# Pell Equation and Class numbers

#### EinStone

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@

MHF Hall of Honor
What do you mean by $$\displaystyle f'(x, y) \sim f(ax + by, cx + dy)$$?

#### EinStone

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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