Shafarevich Exercise 1.10. Find all the automorphisms of the curve $y^2 = x^3 + x^2$

Question

I get stuck on finding all automorphisms of the curve $y^2 = x^3 + x^2$.

I tried to use homogeneous coordinates: $ \zeta \eta^2 = \xi^3 + \zeta \xi^2$, $x = \xi/\zeta, y=\eta/\zeta$.

There is one singular point $(0,0,1)$. Could someone help how to proceed?

Answer 1

Note: I made some silly errors originally, now corrected.

I hope you’ve drawn the picture.

I like to give this curve (but not this question) to Calculus students, when they learn about parametric description of curves. Here, you take a line passing through the origin, $y=tx$, and see that it intersects the curve in three points, as it must, namely the origin counted twice and $(t^2-1,t^3-t)$.

Find the automorphisms of the affine line (or projective line, if you’re working projectively) that leave $\pm1$ fixed (or maybe exchange the two) and you’ve solved the problem.