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.

  • $\begingroup$ so if we have adding on Elliptic curves then Div(C) is equal to curve itself? because every sum is in the curve , for example over complex numbers?! can you say an example to understand? $\endgroup$
    – user371596
    Oct 7 '16 at 10:53
  • $\begingroup$ This formal sum is a starting point. One looks at what are called principal divisors. That is take a rational function on the curve. It leads to a divisor. Zeros of functions specifies the points, and multiplicities specifies the coefficient attached. Same way poles lead to negative coefficients leading to a divisor with sum of coeff = zero. The quotinet of formal sum group modulo principal divisors subgroup (it is a subgroup) will be interesting. $\endgroup$ Oct 7 '16 at 10:58

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