I am working on a chapter about the plane cubics from Basic Algebraic Geometry written Shafarevich.
One of exercises asks following question.
Suppose that two cubics $X_1,X_2$ with equations $y^2=x^3+a_i x+b_i$ for $i=1,2$ are isomorphic. Prove that an isomorphism between $X_1$ and $X_2$ takes their points at infinity to one another is given by a linear map.
I guess the condition involving points at infinity provides that the isomorphism is actually a group isomorphism, when we consider $X_1,X_2$ as groups having their points of infinity as identities.
However, I could not proceed to prove that the isomorphism is given by a linear map.
Thank you for reading and please give any hint to me.
There is a quick answer using Riemann-Roch. Let $\varphi=(f(x,y),g(x,y))$ be such an isomorphism. Since $x$ has a pole of order 2 at infinity, $\varphi$ has to map it to a function with a pole of order 2 at infinity. The space of such functions has dimension 2 by Riemann-Roch, so it is generated by $x$ and $1$. This shows that $f(x,y)=\alpha x +\beta$ for some $\alpha,\beta\in K$ (with $\alpha\neq 0$). Analogously, $y$ has a pole of order 3 at infinity and the space of functions with a pole of order at most 3 at infinity is generated by $1,x,y$.