unique group structure on elliptic curve

Question

Let $C$ be a curve of genus $1$ over a field $k$ with a $k$-rational point $O$, then by the theory of elliptic curve, there is a group structure $(C(k),+)$ on $C$ given by geometric construction or induced by the group structure of Pic$^0(C)$, and these two constructions give the same group structure with $O$ as the zero element.

Then I want to know if there exists another group structure $(C(k),\oplus)$ on $C$ with $O$ as the zero element again, but now there exists some pair $P,Q\in C(k)$ such that $P+Q\neq P\oplus Q$.

Edit: After reading Servaes's answer to this question, I am more concerned with a construction compatible with complex topology or Zariski topology.

Thanks!

Answer 1

Of course; let $\sigma$ be any permutation of $C(k)$ that fixes $O$. Then the operation $$\oplus:\ C(k)\times C(k)\ \longrightarrow\ C(k):\ (x,y)\ \longmapsto\ \sigma^{-1}(\sigma(x)+\sigma(y)),$$ defines a group structure on $C(k)$ with $O$ as its identity element. It is not hard to see that if $|C(k)|>3$ then there exists a permutation $\sigma$ so that the two operations $\oplus$ and $+$ are distinct.

Answer 2

The group structure on any abelian variety is unique up to zero section. See Remark 1.3 in Milne's notes on abelian varieties.