In Jost's Compact Riemann Surfaces, he proves that every compact Riemann surface of genus one can be embedded into $\mathbb{P}^2$. Moreover, he proves the image of the embedding is the zero set of a polynomial of the form $y^2=x^3+ax+b$. I'm fine with his proof up to this point, but then, he claims that the left-hand side has three distinct zeros without any proof. How can one show that these are indeed distinct?

  • $\begingroup$ If I understand you correctly, this is a classical fact about cubics of that form, which are called 'depressed.' en.wikipedia.org/wiki/Cubic_function#Three_real_roots $\endgroup$ – Alfred Yerger Dec 12 '17 at 18:53
  • $\begingroup$ If it has a double root, then $x^3+ax+b = (x-\beta)(x-\alpha)^2$ and the curve becomes $v=u^2 +\beta-\alpha$ with the change of variable $x=v +\alpha, y = uv$, which is isomorphic to $\mathbb{P}^1$ thus of genus $0$. @AlfredYerger $\endgroup$ – reuns Dec 12 '17 at 19:00

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