My aim is to talk about the Shafarevich-Tate group of semi-stable elliptic curves over Q, with an eye towards trying to realize some elements in the Shafarevich-Tate group, concretely, as curves of genus 1 in some appropriate ambient algebraic variety. To prepare for the general subject of arithemtic of elliptic curves and modular curves, one might read the two expository articles by Joseph Silverman and David Rohrlich, which span the pages 17-100 in:
Modular forms and Fermat's Last Theorem,
(eds.: G. Cornell, J. Silverman, G. Stevens), Springer (1997).
For people who do have some background in the arithmetic of modular curves, it would be good to try reading all of Chapter I (skimming Chapters II and III) and becoming reasonably capable of working with the TABLES in:
Algorithms for modular elliptic curves,
J.E. Cremona, Cambridge Univ. Press, 1992.
To get a clearer sense of what I will actually cover in my lecture, you might read the three pages 18-21 and glance at the table at the end of my very rough, preliminary, notes:
Three lectures about the arithmetic of elliptic curves
Barry Mazur. dvi, ps, pdf