In the study of elliptic curves, one must have a solid ground on abstract algebra, algebraic geometry and analysis (modular forms).Would someone who is well-acquainted with the subject give me roughly how much knowledge one should have to understand, for instance, "The Arithmetic of elliptic curves" by Silverman?Also, could somebody suggest some introductory texts on Algebraic geometry and a book, explaining the use of analysis in the study of elliptic curves?Thanks in advance.
The book you are quoting is in fact pretty elementary: a good knowledge of basic algebraic geometry will be more than enough to understand it. You can check for example Hulek's "Elementary algebraic geometry" or M. Reid's "Undergraduate algebraic geometry". In fact Silverman's book is pretty much self-contained by this point of view: the first two chapters should contain all the algebraic geometry that you will need. Of course understanding those chapters is only possible having a solid knowledge of basic abstract algebra, if you have taken 1 or 2 standard algebra courses and you know a bit about Galois theory, you will have no problems.
Modular forms are only mentioned briefly in the appendix. If you want to learn something about that, a very good starting point is Diamond and Shurman's "A first course in modular forms", or if you just want to see them in action with elliptic curves, you can read the first chapter of Silverman's "Advanced topics in the arithmetic of elliptic curves". There are very few analytic concepts that you will need for understanding the book you mentioned; some basic notions of complex analysis are enough. I am no expert at all in the subject, but the little complex analysis I know I learnt it in Rudin's "Real and complex analysis".
You can check out the answers to the following (closely) related MSE question: