Pell Equation and Class numbers

Nov 2009
169
2
Consider quadratic forms \(\displaystyle f(x,y) = \alpha x^2 + \beta xy +\gamma y^2\) for discriminants D. We define the (strict) class number \(\displaystyle h(D)\) using matrices \(\displaystyle M = \left( \begin{array}{ccc}
a & b \\
c & d \end{array} \right) \in SL_2(\mathbb{Z})\) so that \(\displaystyle f'(x, y) \sim f(ax + by, cx + dy);\) then \(\displaystyle h(D)\) is the number of equivalence classes under this equivalence relation.

Similarly, we define the (extended) class number \(\displaystyle h_0(D)\) using matrices \(\displaystyle M \in GL_2(\mathbb{Z})\) and using the equivalence relation \(\displaystyle f'(x, y) \sim (det M)f(ax + by, cx + dy).\)


a) Show that if \(\displaystyle D < 0\), then \(\displaystyle h(D) = h_0(D)\).

b) If \(\displaystyle D > 0\), show that \(\displaystyle h(D) = h_0(D)\) or \(\displaystyle h(D) = 2h_0(D)\) according to whether or not the equation \(\displaystyle t^2 - Du^2 = -4\) has a solution in integers.

I think I proved a), so Im needing help to prove b), but if you have a nice proof for a) I would also like to see it.

I hope someone can help me here.
 
Last edited:

chiph588@

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Sep 2008
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What do you mean by \(\displaystyle f'(x, y) \sim f(ax + by, cx + dy) \)?
 
Nov 2009
169
2
There was a notation error...

Anyway f and f' are equivalent if
\(\displaystyle f(ax +by, cx +dy) = \alpha (ax +by)^2 + \beta (ax +by)(cx +dy) + \gamma (cx +dy)^2 \) \(\displaystyle = (\alpha a^2 + \beta ac + \gamma c^2)x^2 + (2\alpha ab + \beta ad + \beta bc + 2\gamma cd)xy + (\alpha b^2 + \beta bd + \gamma d^2)y^2 = f'(x,y)\)
 
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