Results 1 to 3 of 3

Thread: Pell Equation and Class numbers

  1. #1
    Member
    Joined
    Nov 2009
    Posts
    169

    Pell Equation and Class numbers

    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 by EinStone; May 9th 2010 at 12:57 PM. Reason: notation error
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor chiph588@'s Avatar
    Joined
    Sep 2008
    From
    Champaign, Illinois
    Posts
    1,163
    What do you mean by $\displaystyle f'(x, y) \sim f(ax + by, cx + dy) $?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Nov 2009
    Posts
    169
    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 $ $\displaystyle = (\alpha a^2 + \beta ac + \gamma c^2)x^2 + (2\alpha ab + \beta ad + \beta bc + 2\gamma cd)xy + (\alpha b^2 + \beta bd + \gamma d^2)y^2 = f'(x,y)$
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Pell's equation
    Posted in the Number Theory Forum
    Replies: 7
    Last Post: Apr 25th 2010, 05:22 PM
  2. Question to Pell Equation
    Posted in the Number Theory Forum
    Replies: 9
    Last Post: Apr 2nd 2010, 06:45 PM
  3. Pell's equation
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: Jun 4th 2009, 06:32 AM
  4. simple Pell's Equation
    Posted in the Number Theory Forum
    Replies: 5
    Last Post: May 3rd 2009, 08:42 PM
  5. Pell equation
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: Dec 10th 2008, 12:00 PM

Search Tags


/mathhelpforum @mathhelpforum