* 45 minutes: general talk for number theorists * 30 minutes: specialist part 2 Part I. General 1. Define notion of visibility of H^1 and MW 2. Prove everything visibility using Shapiro 3. Visibility of Mordell-Weil -- E/Q yes; sketch proof -- conjecture: yes in general Part II. Sha 4. Visibility of Sha 5. Visualizing MW in Sha -- surprising nonsquare sha example -- conjecture: yes in general Part III. Modularity? 6. Modularity of elements of Sha 7. Visualizing MW in modular Sha