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How to solve a problem like this:

Find out which elements $N \in \mathbb N$ are represented by the quadratic form $\left \langle 2,3,2 \right \rangle$ in $\mathbb Q$.

The form is $$ f(x,y,z) = 2 x^2 + 3 y^2 + 2 z^2. $$

Do I have to reduce it modulo all primes $p \in \mathbb P\cup \left \{ \infty \right \}$ and use a local-global principle?

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Is this the binary form $2 x^2 + 3 x y + 2 y^2$ or the diagonal ternary form $2 x^2 + 3 y^2 + 2 z^2? $ –  Will Jagy Dec 3 '12 at 22:08
    
It is the diagonal ternary form. –  Lisa Mainhard Dec 3 '12 at 22:40

2 Answers 2

up vote 5 down vote accepted

EDDDITTTT: I notice you also asked about the Hilbert norm residue symbol. So the best thing you can do is get CASSELS and look through the first 60 pages. What you do is begin with $$ 2 x^2 + 3 y^2 + 2 z^2 = n $$ and switch to $$ 2 x^2 + 3 y^2 + 2 z^2 - n t^2 = 0. $$ You especially want the table of the symbol on page 43 and the conditions for isotropy on pages 58-59. I have gone through the whole song and dance for an answer here on MSE, I think it was an indefinite form. If I find it I will put in a link. Isotropy over $p$-adic numbers

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Edit: This answer (wrongly) assumed that the question was about binary quadratic forms.

You should read the first few chapters of the book by Cox, "Primes of the form $x^2+ny^2$", or the chapter on quadratic forms in Davenport's "Higher Arithmetic". The basic steps are these:

  • First, I will talk about representation in $\mathbb{Z}$, since you can always multiply through by denominators. The general answer differs from this one by squares.
  • The discriminant of the form is -7. General theory shows that a number $N$ is properly represented by some form of discriminant $d$ if and only if $d$ is a square modulo $4|N|$. Properly represented means represented by coprime $x,y$. The numbers that are represented in some way, properly or not, are multiples of the properly represented ones by squares (see first point). So the two problems of representation and proper representation are easily translatable into each other.
  • Use quadratic reciprocity to explicate the above condition in the case $d=-7$.
  • Reduction theory shows that every form of discriminant $d<0$ is equivalent by a linear change of variables with determinant 1 to a so-called reduced form. Forms that are equivalent to each other represent the same set of integers. Reduced forms have nice and easy coefficients, among other things they satisfy $0<a\leq \sqrt{|d|/3}$ and $|b|\leq a$. Moreover, $d$ is odd if and only if $b$ is, since $d=b^2-4ac$.
  • Deduce from the reduction theory that there is actually only one reduced form of discriminant -7 (the inequalities imply that $a=1$, $b=\pm 1$, but the $b=-1$ case is equivalent to the $b=1$ case). Thus, the congruence conditions that you got in the third bullet point are actually the necessary and sufficient conditions for a number to be properly representable by the unique equivalence class of forms of discriminant -7.

The theory becomes much more difficult (and interesting) if there are several equivalence classes of the given discriminant. Also, positive discriminant is harder, because reduction theory is harder.

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It's a shame the question was ambiguous and not clarified until after this answer was posted. –  Gerry Myerson Dec 3 '12 at 22:48
    
I also thought it was probably the binary form, otherwise why write it that way out of order. –  Will Jagy Dec 3 '12 at 22:56
    
I'm sorry about confusion. I thought that $\left \langle a_1,a_2,...,a_n \right \rangle$ is the general notation for the quadratic form $a_1 x_1^2+ a_2 x_2^2 + ... + a_n x_n^2$. –  Lisa Mainhard Dec 3 '12 at 23:04

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