For any curve over any discipline, algebraic geometers are thinking about an related group known as the Picard group. It’s a sure quotient of the free abelian group on factors of the curve. It consists of formal sums of factors on the curve modulo these formal sums that come from trying on the zeroes and poles of rational capabilities. It’s a essential instrument within the research of algebraic curves.
Why is an elliptic curve a bunch?

The smaller p2pkh addresses in Bitcoin are derived from a bigger public key. This key’s made out of a scalar non-public key, the general public key’s principally an x and y coordinate on the secp256k1 elliptic curve derived from the scalar non-public key. If the non-public key will not be throughout the curve group you can not derive a legitimate x and y coordinate aka public key from it.
Within the case you introduced s, the scalar shared secret, must be throughout the elliptic curve group as a result of they’re deriving a public key that somebody might declare funds with from s.
Deeper rationalization: What’s the math behind Bitcoin’s elliptic curve?