Let K be a global field and let XK be a geometrically integral and separated K-scheme of finite type (a “K-variety”). Let Σ be a finite non-empty set of places of K, and suppose we are given local points xv ∈ XK(Kv) for each v ∈ Σ. It is natural to ask if we may be able to find x ∈ XK(K) that is arbitrarily close to xv for the natural topology on XK(Kv) (arising from the topology on Kv) for all...

In Ann. of Math. 121 (1985), 111^168, Coleman de¢nes p-adic Abelian integrals on curves. Given a family of curves X/S, a differential o and two sections s and t, one can de¢ne a function lo on S by lo…P† ˆ R t…P† s…P† oP. In this paper, we prove that lo is locally analytic on S. Mathematics Subject Classi¢cations (2000). Primary 14G20; Secondary 14D10, 11G20. Key words. p-adic Abelian integrals...