TITLE: Class number for binary quadratic forms discriminant $\Delta$ to class number $\mathbb Q(\sqrt \Delta)$
QUESTION [7 upvotes]: Jyrki Lahtonen has suggested I write a blog post relating binary quadratic forms to quadratic field class numbers, https://math.stackexchange.com/questions/209512/binary-quadratic-forms-over-z-and-class-numbers-of-quadratic-%EF%AC%81elds/209543#comment1727526_209543
The coincidence part of this is Jyrki bringing up this idea within a couple of weeks of Neil Sloane asking about programming for indefinite forms....
However, I remain a little uncertain about how this works with indefinite forms...from Buell, Binary Quadratic Forms, page 103, the group of binary form classes is isomorphic to the narrow class group of $\mathbb Q ( \sqrt \Delta)$ where $\Delta$ is the discriminant, where I suspect $\Delta$ must be a fundamental discriminant because multiplying by an integer square would not change a field extending $\mathbb Q.$
Then page 103, positive forms we are done, class group and narrow class group are isomorphic. Also done if there is a solution in rational integers to $u^2 - D v^2 = -4.$
Finally the problem: if there is no solution to $u^2 - D v^2 = -4,$ Buell says the class group is the squares of the narrow class group. Buchmann and Vollmer say, page 186, say the class group is a quotient of the narrow class group.
Let's see., examples. I put Positive primes represented by indefinite binary quadratic form  with this in mind. Cohen says that $\mathbb Q(\sqrt {205})$ has class number 2. There are four classes of indefinite binary forms of discriminant 205, and $u^2 - 205 y^2 = -4$ is impossible. So, we went from 4 to 2...
In the paper with Pete Clark, he deliberately made no distinction between indefinite form $f$ and the form $-f.$ So, one possibility here is that we are just dividing by 2 to go from 4 to 2..
Maybe this is the quick version: as far as I can tell, if $1$ and $-1$ are distinct as binary forms of discriminant $\Delta,$ the principal genus has even size, call that $E.$ Suppose there are $G$ genera, so that the total number of classes of binary forms of this discriminant is $EG.$ What is the class number of $\mathbb Q ( \sqrt \Delta)?$ So, question, are the numbers the same for positive forms and when the principal form also represents $-1,$ but if indefinite and the principal form does not represent $-1,$ divide by $2?$
Perhaps I can use this to publicize an elementary trick, generally unknown: an indefinite form $\langle a,b,c \rangle$ with positive $\Delta = b^2 - 4 a c$ not a square is reduced, in the sense of Lagrange, Gauss, and Buell, if and only if:
$$  ac < 0 \; \; \; \;  \mbox{and} \; \; \; \; b > |a + c| $$ 
So, general Question: how to take the class number of binary forms of discriminant $\Delta,$ where either $\Delta \equiv 1 \pmod 4$ is squarefree, or $\Delta \equiv 0 \pmod 4$ and $\Delta/4 \equiv 2,3 \pmod 4$ and this time $\Delta/4$ is squarefree.
To repeat some examples (I've got programs out the wazoo)

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle_All_Reduced 5


Sun Jun 22 20:03:44 PDT 2014


5    factored    5

    1.             1           1          -1   cycle length             2
    2.            -1           1           1   cycle length             2


5    factored    5

    1.             1           1          -1   cycle length             2

  form class number is   1

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ 
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle_All_Reduced 12


Sun Jun 22 20:03:52 PDT 2014


12    factored   2^2 *  3

    1.             1           2          -2   cycle length             2
    2.            -1           2           2   cycle length             2
    3.             2           2          -1   cycle length             2
    4.            -2           2           1   cycle length             2


12    factored   2^2 *  3

    1.             1           2          -2   cycle length             2
    2.            -1           2           2   cycle length             2

  form class number is   2

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ 
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle_All_Reduced 85


Sun Jun 22 20:04:08 PDT 2014


85    factored   5 *  17

    1.             1           9          -1   cycle length             2
    2.            -1           9           1   cycle length             2
    3.             3           7          -3   cycle length             6
    4.            -3           7           3   cycle length             6
    5.             3           5          -5   cycle length             6
    6.            -3           5           5   cycle length             6
    7.             5           5          -3   cycle length             6
    8.            -5           5           3   cycle length             6


85    factored   5 *  17

    1.             1           9          -1   cycle length             2
    2.             3           7          -3   cycle length             6

  form class number is   2

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle_All_Reduced 136


Sun Jun 22 20:04:24 PDT 2014


136    factored   2^3 *  17

    1.             1          10          -9   cycle length             4
    2.            -1          10           9   cycle length             4
    3.             3          10          -3   cycle length             6
    4.            -3          10           3   cycle length             6
    5.             9          10          -1   cycle length             4
    6.            -9          10           1   cycle length             4
    7.             2           8          -9   cycle length             4
    8.            -2           8           9   cycle length             4
    9.             3           8          -6   cycle length             6
   10.            -3           8           6   cycle length             6
   11.             6           8          -3   cycle length             6
   12.            -6           8           3   cycle length             6
   13.             9           8          -2   cycle length             4
   14.            -9           8           2   cycle length             4
   15.             5           6          -5   cycle length             6
   16.            -5           6           5   cycle length             6
   17.             5           4          -6   cycle length             6
   18.            -5           4           6   cycle length             6
   19.             6           4          -5   cycle length             6
   20.            -6           4           5   cycle length             6


136    factored   2^3 *  17

    1.             1          10          -9   cycle length             4
    2.            -1          10           9   cycle length             4
    3.             3          10          -3   cycle length             6
    4.            -3          10           3   cycle length             6

  form class number is   4

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ 
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle_All_Reduced 205


Sun Jun 22 20:04:32 PDT 2014


205    factored   5 *  41

    1.             1          13          -9   cycle length             4
    2.            -1          13           9   cycle length             4
    3.             3          13          -3   cycle length             4
    4.            -3          13           3   cycle length             4
    5.             9          13          -1   cycle length             4
    6.            -9          13           1   cycle length             4
    7.             3          11          -7   cycle length             4
    8.            -3          11           7   cycle length             4
    9.             7          11          -3   cycle length             4
   10.            -7          11           3   cycle length             4
   11.             5           5          -9   cycle length             4
   12.            -5           5           9   cycle length             4
   13.             9           5          -5   cycle length             4
   14.            -9           5           5   cycle length             4
   15.             7           3          -7   cycle length             4
   16.            -7           3           7   cycle length             4


205    factored   5 *  41

    1.             1          13          -9   cycle length             4
    2.            -1          13           9   cycle length             4
    3.             3          13          -3   cycle length             4
    4.            -3          13           3   cycle length             4

  form class number is   4

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ 
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle_All_Reduced 221


Sun Jun 22 20:04:40 PDT 2014


221    factored   13 *  17

    1.             1          13         -13   cycle length             2
    2.            -1          13          13   cycle length             2
    3.            13          13          -1   cycle length             2
    4.           -13          13           1   cycle length             2
    5.             5          11          -5   cycle length             4
    6.            -5          11           5   cycle length             4
    7.             5           9          -7   cycle length             4
    8.            -5           9           7   cycle length             4
    9.             7           9          -5   cycle length             4
   10.            -7           9           5   cycle length             4
   11.             7           5          -7   cycle length             4
   12.            -7           5           7   cycle length             4


221    factored   13 *  17

    1.             1          13         -13   cycle length             2
    2.            -1          13          13   cycle length             2
    3.             5          11          -5   cycle length             4
    4.            -5          11           5   cycle length             4

  form class number is   4

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$


in brief, 210 * 4 = 840, 24 reduced forms but 8 SL2 classes,

840    factored   2^3 * 3 * 5 *  7

    1.             1          28         -14   cycle length             2
    2.            -1          28          14   cycle length             2
    3.             2          28          -7   cycle length             2
    4.            -2          28           7   cycle length             2
    5.             3          24         -22   cycle length             4
    6.            -3          24          22   cycle length             4
    7.             6          24         -11   cycle length             4
    8.            -6          24          11   cycle length             4

  form class number is   8

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$

REPLY [2 votes]: Alright, i have gotten the binary forms part of the story. I may or may not ever know the field side of things, that's life.
From experiments with the website http://www.numbertheory.org/php/classnopos.html and comparison with my own programs, it appeared that the division in half amounted to identifying the distinct forms (whenever the principal form does not represent $-1$)
$$ \langle \alpha, \beta, -\gamma \rangle \; \; \mbox{with} \; \; \langle -\alpha, \beta, \gamma \rangle,   $$ where these are ``reduced'' in the sense of Gauss and Lagrange when
$$   \alpha \gamma > 0 \; \; \mbox{and} \; \; \beta > |\alpha - \gamma|.  $$
So it is needed to show that these really are distinct classes when $1$ and $-1$ are distinct as forms. However, this is not hard. If the two forms above are equivalent, then the opposite of $\langle \alpha, \beta, -\gamma \rangle$ in the form class group is $ \langle \gamma, \beta, -\alpha \rangle. $ Thus the hypothesis amounts to
$$ 1 =\langle \alpha, \beta, -\gamma \rangle \circ \langle \gamma, \beta, -\alpha \rangle.  $$ We are allowed to insist that $\gcd(\alpha,\beta ) = 1,$  important. Can always be arranged, although the result may not be ``reduced'' any longer.
The algorithm of Shanks, Buell Binary Quadratic Forms, pages 64-65, tells us that, with   $\gcd(\alpha,\beta ) = 1,$
$$ \langle \alpha, \beta, -\gamma \rangle \circ \langle \gamma, \beta, -\alpha \rangle = \langle \alpha \gamma, \beta, -1 \rangle.  $$
In short, the hypothesis that $ \langle \alpha, \beta, -\gamma \rangle \; \; \mbox{and} \; \; \langle -\alpha, \beta, \gamma \rangle   $ are equivalent implies directly that the principal form represents $-1.$ Very satisfying from my point of view. Note that we do not need to use Shanks, various books discuss the ``united forms'' approach of Dirichlet, pages 55-57 in Buell. On page 57, he confirms, with $\color{green}{\gcd(a_1,a_2,B)= 1},$ that
$$ \langle a_1, B, a_2C \rangle \circ \langle a_2, B, a_1C \rangle = \langle a_1 a_2,B, C \rangle.  $$ Dirichlet gives the same outcome as the Shanks method, but with no additional $\gcd$ assumptions, using $$ a_1 = \alpha, a_2 = \gamma, B = \beta, C = -1.  $$
This is also Theorem 98 on page 138 of Leonard Eugene Dickson, Introduction to the Theory of Numbers. 
Put another way, when the principal form does not represent $1,$ we get a distinct form that does represent $-1,$ and Dirichlet says 
$$\color{magenta}{ \langle \alpha, \beta, -\gamma \rangle \circ \langle -1, \beta, \alpha \gamma\rangle = \langle -1, \beta, \alpha \gamma\rangle \circ  \langle \alpha, \beta, -\gamma \rangle = \langle -\alpha , \beta, \gamma \rangle}.  $$ As $\langle -1, \beta, \alpha \gamma\rangle$ is not the principal class, the result of the Gauss composition gives a different class from the original, therefore  $\langle \alpha, \beta, -\gamma \rangle$ and $\langle -\alpha , \beta, \gamma \rangle$ must be distinct.