On the Image of the Second l-adic Bloch Map

Abstract

After Jacobians of curves, Prym varieties are perhaps the next most studied abelian varieties. They turn out to be quite useful in a number of contexts. For technical reasons, there does not appear to be any systematic treatment of Prym varieties in characteristic 2, and due to our recent interest in this topic, the purpose of this paper is to fill in that gap. Our main result is a classification of branched covers of curves in characteristic 2 that give rise to Prym varieties. We are also interested in the case of Prym varieties in the relative setting, and so we develop that theory here as well, including an extension of Welters' Criterion. This is proven in Theorem 5.12 and Theorem 6.5, and the full result also allows for the inseparability of f . The abelian scheme P(C/C ′ ) of part ( 1) is constructed as follows. The morphism f : C → C ′ induces a morphism of abelian schemes f * : Pic 0 C ′ /S → Pic 0 C/S ; let Y be its image. The principal polarization of Pic 0 C/S restricts to a polarization on Y, whose exponent we denote by e ( §1.1.1). Using the polarization on Pic 0 C/S we define a norm endomorphism N Y : Pic 0 C/S → Pic 0 C/Swith image Y ( §2.1), and define P(C/C ′ ) to be the image Im([e] X − N Y ). In this context, we say that P(C/C ′ ) is a Prym scheme of exponent e (embedded in Pic 0 C/S ). In the situation of part (2), when the induced polarization on P(C/C ′ ) is the e th multiple of a principal polarization ξ, we call the principally polarized abelian scheme (P(C/C ′ ), ξ) a Prym-Tyurin Prym scheme of exponent e. In much of the literature, as in the beginning of this introduction, the phrase "Prym variety" often refers to case 2(a).We also recover an arithmetic version of a criterion of Welters; here we work over a field K. Theorem (B).Let C be a smooth pointed proper curve over a field K, which we take to be embedded via the Abel-Jacobi map in its principally polarized Jacobian (Pic 0 C/K , Θ C/K ), let Z ֒→ Pic 0 C/K be a sub-abelian variety of dimension g Z with principal polarization Ξ, and let β be the canonical composition