I came across this sentence in one of the material I was reading this below :

A major result is the Hasse-Minkowski Principle, which implies that a curve C has a point over $\mathbb{Q}$ iff it has a point over $\mathbb{R}$ and over every local field $\mathbb{Q}_p$. This also implies the points of genus zero curve over $\mathbb{Q}$ can all be determined easily.

But the local solubility criteria says nothing about how one actually finds a global solution. Can someone clarify this for me.

  • 1
    $\begingroup$ Over $\mathbb Q_p$, solutions are not so hard to find because this is easily reduced (a la Hensel's lemma) to finding them over the finite rings $\mathbb Z/p^n\mathbb Z$ and lifting them to $\mathbb Q_p$. Then by comparing a (finite) number of values of $p$ (often including $p=\infty$), we can reduce this to a finite number of candidates. $\endgroup$ – RKD May 25 '17 at 18:31

You are correct in that the local-to-global principle only let's you decide whether there are global solutions, but it does not offer a way to easily find the rational points. For instance, let $C$ be the Pell equation $$C: x^2-109y^2=-1.$$ Then, it is easy to use the Hasse-Minkowski theorem to deduce that there must be global solutions (you only need to check there are solutions over $\mathbb{R}$ and over $\mathbb{Q}_{109}$, and by Hensel's lemma, you just need to check for solutions in $\mathbb{Z}/109\mathbb{Z}$ for the latter). However, the simplest solution is given by $$x=8890182, \text{ and } y = 851525.$$ The local-to-global solution does not tell you how to find it. There are "elementary methods" to find such a solution (e.g., continued fractions), but what elementary means is on the eye of the beholder.

  • $\begingroup$ thank you for explaining this to me. $\endgroup$ – shahrina ismail Jun 14 '17 at 2:30
  • $\begingroup$ You are welcome! $\endgroup$ – Álvaro Lozano-Robledo Jun 14 '17 at 17:05

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