My understanding of the use of the discriminant in elliptic curve theory is to test whether an elliptic curve in Weierstrass normal form over a field not of characteristic either 2 or 3, $y^{2} = x^{3}+Ax+B$, is smooth, i.e., non-singular. What I am confused about is the relationship between the discriminant and smooth curves. For instance, $y=x^{2}$ over $\mathbb{R}$ has a zero discriminant, yet is a smooth curve.

  • Why does a zero discriminant of an elliptic curve tell us that the curve is not smooth, and why does this not seem to apply likewise to other types of curves, such as, quadratics?

  • Is it perhaps because smoothness implies non-singularity for elliptic curves, but in general this is not the case?

  • $\begingroup$ Discriminant plays a role for equations of the form $y^2=f(x)$, not when it is of the form $y=f(x)$. Discriminant says something about the repeated roots of $f(x)$. For example, $y^2=x^2$ is not smooth, but $y=x^2$ is. $\endgroup$ – Mohan Nov 29 '15 at 2:54
  • $\begingroup$ @Mohan The discriminant is useful in equations of both forms. For equations of the form $y^{2}=x^{3}+Ax+B$ a non-zero discriminant seems to imply a smooth curve. Why? And, for equations of the form $y = f(x)$ the discriminant is also useful, as you said in ascertaining repeated roots, but it also makes an appearance in root formulas. For example, the discriminant of a quadratic $b^{2}-4ac$ in one variable appears in the quadratic formula. $\endgroup$ – Aguila Nov 29 '15 at 18:11

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