We then define ∂ ℓ−1 v to be the composition of the boundary map…”

“…The purpose of this note to establish several compatibilities between certain residue maps inétale and Galois cohomology that arise naturally in the analysis of affine curves with good reduction. Our main result (Theorem 1.1) plays a key role in [4] in the proof of a finiteness statement for the unramified cohomology of function fields of affine curves over number fields, with applications to finiteness properties of the genus of spinor groups of quadratic forms, as well as some other groups, over such fields. We formulate our result below in a rather general setting to make it suitable for other applications.…”

“…Remark 1.2. We note that the isomorphism (3) holds in the following situation, which was needed for the considerations in [4]. Let p be any prime = char k (v) .…”

A generalization of Serre’s condition $$\mathrm {(F)}$$ ( F ) with applications to the finiteness of unramified cohomology

“…We should point out that the finiteness statement of part (a) has been used in [7] and [8] to show that if K is a field of characteristic = 2 that satisfies (F ′ 2 ), then the number of K(C)-isomorphism classes of spinor groups G = Spin n (q) of nondegenerate quadratic forms q over K(C) in n ≥ 5 variables, as well as of groups of some other types, that have good reduction at all geometric places v ∈ V 0 is finite. In fact, this result is part of a general program of studying absolutely almost simple groups having the same maximal tori over the field of definition -we refer the reader to [6] for a detailed overview of these problems and several conjectures.…”