I am starting to read lecture notes on basics of arithmetic geometry by A. V. Sutherland.
In the second lecture, there is a procedure how to decide whether a conic over $\mathbb{Q}$ has a rational point (the lecture notes, p. 2, section 2.3 "Conics over $\mathbb{Q}$").
I am stuck on the part where this problem should be transformed into the problem of finding an element of a quadratic field with a given norm. Frankly, I don't think I understand the approach there at all:
- One starts with a projective conic given by the equation
$$aX^2+bY^2+cZ^2=0, \;\; a, b, c \in \mathbb{Z},\; a>0,\; b, c<0,$$ multiply it by $a$ to get $$a^2X^2+abY^2+acZ^2=0$$ and then, suddenly, one has the equation
$$X^2+abY^2=(-ac)Z^2$$ instead of the (imho correct) equation $$a^2X^2+abY^2=(-ac)Z^2.$$
I don't see whether this is some kind of substitution trick or simply a mistake. What puzzles me the most is that it seems that this form is crucial to the following arguments.
- Moreover, the author further claims that we can WLOG assume that the integers $-ab, -ac$ are square-free. Why should it be this case? What if one, for example, start with a curve given by the following equation: $$4X^2-9Y^2-25Z^2=0 \;.$$ Then how can I transform it into a form where the numbers $-ab, -ac$ are square-free?
Thank you in advance for any help.