Showing papers by "Alice Silverberg published in 1988"

Mordell-Weil groups of generic abelian varieties in the unitary case

Abstract: The generic fibre of a fibre system of polarized abelian varieties with level structure, and with endomorphism structure coming from a CMfield, is defined over the function field of the moduli space for the abelian varieties. We prove that the points on this generic abelian variety which are defined over the function field of the moduli space form a finite group. The methods of proof generalize those of MordellWeil groups of generic abelian varieties, Invent. Math. 81 (1985), 71-106, to which this paper is a sequel. Introduction. In this paper we will consider fibre systems of polarized abelian varieties over C characterized by having a fixed CM-field embedded in their endomorphism algebras. If V is the moduli space for such abelian varieties (with level structure), and W is the fibre variety constructed in [4], then the fibre over the generic point of V is an abelian variety defined over the function field of V. We consider the points of this abelian variety which are defined over C(V). Our result is THEOREM. If dimV > 0, then the Mordell-Weil group of the generic fibre is finite. Here, V is isomorphic to a product of domains of the form {r x s complex matrices Z1 ZtZ > 0} modulo the action of a discrete subgroup of a unitary group. The setting for which the theorem holds is described more precisely in ? 1. The analogous theorem was proved for V a noncompact quotient of the complex upper half plane by Shioda (Theorem 5.1 of [5]). When V is a quotient of a Hilbert-Siegel space Hr and V is the moduli space for abelian varieties whose endomorphism algebras contain a fixed totally real field, CM-field, totally indefinite quaterion algebra over a totally real field, or quaternion algebra over a CM-field, the analogous theorem was proved in [6] (see also [7]). The present paper makes use of new methods to remove the restriction on the base variety V in the CM-field case. I owe many thanks to Goro Shimura for suggesting the problem and giving guidance and advice, and to Robert Indik for the proof of Theorem 6. NOTATION. We will write diag(A, ... , A,) for the matrix with blocks A1,... , A, on the diagonal, I(r) for the identity matrix in GLr(Z), and t-y for the transpose of the matrix -y. Received by the editors March 26, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 11GIO, llG15; Secondary 14K22. The author was partially supported by an NSF Postdoctoral Fellowship. ?1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page