For $m$ a non-zero integer and discriminant $d=b^2-4ac$ congruent to either 0 or 1 modulo 4, show that m is properly represented by some binary quadratic form $f(x)=ax^2+bxy+cy^2$ if and only if the congruence $$z^2\equiv d\ (mod\ 4|m|)$$has a solution.

I have proved the forward half, but cannot prove the congruence implies the proper representation.

  • $\begingroup$ What is $\;z\;$ , anyway? $\endgroup$ – DonAntonio May 12 '19 at 11:22
  • $\begingroup$ z is can be any integer, just a place holder to demonstrate that d is a quadratic residue $\endgroup$ – W M Seath May 12 '19 at 11:23
  • $\begingroup$ Oh...so write so! Write "...iff $\;d\;$ is a quadratic residue modulo $\;m\;$" . $\endgroup$ – DonAntonio May 12 '19 at 11:26

If $$ z^2 = d + 4mt, $$ then $$ z^2 - 4mt = d, $$ and the binary $$ \langle m,z,t \rangle $$ or $$ g(x,y) = m x^2 + z xy + t y^2 $$ gives $$ g(1,0) = m $$

Added: note that they are not saying anything about $\gcd(m,z,t).$ The forms that are in (and contribute to $h(d)$) the form class group are primitive, but this problem may produce an imprimitive form. For that matter, they do not prohibit $d$ from being a square, which allows sort of degenerate forms that factor, such as $x^2 - y^2$


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