In the definition of Divisor of a curve (Arithmetic of Elliptic Curves, Silverman) what is the definition of "formal sum"? Adding points naturally like: $((2,3)+(1,1)=(3,4))$ or adding like adding group law in elliptic curve? Thanks
Neither. Don't write co-ordinates. Use symbols. So a formal sum of $P$ and $Q$ is $P+Q$, nothing more nothing less. In particular if $P=Q$ and if $P$ is a point of order 2, $P+Q$, is still $P+Q$ and not the identity element of the curve. No simplification. Everything is considered a new expression. Another way of stating is, it is a free abelian group on the elements of the curve.