CHOW’S K/k-IMAGE AND K/k-TRACE, AND THE LANG–NÉRON THEOREM

Let K/k be an extension of fields, and assume that it is primary: the algebraic closure of k in K is purely inseparable over k. The most interesting case in practice is when K/k is a regular extension: K/k is separable and k is algebraically closed in K. Regularity is automatic if k is perfect. (For K/k finitely generated, regularity is equivalent to K arising as the function field of a smooth and geometrically connected k-scheme.) In the theory of abelian varieties over finitely generated regular extensions K/k with respect to some field of “constants” k, there is a generalization of the Mordell–Weil theorem, due to Néron [26] (in his thesis) and Lang–Néron [19], and in this theorem a crucial role is played by the K/k-trace and the K/k-image of an abelian variety A over K. These constructions are also ubiquitous in many problems concerning families of abelian varieties. (The family is parameterized by a nice base V over k, and K = k(V ).) For an arbitrary primary extension of fields K/k, the K/k-trace of A is a final object in the category of pairs (B, f) consisting of an abelian variety B over k equipped with a K-map of abelian varieties f : BK → A, where BK denotes the scalar extension B⊗kK; we write (TrK/k(A), τA,K/k) to denote such a final object (if it exists). Likewise, the K/k-image of A is an initial object in the category of pairs (B, f) consisting of an abelian variety B over k equipped with a K-map of abelian varieties f : A→ BK ; we write (ImK/k(A), λA,K/k) to denote such an object (if it exists). Roughly speaking, the K/k-image is the largest quotient of A that can be defined over k, and the K/k-trace is the largest abelian subvariety of A that can be defined over k. A precise description along these lines requires some care in positive characteristic. These concepts are due to Chow ([3], [4]). Despite the importance of Chow’s K/k-trace and K/k-image and the Lang–Néron theorem in arithmetic geometry, unfortunately no detailed general reference on these topics has been available entirely in the language of schemes. The papers of Chow ([3], [4]) and the book on abelian varieties by Lang [18] discuss the K/k-image and K/k-trace and develop their properties, but entirely in Weil’s framework [34]. Similarly, in Lang’s modern book [20] the Lang–Néron theorem is proved in Weil’s language. In connection with my work in [5], where the Lang-Néron theorem plays a crucial role, I was motivated to write this expository account of a scheme-theoretic approach to Chow’s results and the Lang–Néron theorem. In some instances the old and new methods are expressing similar ideas, but in other cases where we make extensive use of infinitesimal or flat descent methods it is less clear how much overlap there is. For example, our use of infinitesimal group schemes in the proof of the fundamental Chow regularity theorem (Theorem 5.5) replaces the ineffective “sufficiently large” aspect of the original version of the theorem (as in [3, Cor. to Thm. 8] and [18, VIII, Thm. 3]) with a simple explicit lower bound. We begin in §2 with some intuition and examples related to Chow’s work and the Lang–Néron theorem (including a precise statement of the latter). In §3 we summarize some background facts and terminology from algebraic geometry (centered largely on Grothendieck’s descent theory and group schemes) and prove some other additional results for convenient reference later; some of the topics discussed in §3 are used in §2. In our development of the K/k-image in §4, we prove that the canonical map λA,K/k : A→ ImK/k(A)K is surjective with connected kernel that may be non-smooth in positive characteristic (Example 4.4). The behavior of the K/k-image with respect to extension of the ground field k is treated in §5. The key result