This is a superbly written introduction to elliptic curves. I like the straight-forward language. I dread the stiff elaborations, one finds in some german books with awkward idioms etc..

I found it fascinating, how the elements of general theory, explicit formulae and geometric ideas (the group law on an elliptic curve is constructed via means of geometry) are interwoven.

However, if you want to get a glimpse of such fundamental theorems like the Mordell-Weil theorem, you will need a solid understanding of the basics of algebraic number theory.

Also, if the author tells you "it is clear", it may take you two or three pages of your own thoughts and scribblings to actually see, why it is "clear". Sometimes it really is clear, but sometimes he might be referring to basic results from algebraic number theory. For example in VIII.$1 Proposition 1.6, a field is constructed, which is unramified outside a certain set of places of the number field K. The notion "It is clear .... is unramified if and only if ord_v(a) = 0 ..." had me puzzled for a while, until it dawned on me, that I needed a certain separability criterium for the polynomial to show what was needed.

All in all, still a great book.

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