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The Manin obstruction explains why the Hasse principle doesn't work for non-quadratic forms. Let's write the notation first;

$V(\mathbb{Q})$ is variety for rational numbers.

$V(A_\mathbb{Q})$ is variety for adèlic points (i.e., it contains real solution and p-adic solution for all $p$).

It's clear that $V(\mathbb{Q}) \subset V(A_\mathbb{Q})$.

Manin obsruction says, we can find new variety set $V(A_\mathbb{Q})^{\text{Br}}$ such that $V(\mathbb{Q}) \subset V(A_\mathbb{Q})^{\text{Br}} \subset V(A_\mathbb{Q})$. If $V(A_\mathbb{Q})$ is not empty but $V(A_\mathbb{Q})^{\text{Br}}$ is empty, then $V(\mathbb{Q})$ must be empty. So we can conclude the defining equations have no rational solution.

But why doesn't it work for quadratic forms? I estimate that, for quadratic forms, $V(A_\mathbb{Q})^{\text{Br}}$ must be equal to $V(A_\mathbb{Q})$ or $V(\mathbb{Q})$. Is this approach correct?

I could not find an explanation of this in the article and books I have examined. I think, I can use this for my thesis. Therefore I have to explain why it doesn't work or I must have a reference for this.

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    $\begingroup$ What are the article(s) and books you examined? $\endgroup$
    – LSpice
    Commented Jun 26, 2020 at 15:21
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    $\begingroup$ Introduction to Modern Number Theory- Yuri Ivanovic Manin, Alexei A. Panchishkin The Brauer-Manin Obstruction to the Hasse Principle - Boris Lerner Survey of Diophantine Geometry- Serge Lang I examine this source but I can't find. $\endgroup$
    – Lord Haydari
    Commented Jun 26, 2020 at 16:44

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