Class number for binary quadratic forms discriminant $\Delta$ to class number $\mathbb Q(\sqrt \Delta)$

Question

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)

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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$

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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$ 

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Answer 1

It seems to me that what Buell says about the narrow class group is not quite right (it's hard for me to say, as I don't have a copy of it). Magma tells me that in $\mathbb{Q}(\sqrt{210})$, the narrow class group is $(\mathbb{Z}/2\mathbb{Z})^{3}$ and the ideal class group is $(\mathbb{Z}/2\mathbb{Z})^{2}$. The squares in the narrow class group would be trivial.

There is a short section (less than a page) in David Cox's "Primes of the form $x^2 + ny^2$" that starts on page 128 where he discusses what happens in the real quadratic case, leaving the task of checking the details to the reader (in a sequence of exercises). Cox states that if you want to stick with the traditional notion of equivalence of forms, then the group of forms is isomorphic to the narrow class group. On the other hand, you can change the notion of equivalence, and say that $f(x,y)$ and $g(x,y)$ are equivalent if there is a matrix $\left(\begin{matrix} p & q \\ r & s \end{matrix} \right) \in {\rm GL}(2,\mathbb{Z})$ so that $f(x,y) = \det \left(\begin{matrix} p & q \\ r & s \end{matrix}\right) g(px+qy,rx+sy)$, then the group of forms is isomorphic to the class group.

Cox also shows that the class group is a quotient of the narrow class group, and the kernel of that map has order $1$ if there is the norm of the fundamental unit in $\mathcal{O}_{K}$ is $-1$, and the kernel has order $2$ otherwise. In short the answer to your question is yes.

Answer 2

I have always been fuzzier on the theory of indefinite binary forms than the definite theory. This may come from the fact that I got to learn the definite theory by teaching a course out of Cox's book, and then I led some student research projects on the definite case. (Actually, just in the last week I spoke with my coauthors about extending our Geometry of Numbers approach to studying indefinite integral quadratic forms.) My understanding is that the indefinite theory of binary forms is totally worked out, and indeed was worked out by Gauss in his Disquisitiones Arithmeticae....which however does not make it easy to read.

The most I ever thought about indefinite binary forms was in writing Sections 3.2 and 5.2 of our paper (to appear, Acta Arithmetica). For everyone else: most of this paper concerns positive integral quadratic forms, but when it came to binary forms I knew the results should carry over to the indefinite case as well, so we wrote (okay, I wrote this part) it so as to cover that case as well. A statement of the correspondence between the two kinds of class groups of indefinite binary quadratic forms appears as Theorem 5.4b) there. In particular it applies to nonmaximal orders as well: $\operatorname{Pic}^+(R_D)$ is the narrow Picard group of the quadratic order of discriminant $D$: invertible fractional ideals modulo principal ideals with totally positive generators. Referring to the narrow class group "of $\mathbb{Q}(\sqrt{\Delta})$" may just be notationally sloppy. I'm not sure: Buell is not one of my references on this subject.

The distinction between the narrow class group and the class group of a real quadratic field is correctly reported in your question: the two groups are quotients of the same group by two subgroups which are either equal or one has index 2 in the other. Equality of the two groups occurs precisely when the fundamental unit has norm $-1$, which can be expressed in terms of the negative Pell equation as you have and in many other ways: e.g. continued fractions [not my thing!!] and the existence of units of all four possible sign combinations with respect to the two embeddings. I make some remarks about this in the context of classfield theory in $\S$ 3.3.4 of these notes from Algebraic Number Theory II. (Don't expect too much: you'll be disappointed.)

What are some good references for this? I like:

• Daniel Flath's Introduction to Number Theory. (This is cited in our paper.) Flath's ambition is no more and no less than to cover the main results on binary forms treated in Gauss's Disquisitiones Arithmeticae. I found it to be very helpful.

• Franz Halter-Koch's Quadratic Irrationals: An Introduction to Classical Number Theory. This recent text really takes its time and goes deep on this material. In particular he treats the "class semigroups" that arise when one consider not-necessarily-invertible ideal classes in non-maximal quadratic orders. Any student of mine will tell you that I am slightly obsessed with the "ideal class monoids" that arise in this way.

You also know that I have my eye on more than just the integers as a base ring, and I am kind of surprised that the theory of class groups of binary forms has not been more explicitly worked out over an S-integer ring in a global field. (Or maybe someone will tell me that I'm wrong about this: I hope so!) This is more puzzling because there are papers which treat, in principle, far more general cases, namely:

• Irving Kaplansky, Composition of binary quadratic forms. Studia Math. 31 1968 523–530.

treats the case of binary forms over a Bezout domain (hence any PID).

• Martin Kneser, Composition of binary quadratic forms. J. Number Theory 15 (3) (1982) 406–413.

works over an arbitrary commutative ring.

• Melanie Matchett Wood, Gauss composition over an arbitrary base. Adv. Math. 226 (2011), no. 2, 1756–1771.

treats an amazingly general case: the base is an arbitrary scheme, and several kinds of degenerate quadratic structures are naturally included. (This paper is hard to read without some facility in algebraic geometry. I gave it to a student once before I knew about the work of Kaplansky and Kneser. I won't do that again: Wood's paper navigates a lot of technicalities so as to be able to do the general case.)

Nevertheless I do not know of anything like the treatment of ideal and bi-idoneal forms over (even) the ring of integers of an arbitrary number field. Well, there is an endless supply of students looking for projects...

Answer 3

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.