Results 1 to 3 of 3

Math Help - Pell Equation and Class numbers

  1. #1
    Member
    Joined
    Nov 2009
    Posts
    169

    Pell Equation and Class numbers

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

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


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

    b) If D > 0, show that h(D) = h_0(D) or h(D) = 2h_0(D) according to whether or not the equation 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 01:57 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  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
    f(ax +by, cx +dy) = \alpha (ax +by)^2 + \beta (ax +by)(cx +dy) + \gamma  (cx +dy)^2 = (\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: April 25th 2010, 06:22 PM
  2. Question to Pell Equation
    Posted in the Number Theory Forum
    Replies: 9
    Last Post: April 2nd 2010, 07:45 PM
  3. Pell's equation
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: June 4th 2009, 07:32 AM
  4. simple Pell's Equation
    Posted in the Number Theory Forum
    Replies: 5
    Last Post: May 3rd 2009, 09:42 PM
  5. Pell equation
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: December 10th 2008, 01:00 PM

Search Tags


/mathhelpforum @mathhelpforum